A Noncommutative Sigma Model

We replaced the classical string theory notions of parameter space and world-time with noncommutative tori and consider maps between these spaces. The dynamics of mappings between different noncommutative tori were studied and a noncommutative generalization of the Polyakov action was derived. The quantum torus was studied in detail as well as *-homomorphisms between different quantum tori. A finite dimensional representation of the quantum torus was studied and the partition function and other path integrals were calculated. At the end we proved existence theorems for mappings between different noncommutative tori.

"In the past decades theoretical physicists have been using ever more sophisticated mathematics to model the universe and its fundamental forces. Quantum Theory and Geometry are the two main ingredients but there are different schools of thought on how to fuse them together. Einstein, with his success in General Relativity, argued for the primacy of Geometry and Dirac said we should be guided by beauty. I belong to this camp and am tentatively exploring some new ideas." -Sir Michael Atiyah (1 February 2011, Salle 5 of College de France) ACKNOWLEDGMENTS "If you attack a mathematical problem directly, very often you come to a dead end, nothing you do seems to work and you feel that if only you could peer round the corner there might be an easy solution. There is nothing like having somebody else beside you, because he can usually peer round the corner." -Sir Micheal Atiyah There are numerous people who helped me peer around the corner, but the person to whom I owe the most gratitude with respect to this thesis is Dr. Rocco Duvenhage. Patience is without a doubt one of his virtues. I would like to thank the physics department for not ostracizing the small assembly of mathematicians (Rocco and myself) with whom you share these humble corridors. Discussions with Danie van Wyk and Gusti van Zyl proved valuable. Needless to say that without a steady supply of coffee from the lab this thesis would not have been possible. I would like to express my sincere acknowledgements to NITheP for their financial assistance regarding my studies and travel expenses. Last but not least I would like to thank Lerinza who spent a countably infinite number of evenings with me while working on this thesis. In string theory we are interested in mappings from a two dimensional parameter space which we call Σ to our spacetime, X. The parameter space has coordinates σ and τ , which are roughly related to position along the string and the time on the string respectively. If we assume that spacetime has d spatial coordinates and one time coordinate we can write any spacetime coordinate as x = (x 0 , x 1 , · · · , x d ).
A string in spacetime at time τ can then be seen as the one dimensional collection of points joining ϕ(τ, 0) to ϕ(τ, σ 1 ) through the mapping functions.
Here we took σ 1 to be the upper limit of the σ-coordinate in parameter space. The world lines of the string endpoints have constant values of σ and we can then parametrize the string by τ .
In studying the dynamics that arise in string theory one usually starts out with the Nambu-Goto action. In Lagrangian mechanics the action of xi 0.1. STRING THEORY a free point particle is proportional to its proper time which we can also see as the length of its world-line in spacetime. The Nambu-Goto action generalizes this idea to a one dimensional object which sweeps out a two dimensional surface in spacetime. The Nambu-Goto action is then defined to be the proper area of the surface swept out by the one dimensional object. The Nambu-Goto action can be written as: The action we will mostly be concerned with is known as the Polyakov action. The Polyakov and Nambu-Goto actions give the same equations of motion for the relativistic string and are in fact equivalent. However in the framework of modern string theory as well as in this thesis the Polyakov action will be a more natural choice to work with. For completeness sake we state the Polyakov action here: Notice that this equation makes use of Einsteinian notation where repeated indices in the subscript and superscript imply summation. h αβ and η µν are the metrics of repectively the parameter space and world-time. The reader should not be concerned with these metrics as we will only consider the case of flat Euclidean space and make the necessary simplifying assumptions in which case the Polyakov action reduces to The notion of compactification the parameter space also plays a major role in string theory and will also help in visualizing the process which we are going to follow in this thesis. By identifying τ i and τ f lines in parameter space we obtain a cylinder. We can then identify the σ = 0 and σ = σ 1 lines which correspond the to top and bottom boundaries of the cylinder to form a torus. The reason we are mentioning compactifying of the parameter space is because we would like to replace the notions of classical string theory regarding parameter space and spacetime with noncommutative spaces, in particular noncommutative C * -algebras.
The quintessential example of a noncommutative space is the quantum torus. So it makes sense to visualise the parameter space as being compactified into a torus and to generalize the results of classical string theory on these classical spaces to more general C * -algebras. In this thesis we replace both parameter space and spacetime by different noncommutative tori and consider *-homomorphisms between these noncommutative C * -algebras xiii 0. 2

. STRUCTURE OF THIS DISSERTATION
which play the role of the mapping functions. In this way we develop the theory of a noncommutative σ-model. Noncommutative versions of the mapping functions and the Polyakov action are also derived.

Structure of this Dissertation
Starting off in Chapter 1 we consider the C * -algebra generated by a collection of operators. Following these ideas we introduce the quantum torus and study some of its more interesting properties, such as the fact that it can be written as a crossed product which will greatly aid us in determining its K-theory and the unique trace on the quantum torus. The final section of Chapter 1 deals with a finite dimensional representation of the quantum torus. We consider the special case of two n by n unitary matrices satisfying the commutation relation that defines the quantum torus. In Chapter 2 we introduce the concept of a σ-model and show that we can generalize this classical notion to the noncommutative world. This is primarily done by showing that the Polyakov action can be written in a noncommutative manner where integration is replaced by taking the trace and norms squared are replaced by taking the trace of the product of an operator with its adjoint. We explore a finite dimensional σ-model based on the construction of the finite dimensional representation of the quantum torus. In this case we are able to determine an explicit partition function of our finite dimensional representation and calculate minima and maxima for different parameters. If we make certain assumptions regarding our system we are also able to calculate thermodynamic quantities such as the expectation value of the energy as well as the entropy of the finite dimensional σ-model. Up until this point we have assumed the existence of such *-homomorphisms between different quantum tori. To be able to prove the existence of such *-homomorphisms we require some knowledge regarding the K-theory of C * -algebras and Morita Equivalence, an overview of these topics can be found in Chapter 3. In order to calculate the K-groups of the quantum torus in Chapter 3 we require an extremely deep result from K-theory known as the Pimsner-Voiculescu-sequence [16, Theorem 2.4]. We state the Pimsner-Voiculescu-sequence without proof and only use it to determine the K-theory of the quantum torus. The most technical section of this thesis is found in Chapter 4 where we prove the existence of *-homomorphisms between different quantum tori. In Chapter 4 we make use of a fundamental paper by Rieffel [18] and use his results without proof. In the appendix the reader can find some information regarding short exact sequences. xiv CHAPTER 1

THE QUANTUM TORUS
"Young man, in mathematics you don't understand things. You just get used to them." -John von Neumann's reply to Felix T. Smith who had said "I'm afraid I don't understand the method of characteristics." In order to construct a noncommutative σ-model we require a noncommutative space onto which our world sheet will be mapped. The quintessential example of a noncommutative space in the irrational rotation algebra or quantum torus. Here we present a detailed account of the quantum torus.

C * -Algebra Generated by a set of operators
We start off this section with some general background which will serve as preparation in the construction of the quantum torus. Consider the set O ⊂ B with B a C * -algebra. Let A 0 be the set of all finite linear combinations of finite products of elements in O ∪ O * .
We now proceed to show that A 0 is a subspace of B. Consider the elements a, b ∈ A 0 which we can write as where a j i , b j i ∈ O ∪ O * and α i , β i ∈ C. When we consider the sum of any two elements of such form we find

C * -ALGEBRA GENERATED BY A SET OF OPERATORS
which is once again a finite linear combination of finite products of elements in O ∪ O * . Similarly when considering scalar multiplication we find that which is of course a finite linear combination of finite products of elements Let A denote the closure of A 0 . We show that A forms a subspace of B. Consider any a and b in A and let {a n } and {b n } be two sequences in A 0 which respectively converge to a and b in A. We can now also consider the sequence {a n + b n } and show that it converges to a + b lim n→∞ (a n + b n ) = lim n→∞ a n + lim So a + b ∈ A. Next consider the sequence {αa n } with α ∈ C lim n→∞ αa n = α lim n→∞ a n = αa So αa n ∈ A. Since we have closure under vector addition and scalar multiplication A is a vector subspace of B.
Next we show that A 0 is an algebra. Consider the elements a and b as defined earlier. Then which according to the definition is again in A 0 since it is a finite linear combination of finite products of elements if O ∪ O * . Now we show that A also forms an algebra. Consider any a and b in A and any two sequences {a n } and {b n } in A 0 that converge respectively to a and b in A. So given > 0, there is some N such that a n − a < 2M , b n − b < 2M for n ≥ N , where we define M := max{ a n , b }. We will show that the sequence {a n b n } converges to ab ∈ A: a n b n − ab = a n b n − a n b + a n b − ab ≤ a n b n − b + b a n − a ≤ M 2M + M 2M =

CHAPTER 1. THE QUANTUM TORUS
So ab ∈ A and we once again have the bilinear map with A in the place of A 0 . This implies that A is an algebra.
We can define the involution on A as that inherited from B. For any a in A there is a sequence {a n } in A 0 converging to a in A. Recalling that A 0 is defined as the set of all finite linear combinations of finite products of elements in O ∪ O * . This implies that for each n, {a * n } is in A 0 . Now, since A is the closure of A 0 it follows that a * is in A.
Now since A is a closed, self-adjoint *-subalgebra of the C * -algebra B, A is a C * -algebra in its own right. We will refer to A as the C * -subalgebra of B generated by O. Sometimes we will make use of the notation C * (O) to emphasize that we are considering the C * -algebra generated by O.

The Quantum Torus
Topologically we can define the classical two-torus, T 2 by the quotient space The underlying Hilbert space we will be dealing with here is L 2 (T 2 ) and we define the following operators  [23, p. 109] Let θ ∈ [0, 1). Let A θ be the C * -subalgebra of B(L 2 (T 2 )) generated by two unitary operators {U, V }, satisfying the commutation relation: Proof. In this limiting case where θ = 0 the operators U and V reduce to So we can think of them simply as multiplication by an exponential function. Now A 0 is the commutative unital C * -algebra generated by U and V . A 0 clearly separates the points of T 2 and by the Stone-Weierstrass Theorem we can conclude that A 0 is dense in C T 2 . However C T 2 = A 0 = A 0 which concludes the proof.
The reader will notice that equation (1.2.1) has the exact form of the Weyl commutation relations.

The n-Dimensional Quantum Torus
For the remainder of the thesis we will use the quantum torus as defined in the previous section, however we can extend the results to a more general case. In this section we extend the quantum torus generated by two unitary operators with the commutation relation (1.2.1) to a "higher dimensional" version. A similar construction used to describe the normal quantum torus can now be followed to construct any quantum n-torus. We define the generating operators as follows for every j = 1, . . . , n and f ∈ L 2 (T n ) where T n is the n-torus which we define topologically T n := R n /2πZ n and where each z j,i is some constant. Proof. Just as in the case of the two-torus we define the operators A j in the following way

THE N-DIMENSIONAL QUANTUM TORUS
Here x denotes the n-dimensional vector on the n-torus and z j the constant we add to each component of x under the action of the U operator. This shows that U * j U j = I. This shows that U j is unitary for every j = 1, . . . , n.
We can also determine the commutation relation between any two of these U j operators. Let and U l f (x 1 , · · · , x n ) = e ix l f (x 1 + z l,1 , · · · , x n + z l,n ) := h(x 1 , · · · , x n ).
Similarly we can determine This enables us to determine the commutation relation between any U k and U l , which we find to be We are now able to generalize our earlier definition in the following way.

CHAPTER 1. THE QUANTUM TORUS
For the remainder of this thesis we will be working with the normal quantum two torus, the C * -algebra generated by two unitary operators U and V satisfying the commutation relation (1.2.1).

The Quantum Torus as a Crossed Product
This section is based on the work done by Dana Williams in his book [24]. We will follow the broad outline laid out in his book to construct the crossed product of the quantum torus. We include the necessary lemmas and propositions from [24] some of which, without proof, since the reader can easily find them in the book. For most of this section we will simply use the results mentioned in [24] to show that the quantum torus can be written as a crossed product of two C * -algebras.
consisting of a C * -algebra A, a locally compact group G and a continuous homomorphism α : G → AutA.
The ideas of C * -dynamical systems will occur frequently in the sections that follow. Definition 1.4.2. (C(X), C 0 (X) and C c (X)) If X is a locally compact Hausdorff space, then C(X), C 0 (X) and C c (X) denote, respectively, the algebra of all continuous complex-valued functions on X, the subalgebra of all bounded complex valued functions vanishing at infinity and the subalgebra of C 0 (X) consisting of functions with compact support.

Definition 1.4.3. (Covariant Representation)
Let (A, G, α) be a C * -dynamical system. Then a covariant representation of (A, G, α) is a pair (π, U ) consisting of a representation π : A → B(H) and a unitary representation U : G → U (H) on the same Hilbert space H, such that for any a ∈ A π (α s (a)) = U s π(a)U * s . Here we define the notation U (s) := U s and similarly α(s)(a) := α s (a).
We state the next three lemmas without proof (The interested reader can be find details regarding the proofs in [ is dense in C c (G, D) in the inductive limit topology, and therefore for the topology on C c (G, D) induced by the L 1 -norm.

THE QUANTUM TORUS AS A CROSSED PRODUCT
defines a *-representation of C c (G, A) on H called the integrated form of (π, U ). We call π U the integrated form of (π, U ).
Lemma 1.4.6. Suppose that (A, G, α) is a dynamical system and that for each f ∈ C c (G, A) we define Then · u is a norm on C c (G, A) called the universal norm. The completion of C c (G, A) with respect to · u is a C * -algebra called the crossed product of A by G and denoted by A α G.
It is possible to describe the C * -algebra structure on C c (G, C 0 (X)) in terms of functions on G × X. We make the identification where f ∈ C c (G, C 0 (X)). Then clearly ψ f (s, ·) = f (s) ∈ C 0 (X). It is clear that we have the following inclusions with µ the Haar measure of the group G. Note that if the group is not abelian, there will be both a left and a right Haar measure. For our puroposes the left Haar measure will be in view.
The function ∆ is independent of choice of Haar measure and is called the modular function on G.
It is clear that for any f and g in C c (G, A) and (r, s) ∈ G × G we have f (r)α r (g(r −1 s)) ∈ A. Since both f and g are in C c (G, A) it is easy to see that the mapping (s, r) → f (r)α r (g(r −1 s)) is in C c (G × G, A). We also observe that f * g(s) := G f (r)α r (g(r −1 s))dµ(r)

CHAPTER 1. THE QUANTUM TORUS
defines ans element of C c (G, A). We call the above star product the convolution of f and g. It can be shown using [24, Proposition 1.105, Lemma 1.92] that for all f, g, h ∈ C c (G, A) this convolution is associative .
The *-mapping is an involution on C c (G, A).
Proof. The following calculation proves the lemma: The convolution together with the involution defined in the previous lemma shows that C c (G, A) is a *-algebra. Remark 1.4.9. We know that C c (Z, C c (T)) has a dense *-subalgebra C c (Z× T) where the convolution product is given by the finite sum and the involution by If ϕ ∈ C(T) and h ∈ C c (Z), then we define ϕ ⊗ h as an element of C c (Z × T) by writing ϕ ⊗ h(n, z) = ϕ(z)h(n).
Let δ n be the function on Z which is equal to 1 at n and zero elsewhere.

THE QUANTUM TORUS AS A CROSSED PRODUCT
Proof. Let ϕ ∈ C(T) and h ∈ C c (Z) and consider A similar calculation shows that (1 ⊗ δ 0 ) * (ϕ ⊗ h) = ϕ ⊗ h. We also compute the involution of 1 ⊗ δ 0 . Proof. We perform the following calculation With a similar calculation we can show that u * u * = 1 ⊗ δ 0 . This shows that u is indeed unitary.

CHAPTER 1. THE QUANTUM TORUS
Proof. The lemma follows from the following calculation With a similar calculation we can show that v * v * = 1 ⊗ δ 0 . This shows that v is indeed unitary. Proof. We will calculate the left hand side first, Similarly we calculate the right hand side, = zδ 1 (n).
Hence we have uv = e 2πiθ vu.

THE QUANTUM TORUS AS A CROSSED PRODUCT
Lemma 1.4.14. Let u = 1⊗δ 1 as defined previously. Let i C(T) (ϕ) := ϕ⊗δ 0 for all ϕ ∈ C(T). Then Proof. Let us first consider the case where n = 1.
holds for any k ∈ Z. We want to show that it is also true for k + 1.
Hence, the statement of the lemma follows from mathematical induction.
Lemma 1.4. 15. Let (C(T), Z, α) be a dynamical system and let u and i C(T) be as in the previous lemma. Then

CHAPTER 1. THE QUANTUM TORUS
We compute Finally, let us consider the right hand side: This concludes the proof.
Remark 1.4.16. By Lemma 1.4.4 we see that is dense in C c (Z, C(T)), and hence dense in C c (Z × T). Furthermore, since ι T separates the points of T, which is clearly compact, and the subalgebra {ι T : ι T (z) = z for all z ∈ T} clearly contains all the constant functions in C(T) we conclude by the Stone-Weierstrass Theorem that ι T generates C(T) as a C * -algebra. In particular we see that span {v n : n ∈ Z} is a dense subalgebra of i C(T) (C(T)).
Theorem 1.4.17. The quantum torus A θ can be written as the crossed product C(T) α Z.
Proof. From Lemma 1.4.6 we know that C(T) α Z is the completion of C c (Z, C(T)) in the universal norm. By Remark 1.4.16 it follows that the unitary operators u = 1⊗δ 1 and v = ι T ⊗δ 0 generate C(T) α Z. By Lemma 1.4.13 we see that these unitary operators satisfy the commutation relations uv = e 2πiθ vu with θ irrational. Hence from the definition of the quantum torus (see Definition 1.2.2) we conclude that

THE QUANTUM TORUS AS A CROSSED PRODUCT
We state the next lemma without proof. Details can be found in [24, p. 96].
Lemma 1.4.18. Suppose that Z acts on T by an irrational rotation n · z := e 2πiθ z.
Then for every z ∈ T, the orbit Z · z is dense in T. Proof. Note that since U and V are unitary, their spectra σ(U ) and σ(V ) are subsets of T. Let I denote the identity operator in C * (U, V ) = A θ . Note that We could have switched the roles of U and V to find the relation for σ(U ). Now, since σ(V ) must be nonempty, by Lemma 1.4.18 σ(V ) has to contain a dense subset of T. But since σ(V ) is closed the lemma follows.
Using the previous lemmas and propositions we can prove an important theorem in the theory of quantum tori known as the universal property of the quantum torus.
Theorem 1.4.20. (Universal Property of the Quantum Torus for θ ∈ R/Q) Let B be a C * -algebra with unitaries u, v such that vu = e 2πiθ uv. Let A θ be as in Definition 1.2.2 but with θ irrational. Then there exists a *isomorphism ϕ : Proof. Since any C * -algebra can be seen as a *-subalgebra of B(H) for some Hilbert space H we only have to consider unitary operators U and V in B(H) satisfying the commutation relation which maps ι T to V . With H a Hilbert space let W : Z → U (H) be given by W n := U n .
where α n came from the dynamical system (C(T), Z, α). We clearly see that (π, W ) is a covariant representation of (C(T), Z, α). Furthermore, let u = 1 ⊗ δ 1 . As in the previous lemmas, we then have the following Similarly, let v = ι T ⊗ δ 0 . As in the previous lemmas, we then have the following Then clearly L := π W is a *-isomorphism of A θ into C * (U, V ).
The following result will be of importance in Chapter 3 where we are going to study Morita equivalence of different quantum tori. Lemma 1.4.21. A θ is isomorphic to A θ+c where c is some integer.

TRACE OF THE QUANTUM TORUS
Proof. A θ is generated by unitary operators U and V satisfying the commutation relation Similarly the quantum torus A θ+c is generated by the unitary operatorsŨ andṼ which satisfy the commutation relatioñ So by that universal property of quantum tori we can now conclude the A θ ∼ = A θ+c .

Trace of the Quantum Torus
Classically we my define an example of a state on C(T 2 ) by We would like to extend this definition to the noncommutative torus A θ .
Let Ω = I ∈ L 2 (T 2 ) be the identity. Then for any f ∈ L 2 (T 2 ) we can write Ω, f Ω =  Proof. In order to prove this theorem we need to show that τ is a tracial positive linear functional. Let A and B be elements in A θ . Linearity is trivial due to the inner product. If A is a positive element in A θ we can write A = S * S for some S ∈ A θ . Substituting this into the definition we find τ (S * S) = Ω, S * SΩ Let B be the *-algebra generated by {U, V }. We want to show that τ (ab) = τ (ba) for any

TRACE OF THE QUANTUM TORUS
Recall that the elements of B consist of finite linear combinations of elements of the form U m V n and that the above result holds for any a, b ∈ B. Since B is a dense *-subalgebra of the C * -algebra A θ we can extend the linear functional (1.5.1) to the whole quantum torus. Proof.
This implies that x = 0 and Adu is injective. Let a ∈ A be arbitrary So Adu is surjective. We conclude that Adu is a bijective *-homomorphism and hence a *-isomorphism.
Remark 1.5.3. The ideas and methods of dynamical systems constantly come into play throughout this thesis. In the following lemma we will use a dynamical system approach to show that the canonical trace τ on the quantum torus is unique. Consider the two *-automorphisms α(a) := U aU * , β(a) := V aV * .
We observe that for any trace ω on A θ we have and a similar result for β. Hence any trace on A θ is invariant with respect to the above *-automorphisms.

CHAPTER 1. THE QUANTUM TORUS
Lemma 1.5.4. Any state on A θ that is invariant under both α and β as defined above has to be the canonical trace τ on A θ .
Proof. Since all elements of A θ can be expressed as finite products of elements of the form U m V n , where U and V are the two unitary operators generating the quantum torus, we only have to consider the action of the two *-automorphisms on Let ω be an arbitrary state on A θ that is invariant under α and β. We then have These solutions can only be realised when either both m and n are zero or ω (U m V n ) = 0. Comparing these to

The Koopman Construction and the natural action of R 2 on A θ
In this section we study the natural action of R 2 on the quantum torus in preparation for the following section in which we study the smooth elements of the quantum torus. Ideas from dynamical systems such as those used in section 1.5 will once again play a major role in the present section, as well as those that follow. We are familiar with the time evolution of operators in quantum mechanics, if A is some operator on B(H), we can describe the time evolution where A(0) denotes the original operator and A(t) the same operator at some later time. U is a unitary group which in quantum mechanics is normally written as with H = H * the Hamiltonian and t the time. In the next section we will use this point of view to describe dynamical systems on the quantum torus.
Let µ be a probability measure on the measure space (X, Σ). Define the operator T in the following way: Proof. We can define the operator

Now clearly we have
So W has an inverse. Let f, g ∈ H, the inner product is defined by Now we consider hence W * W = I and W is unitary.
We would like to consider the action of the classical torus, T 2 = R 2 /2πZ 2 on the quantum torus, however it is enough to consider only the action of R 2 on the quantum torus since the contribution from Z 2 does not play any part. For convenience we state the generating operators of the quantum torus below: We define the mapping We are interested in square integrable functions on T 2 . However any f ∈ L(T 2 ) is an element of the class of measurable functions f : |f | 2 dµ < ∞ with respect to the normalized Haar measure on T 2 . In our case we may identiry the Haar measure with the Lebesgue measure on [0, 1) This implies that f = g almost everywhere except maybe on some set of zero measure. Hence the same holds for the composition f • ϕ r,s = g • ϕ r,s . This enables us to well define the operator be an element of Σ. Then according to our construction we have After applying ϕ −r,−s to Y , we find .
. So ϕ r,s satisfies all the conditions of Lemma 1.6.1 and we can conclude that W r,s is unitary.
Proof. Let f ∈ L 2 (T 2 ) and consider the following equalities

THE KOOPMAN CONSTRUCTION AND THE NATURAL ACTION OF R 2 ON A θ
We also have Similarly we find that W r,s W 0,0 = W r,s . Since we know that W r,s is unitary the lemma follows.
This is an example of the Koopman construction. Since W r,s is unitary we can use it to determine the time evolution of a dynamical system, similar to how time evolution arises in elementary quantum mechanics. Now we define a mapping on A θ which describes the evolution Suppose a and b are two distinct elements of A θ and consider So α r,s is well defined. Now we are able to consider the "time" evolution of the generating operators of the quantum torus. We see that

CHAPTER 1. THE QUANTUM TORUS
and finally We can perform a similar calculation to obtain the "time" evolution of V . The final results are The previous construction now enables us to construct a mapping from R 2 to A θ in a natural way, namely where a ∈ A θ .
Proof. Let r l and s l be sequences in T respectively converging to r and s. Let B be the *-algebra generated by {U, V }, the generating operators of the quantum torus. Every element of B is a finite linear combination of terms of the form U m V n , so By the uniqueness of the limit α is continuous on B. We know that B is dense in A θ and we can extend the continuity to the whole of A θ . Let (r n ) and (s n ) be two sequences in T respectively converging to r and s. Let a ∈ A θ . Then we can find an upper bound for α r,s Given b ∈ B and > 0, then for some n large enough, we have

DERIVATIONS AND THE SMOOTH ALGEBRA
Now we show that the continuity can be extended to the whole of A θ : To conclude, we have a C * -algebra A θ , a group R 2 and a continuous homomorphism (r, s) → α r,s (a) for some s ∈ A θ and according to Definition Remark 1.7.2. Let A(0) be an operator in B(H) with H our Hillbert space. When we consider the Heisenberg picture of quantum mechanics we can determine the time evolution of A(0) in the following familiar fashion: We would like to focus on a particular point of interest which will suppress the exponential functions. This happens when t = 0.
If we define

CHAPTER 1. THE QUANTUM TORUS
and take two operators A and B on B(H) where H is the Hilbert space in question, we calculate This formal calculation suggests that indeed δ is a derivation. Now using a similar approach to the one used in quantum mechanics we will define derivations on the irrational rotation algebra which reduce to normal partial derivatives in the classical scenario.
We require a definition for derivatives of operator valued functions. Let A be a normed algebra and consider a mapping Note that sometimes we make use of the notation d dx f (x) to denote the derivative of f at x and sometimes switch between the two notations. Furthermore note that the existence of f (x) implies the continuity of f at x. Proposition 1.7.3. The "product rule", "sum rule" as well as the rule for scalar multiplication holds for the above definition of the derivative of an operator valued function. Furthermore we also have

DERIVATIONS AND THE SMOOTH ALGEBRA
Proof. Let A be a normed algebra and consider two mappings f, g : R → A. We would like to show that Consider the definition ) . Now if we take the limit as h goes to zero. We find that Hence by the uniqueness of the limit we can conclude that the product rule holds for the derivative of operator valued functions. Now suppose that both f (x) and g (x) exist, we would like to show that Consider the following: Taking the limit as h goes to zero we find that the "sum rule" holds. The scalar multiplication result follows trivially.

CHAPTER 1. THE QUANTUM TORUS
By the uniqueness of the limit we conclude that d Remark 1.7.4. Since we are considering functions from R to some noncommutative algebra we can simplify (1.7.1) to If we write the derivative as above the connection with the normal definition of the derivative becomes clear.
Remark 1.7.6. For ease of calculation it will be useful to write these partial derivatives in terms of the accent notation. For this purpose we define where all the x i 's are fixed except for x j .

DERIVATIONS AND THE SMOOTH ALGEBRA
are of C ∞ class, hence so is the mapping Let k ∈ C and a ∈ A ∞ θ and consider (r, s) → α r,s (ka) = kα r,s (a).
But (r, s) → α r,s (a) is of class C ∞ , so clearly scalar multiplication also holds. Similarly using the fact that α r,s is a *-automorphism we can consider the mapping By Proposition 1.7.3 we know that the "product rule" holds for operator valued functions of R and we can conclude that the mapping is of C ∞ class. A ∞ θ inherits its involution directly from A θ . We can consider the mapping (r, s) → α r,s (a * ) = α r,s (a) * which implies that a * is of C ∞ class. This shows that A ∞ θ is indeed a *-algebra.
The next proposition shows that the smooth irrational rotation algebra is not empty. Lemma 1.7.9. Let B be the *-algebra generated by the generating unitaries U and V of the quantum torus. Then B is contained in A ∞ θ .
Proof. We can consider the mapping

CHAPTER 1. THE QUANTUM TORUS
We substitute the above result into the definition for the derivative of an operator valued function to obtain Performing similar calculations we can find the s-derivative of U m V n and we can extend the results to partial derivatives of any order. By the uniqueness of the limit we can conclude that U m V n ∈ A ∞ θ . Hence A ∞ θ contains all finite linear combinations of elements of the form U m V n and hence contains B.
Proof. Let B be the *-algebra generated by the unitaries U and V that generate the quantum torus. By Lemma 1.7.9 we know that B is contained We will now define derivations on A θ in accordance with Remark 1.7.2.

DERIVATIONS AND THE SMOOTH ALGEBRA A ∞ θ
A similar arguments holds for δ 2 . Linearity follows trivially from Let λ ∈ C and consider Similar calculations holds for δ 2 . Hence we have δ j (λa) = λδ j (a). Furthermore we have and a similar calculation holds for δ 2 . Hence we have δ j (a * ) = δ j (a) * . Lastly we need to show that the mapping is of C ∞ class. We will first consider the case of δ 1 (a) Here we have a problem, it is not known if the partial derivatives ∂ r and ∂ t commute. The problem then is with the term of the form We know that a ∈ A ∞ θ and hence is of class C ∞ . Furthermore since α t,s is a *-isomorphism from A ∞ θ to itself, α t,s (a) also is of class C ∞ and the same holds for ∂ t α t,s (a). Hence the mapping is of class C ∞ and there exists an operator ∂ p α p,q (∂ t α t,s (a)) such that .

CHAPTER 1. THE QUANTUM TORUS
Then in particular we can choose p = r and q = 0, then is of class C ∞ and ∂ r α r,0 (∂ t α t,s (a)) exists. Hence, we see that The same can be shown in the case of δ 2 (a). And a similar procedure follows for the higher order derivatives.
Remark 1.7.12. According to the previous theorem we can write the derivations in the following form: We may ask if the above constructions reduce to their respective classical counterparts in the case where θ is zero and A θ ∼ = C(T 2 ) as in Proposition 1.2.3. For this let us first consider the time evolution in the classical limit. Let h = U * r,s and l = gh. Suppose g ∈ C(T 2 ) and consider the following: Clearly we observe that τ r,s (g) = g • ϕ r,s which is exactly the evolution we would expect to find classically. Similarly we can show that in the classical limit the derivations as defined above reduce to the corresponding partial derivatives. This result is also mentioned in [20]. Consider the derivation δ 1 in the classical limit: So we see that δ 1 = ∂ ∂x in the classical limit. Similarly we can show that we also have in the classical limit. Proof. Let B be the *-algebra generated by the generating unitaries U and V of the quantum torus. Since every element of B is a finite linear combination of terms of the form Y m V n we only need to concern ourselves with these terms. Observe that Since B is dense in A θ we can extend this result to the whole of A θ and the result follows.
We have found an analog of partial derivatives on the quantum torus. We can consider these mappings as the infinitesimal generators of the action of T 2 on the quantum torus. Lemma 1.7.14. Let τ be the unique trace on A θ and let δ j with j = 1, 2 be the derivations defined above. Then τ • δ j = 0.
Proof. As usual let B denote the *-algebra generated by the generating unitaries U and V of the quantum torus. Since all elements of B are finite linear combinations of terms of the form U m V n we only need to concern ourselves with terms of this form. Let us first consider the derivation δ 1 .

CHAPTER 1. THE QUANTUM TORUS
Observe that Now substitute this result into the unique trace of the quantum torus to get A similar calculation can be performed for δ 2 . Since B is dense in A θ we can extend the result to the whole of A θ . This then concludes the proof.
The next lemma can be seen as the noncommutative generalization of integration by parts to irrational rotation algebras Proof. From Lemma 1.7.14 we have The result clearly follows from the above calculation.

Finite Dimensional Representation
In this section we construct a matrix model generated by two unitary matrices which satisfy the same commutation relation as the two unitary operators that generate the quantum torus. The noncommutative behaviour of the matrices is then naturally encoded into the model and it serves as a good example of how we can think about noncommutative actions. We follow the outline of [11]. We can choose an orthonormal basis |j 1 with 0 ≤ j ≤ n − 1 of C n and set |n 1 = |0 1 . Introduce the unitary matrices u, v in M n (C) by their respective action on these basis vectors by setting u|j 1 = q j |j 1 , v|j 1 = |j + 1 1 .

FINITE DIMENSIONAL REPRESENTATION
Here q is some complex number such that |q| = 1 and we assume q n = 1. They satisfy (1.8.1) uv = qvu, u n = 1, v n = 1.
We can choose another orthonormal basis |j 2 in which v is diagonal. These two bases can be related by the Fourier transform Where k and r are non-negative real numbers, introduce hermitian matrices x and y by It can be shown that for q = e ik/r 2 we have where the parameters r and k must be related by Proof. Since A := M n (C) is a von Neumann algebra we must have A = A , where A is the bicommutant of A [14, p,115]. Let C * (u, v) be the C * -algebra generated by the unitary matrices u and v. We will show that the commutant of C * (u, v) consists only of scalar multiples of the n × n identity matrix. This in turn implies that C * (u, v) must be all of A.
Consider the C * -subalgebra C * (u) ⊂ C * (u, v). Since q is the an n-th root of unity, which we can also assume to be q = e 2πi n , C * (u) contains all the diagonal matrices. The easiest way to see this is to note that the diagonal entries 1, q, · · · , q n−1 of u are all distinct. Hencegiven a polynomial p for which p(q j ) = δ ij for some fixed 0 ≤ i ≤ n − 1, we then have that E ii = p(u) ∈ C * (u). When a matrix commutes with all diagonal matrices it also has to be a diagonal matrix. Furthermore, diagonal matrices that commute with v have to be scalar multiples of the n×n identity matrix. This implies that the commutant of C * (u, v) is the scalars. This completes the proof.

CHAPTER 1. THE QUANTUM TORUS
We can determine the commutation relations Where we define the projections Define the derivations Due to equation (1.8.1) we can think of M n (C) as a finite dimensional representation of the quantum torus. Here we generalize the Polyakov action to the case where parameter space and world-time are both replaced by different noncommutative tori and show that in the classical limit the noncommutative action reduces to its classical counterpart. This chapter plays a key role, as it shows how the mathematics and physics come together to form a noncommutative sigma model. To explore these results we consider an example not explored by the previous two authors. Here we create an example of such a noncommutative action in the case of the space of 2 × 2 matrices with complex entries, denoted by M 2 (C). In this special case we construct a specific partition function and calculate noncommutative path integrals.

Noncommutative Action
Recall that in classical string theory we can describe the string dynamics using the Polyakov action [25, p. 583] given by where σ 1 and σ 2 play the roles of σ and τ respectively. We use Einsteinian notation, where repeated indices imply summation. For example using the

NONCOMMUTATIVE ACTION
Euclidean metric we can write n µ=1 X µ X µ = X µ X µ .
Let us simplify equation (2.1.1) by setting all the constants equal to one and looking specifically at the Euclidean metric together with the mapping where the X j (σ, τ ) are called the mapping functions and map R into itself.
Here Σ plays the role of the two dimensional parameter space and R n that of world-time. Using the framework just described we write the Polyakov action as In order to generalize the action to the noncommutative realm we have to work in an algebraic picture. For this purpose, let f ∈ C(R n ). Then we define Consider the following calculation where in the final step we employ the shorthand notation X µ := X µ • g. Since we can write e iX µ •g = e iX µ • g we see that in the context of *-algebras we can regard e iX µ as a unitary element. This leads us to define the unitaries The calculation that led to equation (2.1.3) then enables us to write the Polyakov action as where ϕ is as defined in equation (2.1.2).

CHAPTER 2. σ-MODELS
We would like to generalize the Polyakov action to the case where our parameter space and world-time both become noncommutative C * -algebras and the natural way to proceed is to make use of equation (2.1.4). In order to justify the noncommutative version of the Polyakov action let us first consider how this action would behave on the classical torus.
We have seen in Proposition 1.2.3 that when θ = 0 we have A 0 ∼ = C(T 2 ). This will be the case we are interested in to show that the noncommutative version of the Polyakov action reduces to it's classical form in the θ = 0 limit. Let us now suppose that parameter space as well as world-time can both be described by T 2 . The mapping g : Σ → X then becomes g : T 2 → T 2 and as in equation (2.1.2) we have Recall that in the classical limit we can write the generating unitaries of the quantum torus, U and V ; as In deriving equation (2.1.4) we obtained n unitaries e iX µ , however now we are interested in T 2 = R 2 /2πZ 2 and so only two unitaries are needed and they are U and V , the generating unitaries of the quantum torus in the classical limit. We saw in subsection 1.7.1 that in the classical limit where θ = 0, the derivations we constructed in section 1.7 reduce to normal partial derivatives. In our present case ∂ σ 1 = δ 1 and ∂ σ 2 = δ 2 . Now we can write equation (2.1.4) as From Theorem 1.5.1 we know that we can write the above expression for the Polyakov action as in the following claim: Claim 2.1.1. The natural generalization of the Polyakov action to noncommutative C * -algebra, A θ is S(ϕ) = τ [δ 1 (ϕ(U )) * δ 1 (ϕ(U )) + δ 2 (ϕ(U )) * δ 2 (ϕ(U )) +δ 1 (ϕ(V )) * δ 1 (ϕ(V )) + δ 2 (ϕ(V )) * δ 2 (ϕ(V ))] . (2.1.5)

NONCOMMUTATIVE ACTION
This can also be written as where we denote U 1 := U, U 2 := V.
When we consider the manipulations done on the Polyakov action in the classical limit we can clearly see that equation (2.1.5) reduces to the normal Polyakov action. In what follows we will use equation (2.1.5) when considering the action on mappings between different noncommutative tori. As we have mentioned earlier, the existence of *-homomorphism A Θ → A θ is not obvious and so far we have merely assumed the existence of such maps. In chapter 4 we will study the existence of such mappings. Proof. According to equation (1.7.3) we have This completes the proof.
Proposition 2.1.3. (More general, but still Θ = θ) Let U and V be the generating unitaries of the quantum torus. Consider the mapping defined by . ϕ is a well defined *-isomorphism and S(ϕ) = 4π 2 (p 2 + q 2 + r 2 + s 2 ).
Proof. From the definition of ϕ A we have ⇐⇒ U p+r V q+s = e 2πiθ e 2πiθ(qr−ps) U p+r V q+s .

CHAPTER 2. σ-MODELS
But since A ∈ SL(2, C) we have ps − qr = 1 which implies that ϕ A is well defined. That ϕ A is a *-isomorphism follows from the universal property of the quantum torus (Theorem 1.4.20).

The Two Dimensional Representation
In this section we look at an example not included in [13]. We would like to construct an example which shows what can be done in the simplest case which exhibits clear noncommutative behaviour. The natural choice would be to consider the space of two by two matrices with complex entries. We considered the general n × n case in section 1.8 and in this section we will continue with the notation introduced for the general case. Let q = e i k r 2 = e iπ = −1. Then for the two by two case we can write these matrices explicitly as where they satisfy the form of equation (1.8.3). These matrices satisfy the commutation relation uv = −uv.
From Proposition 1.8.1 we know that M 2 (C) is generated by these two unitary matrices u and v and for the remainder of the section we will concern ourselves with the case M 2 (C) which we regard as the finite dimensional representation of the quantum torus. Similarly we can write down expressions for the matrices x and y as in equation (1.8.2) x = k r 0 0 0 1 , y = k 2r 1 −1 −1 1 .

THE TWO DIMENSIONAL REPRESENTATION
Likewise, the derivations (1.8.4) can be written as Using these formulas for the matrices together with the method used by Mathai and Rosenberg [13] and equation (2.1.5) we are able to determine a specific action for the finite dimensional representation. is a closed, two sided ideal of M n (C). But since M n (C) is simple ker(ϕ) must be either 0 or M n (C). If ker(ϕ) = M n (C) then ϕ has to be identically zero. But this is not possible since ϕ is unital, and hence ker(ϕ) = 0. Suppose ϕ(a) = ϕ(b) for some a, b ∈ M n (C). Then ϕ(a − b) = 0. But this can only occur for a = b. Hence ϕ is a bijective *-homomorphism and therefore a *-isomorphism, which in this case implies it is in fact a *-automorphism.
We use equation (2.1.6) to perform computations and observe for conveniece that where we denote u 1 := U, u 2 := V.

CHAPTER 2. σ-MODELS
where W is some unitary operator in K(H) [17]. Therefore these types of mappings are all that we have to consider when calculating the actions of the *-automorphisms. The action can then be rewritten in the form If we are now able to parametrize all the unitary two by two matrices with complex entries we will be able to determine the action for any *automorphism of M 2 (C).

Parametrization of SU (2)
There has been some literature regarding the parametrization of SU (N ) [22], but for now we are only interested in the SU (2) case. This case has been studied in [7], however for completeness we give a brief recollection of the main constructions. By definition SU (2) consists of unitary two by two matrices g with complex entries which adhere to the constraint det g = 1.
We can write g ∈ SU (2) as g = a b c d .
Since the row (and column) vectors of unitary matrices are orthonormal we can always choose the row vectors such that d = a and c = −b. Then aa + bb and the det g = 1 condition becomes |a| 2 + |b| 2 = 1.

PATH INTEGRALS
This is just the unit sphere S 3 centered at the origin in four dimensional real Euclidean space. The matrix g can now be rewritten as Using Euler angles we determine the parametrization of the unit sphere, S 3 as in [7] u 1 = cos This implies that we can write any unitary matrix g ∈ SU (2) as a function of the Euler angles as follows: We can also find the invariant volume element or measure as in [7] dµ(g) = 1 16π 2 sin θdθdφdψ which is normalized such that SU (2) dµ(g) = 1.
This parametrization gives us all the elements of SU (2) exactly once. We can now use this to express the action of any *-automorphism of M 2 (C) as a function of φ, θ and ψ.

Path Integrals
In string theory the path integral is an integral over the space of all worldsheets of the string. In the noncommutative case, specifically M n (C), the world-sheet is the space of all unital *-homomorphisms ϕ : M n (C) → M n (C), described by ϕ(·) = U (·)U * where U is in SU (n). We only need to consider SU (n) since we can take any matrix in U (n) and multiply it by a certain factor in such a way that its determinant becomes 1. In [13] the authors try to calculate noncommutative CHAPTER 2. σ-MODELS path integrals but have to make numerous simplifying assumption. In the end they find a semi classical approximation by summing over the critical points. In this section we calculate explicit noncommutative path integrals in the case of M 2 (C). Since we are now able to determine the action for any *-automorphism of M 2 (C) we can proceed to study the noncommutative path integral. After the parametrization we can write the action of any *-automorphism ϕ of M 2 (C) as Tr [δ k (gu l g * ) * δ k (gu l g * )] = 1 16r 2 e −2i(φ+ψ) −4 −1 + e 4iφ −1 + e 4iψ cos(θ)− 1 + e 2iφ 2 1 + e 2iψ 2 cos(2θ)+ 4e 2i(φ+ψ) (21 + cos(2φ)(1 − 3 cos(2ψ)) + cos(2ψ)) .
Mathematica was used to perform the calculations which yielded the result of the second line. We are able to determine the relationship between the minima ( Figure 3) and maxima (Figure 4) of the Polyakov action with respect to r. The minima and maxima can respectively be described by We can also consider the path integral associated with this particular Polyakov action. We can write the partition function as the following path integral A graphical representation of the partition function, Z(g) as a function of r is given below ( Figure 5).
Classically we would be very interested in the critical points of the action since they should satisfy the classical Euler-Lagrange equations and describe the physical trajectories of particles. A strange phenomena that we see in the two by two matrix scenario is that these maxima and minima occur infinitely many times as can be seen by the contour plots in figures 6 and 7.
In normal statistical mechanics we can calculate the expectation values of the energy and the variance of the energy by respectively, where β = 1 k B T is the inverse temperature. The only free parameter we have at our disposal is r, which was introduced in equation   (1.8.2). From a dimensional analysis point of view r will have the dimensions of length. Furthermore, in special relativity time also has the dimensions of length, so heuristically at least, the role of time can then also be played by r in the above context. It is a familiar result from statistical mechanics that we can write β = it, the "length" of the system in imaginary time. Then, at least heuristically we can treat r as the inverse temperature β. This enables us to determine the "expectation value of the energy of our string" of which we give a graphical representation in figure 8. In Principle we are now able to calculate various other thermodynamic quantities and study their respective behaviour as a function of r, our "inverse temperature". If we are so bold as to assume that r is indeed the inverse temperature r = β = 1 k B T , then we are able to compute the specific heat C v and entropy S of our system as functions of temperature. For ease of computation the calculations were performed in units of k B . Figures 10 and 11 show the respective graphical representations.  C v Figure 10. The specific heat of the finite dimensional representation as functions of "temperature", T = 1 r . The oscillating terms appearing form T = 4 and onward occur due to convergence problems, however when studying the T → ∞ limit we observe that the specific heat tends to zero.  Figure 11. The entropy as a function of "temperature" T .

CHAPTER 2. σ-MODELS
The study of noncommutative path integrals is in its infancy and the fundamentals of the theory are being developed from a C * -algebraic point of view. The Mathai-Rosenberg σ-model [13] is an interesting noncommutative σ-model to study since we can create a finite dimensional representation which highlights the basic ideas of the model and enables us to calculate specific thermodynamic quantities. Specifically, if we make the assumption that r, which was introduced in equation (1.8.2), plays the role of the inverse temperature then we can determine the specific heat as well as the entropy of the finite dimensional representation as a function of the "temperature". Even when making this bold assumption we do not observe clear phase transitions in the system. This can be ascribed to the fact that interactions have not yet been included into the model. In order to study interactions from the point of view of C * -algebras we have to study the differential geometry of noncommutative spaces, the foundations of the most widely accepted approach having been laid by Connes in his ground breaking book [3]. The introduction of interactions in the Mathai-Rosenberg σ-model might be a suitable project for future work.
When considering higher dimensional representations the thermodynamic quantities have the same general shape as in the M 2 (C) case, which leads us to make the conjecture that when taking the "thermodynamic" limit, which in this case can be seen as n → ∞, the finite dimensional representation approximates the behaviour of the case in Section 1.2. In the C * -algebraic framework there is still room available for improvement and the finite dimensional representation might be able to shed some light on the matter.

K-THEORY AND MORITA EQUIVALENCE
"The most useful piece of advice I would give to a mathematics student is always to suspect an impressive sounding Theorem if it does not have a special case which is both simple and non-trivial." -Michael Atiyah In order to study noncommutative σ-models there has to exist mappings between the target space and world sheet. Without such mappings we are not able to construct an action and all dynamics will be lost. In this chapter we summarize the basic ideas regarding K-theory and Morita equivalence that will be used in proving the existence theorems that appear in the following chapter. For completeness sake we show the construction of the K 0 and K 1 groups. This will give the reader some background information regarding the origin of the abelian cancellative groups used in K-theory.

K-Theory
We follow the same procedure as [14,Chapter 7] when dealing with the K-theory of C * -algebras. Letting A be any unital *-algebra, we define the following set of projections  But (x + y, y + x) (z, t) if x + y + t = z + x + y which implies that z = t because the cancellation property holds. We can easily see that the map We can also define maps between different K 0 groups using *-homomorphisms. Consider the *-homomorphism ϕ : A → B between *-algebras A and B. If a := (a ij ) is an m × n matrix with entries in A we can now extend ϕ to the corresponding matrix algebra.
where (ϕ(a ij )) is an m × n matrix with entries in B. Consider now another n × p matrix, b with entries in A, clearly ab is a m × p matrix with entries in A.
In a similar way we can also show that in the case where m = n, we have ϕ(a * ) = ϕ(a) * , so ϕ : M n (A) → M n (B) is a *-homomorphism. The proof follows trivially from the argument just before the lemma. The proof follows from standard matrix manipulation and Lemma 3.1.4.

K-THEORY
Lemma 3.1.6. Let A and B be unital C * -algebras, then is a well defined group homomorphism.
Proof. The above two results guarantee that From matrix calculations we can show that which then implies that Hence ϕ * is a homomorphism. These results can then be extended to show that ϕ * : is a well defined homomorphism. Finally, the properties of the Grothendieck group G(N ) then imply that the map is a well defined group homomorphism. where p ∈ P [ A]. This enables us to use the theory developed earlier to treat the nonunital case by extending any nonunital C * -algebra to its unitization. Here A[0, 1] is a shorthand notation for C([0, 1], A).
Lemma 3.1.9. Let A be a C * -algebra. The cone, C(A) is a C * -algebra.
Proof. It is clear that C(A) is a *-algebra. It only remains to check that C(A) is complete and satisfies the C * -algebra norm. Let {f n } be a Cauchy sequence in C(A). Given > 0, then there is an N such that whenever n, m > N . Fix n > N and select some δ > 0 for which we have Letting m → ∞ we get f n − f ≤ 3 . Then we can write which shows that f is continuous. Next note that < .
Hence f n converges to f in C(A). Lastly we need to verify that f is in C(A). Clearly 0 = lim n→∞ f n (1) = f (1).

K-THEORY
So C(A) is complete and hence a Banach algebra. It remains to verify that the norm is a C * -algebra norm. Let f ∈ C(A) This concludes the proof. Proof. We will show that S(A) is a closed, two sided ideal in C(A).

Involution on S(A) is inherited from C(A). For any f ∈ S(A), g ∈ C(A) and
x ∈ [0, 1] we have This shows that S(A) is a closed ideal of the C * -algebra C(A) and hence is a C * -algebra in its own right.
Notice that S(A) is a nonunital C * -algebra and when we want to calculate its K-theory we have to make use of Remark 3.1.7.   A fundamental result from K-theory is the six term exact sequence, also known as the PV-sequence, which will be used in determining the K-theory of the quantum torus. The results are given below without proof.

4]
Suppose α is a *-automorphism of the C * -algebra A. Then there is a six term exact sequence When determining the K-theory of the quantum torus we refer to Theorem 3.1.14 as the PV short exact sequence. Regarding the notation used in Theorem 3.1.14, i : A → A α Z is the inclusion map, Id the identity map and α any *-automorphism of the C * -algebra A. Since α is a *-automorphism, α * is a group-homomorphism, which in turn ensutes that Id * − α * is the difference of two group-homomorphisms and hence a group-homomorphism in its own right.
3.1.1. K-Theory of the Quantum Torus. In order to fully understand the C * -algebra which is the quantum torus we need to determine its K-theory. Determining the K-theory of the quantum torus was a difficult task, however Rieffel, Pimsner and Voiculescu succeeded in calculating it in their articles [15,16,18,19]. The main tool which we will be using to find the K-theory of the quantum torus is the PV-sequence which we introduced in Theorem 3.1.14.
Proposition 3.1.15. Let θ be some irrational number. The K 0 group of A θ is Z ⊕ Z.
Let us consider the PV-short exact sequence for the C * -algebra C(T). Using Theorem 3.1.14 we find the following six term exact sequence for By Theorem 1.4.17 we know that we can write A θ = C(T) α Z.
Let us now substitute this result into the above expression, then we find ? So now we are left with finding the K-theory of C(T). Let and let j : S → C(T) be the inclusion, where S is the suspension (see Definition 3.1.8) of C(T). We follow [14, Example 7.5.1] to find the K-theory of C(T). Clearly im(j) = ker( ) and we can define a *-homomorphism where f z (x) = z for any z ∈ C and x ∈ T. In other words f z is the constant function identically equal to z. This implies that we can write So, is a *-homomorphism and = Id. This implies that the diagram is a split short exact sequence of C * -algebras. By [14, Theorem 6.5.2] we know that for any arbitrary C * -algebra A is a split short exact sequence. By [14,Remarks 7.5.3,7.5.4] we know that for i = 0, 1

CHAPTER 3. K-THEORY AND MORITA EQUIVALENCE
is a split short exact sequence of K-groups. Thus, by Lemma A.0.11 we have Now in particular, if we replace the C * -algebra A by C we have Substituting the above result back into equation (3.1.3) we find Recall that for some irrational θ we define where R θ is just rotation by θ, and that we can write the quantum torus as the crossed product A θ = C(T) α Z as in Theorem 1.4.17. On the level of K 0 -groups α gives Since α is a *-automorphism α * is a semigroup automorphism. Since K 0 (A 0 ) = Z it implies that K 0 (A 0 ) + = {0, 1, 2, · · · } = N 0 because K 0 (A 0 ) + is the only semigroup with identity such that Z = {m − n : m, n ∈ K 0 (A 0 ) + }. So in effect the automorphism α * maps N 0 to itself. Now suppose that α * (1) = m, then α * (n) = mn. So the only way for α * to be surjective is when m = 1, hence α * = Id N 0 . If we now extend α * to Z then clearly α * = Id Z , or more precisely We would like to extend the above reasoning to the K 1 -groups and work with group homomorphisms of the form From the definition of the K 1 -group and using the K 0 -notation for the unitization we write α * : K 0 (S(A 0 )) → K 0 (S(A 0 )).

K-THEORY
Consider α * defined on K 0 S(A 0 ) and restricted to K 0 S(A 0 ) Remark 3.1.16. Suppose we have a *-homomorphism then we can write We can extend this to From [14, p. 262] we know that K 0 (S(A 0 )) = Z. Then from the definition of K 0 (S(A 0 )) we have K (S(A 0 )) := ker (τ * ) which is a subgroup of So we have to have since it has to contain all the non negative elements. Let m ∈ N 0 . Then α * (m) = α * (m) ∈ K 0 S(A 0 )) + is non negative, but from [14, p. 262] we know that α * (m) ∈ Z = K 0 (S(A 0 )) if α * (m) ∈ N 0 = K 0 (S(A 0 )) + . Hence, similarl to the case of we find that α * is also the identity homomorphism in the case of If we now consider equation (3.1.4) together with the above reasing considering the α * homomorphisms we see that Id * − α * = 0 for both the K 0 and K 1 groups and we have the following split short exact sequence of C * -algebras Lemma A.0.11 then implies that This proves Proposition 3.1.15.

CHAPTER 3. K-THEORY AND MORITA EQUIVALENCE
3.1.2. Traces and K 0 . In chapter 4 we will be using the unique trace of the quantum torus to define mappings onto the K-groups of the quantum torus. The following proposition then naturally comes into play.
Proposition 3.1.17. A faithful trace on any C * -algebra induces a homomorphism of the K 0 -groups.
Proof. Let A be a C * -algebra with faithful trace denoted by τ A : A → C. Now we define a mappingτ A : K 0 (A) → C by where p = (p kl ) ∈ P [A] is an n × n matrix as defined in equation (3.1.1). The middle term in equation 3.1.5 represents an extension of the trace to M n (A). Such an extension will be made where convenient, and the same notation will be used. Let [p] = [q] ∈ K 0 (A). We would like to show that τ A is well defined. From Definition 3.1.2 there exists an m × n matrix u with entries in A and an integer r such that (3.1.6) I r ⊕ p = u * u and I r ⊕ q = uu * .
The square matrices u * u and uu * are in different matrix algebras over A.
Let us first consider u * u  We can then write the trace of u * u as and the same result holds for uu * after making use of the cyclic property of the faithful trace τ A u * ij u ij = τ u ij u * ij . We determine the traces of (3.1.6) Furthermore from standard properties of the trace we see that Clearly this implies that the traces of the matrices p and q are the same.

K-THEORY
p ∈ M n (A) and q ∈ M m (A) and consider Soτ A is a group homomorphism.   The following definition follows naturally from the previous Lemma.
Remark 3.1.22. We know that the isomorphism ψ : K 0 (A θ ) → Z + θZ is induced by the trace on the quantum torus. We can now easily determine what element of K 0 (A θ ) will be mapped to 1: A θ ]) = τ (I

CHAPTER 3. K-THEORY AND MORITA EQUIVALENCE
Where I A θ denotes the 1 × 1 matrix with the identity elements of A θ along the diagonal.
Lemma 3.1.23. Let G and H be additive subgroups of R both containing the number 1. If η : G → H is an order preserving group homomorphism such that η(1) = 1, then η(g) = g for any g ∈ G.
Proof. Since η(1) = 1, we can write η(Z) = Z, and so the result is trivial for integer multiples of the identity. Suppose that for some g ∈ G we have η(g) > g, in other words η(g) − g > 0. Then there exists a natural number n such that Hence there exists an integer m such that η(ng) = nη(g) > m > ng.
But since m ∈ Z we have m = η(m) > η(ng) which implies that η(ng) > η(ng) which is a contradiction, so we cannot have η(g) > g for some g ∈ G.
Now suppose that η(g) < g for all g ∈ G. Since η is a homomorphism we can write η(−g) > −g which reduces to the previous argument and we once again have a contradiction. Hence we conclude that η(g) = g for any g ∈ G.
We can extend the above result to the following more general scenario.
Lemma 3.1.24. Let G and H be additive subgoups of R both containing the number p. If η : G → H is an order preserving group homomorphism such that η(p) = rp with r ∈ Z then η(g) = rg for any g ∈ G.
Proof. Suppose that for some g ∈ G we have η(g) > rg, in other words we can write η(g) − rg > 0. Then there exists a natural number n such that Hence there exists an integer m such that nη(g) > mp > nrg.
Conversely, suppose that η(g) < rg for some g ∈ G. Since η is a homomorphism, we can write this as η(−g) > r(−g) and then the above argument

MORITA EQUIVALENCE
once again leads to a contradiction. In the end we conclude that η(g) = rg for all g ∈ G.
There remains a final scenario regarding the group homomorphisms which is of importance.
Lemma 3.1.25. Let G and H be additive subgoups of R. If η : G → H is an order preserving group homomorphism such that η(1) = 0 then η(g) = 0 for any g ∈ G.
Proof. The proof is similar to those of Lemmas 3.1.24 and 3.1.23.
Proof. We have ϕ(I A Θ ) = I A θ since ϕ is unital. We know that for any So in particular we have Then clearly the class of the identity is sent to the class of the identity. Furthermore from Lemma (3.1.6) we know that ϕ * is a group homomorphism. It only remains to show that the order is preserved. Let 0 < g ∈ Z + ΘZ. Then according to Lemma 3.1.20 there is some n × n matrix p with entries in A Θ which is a projection such that ψ([p]) = τ (p) = g. Furthermore, we have and ϕ * is an order preserving group homomorphism sending the class of the identity to the class of the identity. Lemma 3.1.23 then implies that ϕ * (g) = g.

Morita Equivalence
This section follows closely the wonderful book [17] by Iain Raeburn and Dana Williams. The aim is not to prove all the theorems, but rather to put forth the basics of the theory. Morita equivalence will only be used in proving the existence of *-homomorphisms of the form
Let A be a C * -algebra. By a right A-module, we shall mean a vector space X together with the linear pairing (x, a) → x · a : X × A → X.
We will often write X A to emphasize the fact that we are viewing X as a right A-module. [17, p. 8] A right inner product A-module is a (right) A-module X with a pairing ·, · A : X × X → A such that the following five properties hold: x, x A ≥ 0 (As an element of A) 5.
x, x A = 0 implies that x = 0 for all x, y, z ∈ X, λ, µ ∈ C and a ∈ A Remark 3.2.3. From conditions (1) and (3) we can show that ·, · A is conjugate linear in the first variable: λx + µy, z A = z, λx + µy * A = (λ z, x A + µ z, y A ) * = λ x, z A + µ y, z A Using (2) and (3) we can also show the following: Proof. Let a ∈ A be arbitrar. Then we know from the above remark that a x, y A = x · a * , y A .
But from Definition 3.2.1 we know that x · a * ∈ X and hence a x, y A ∈ I. Similarly from property (2) of Definition 3.2.2 we know that x, y A a ∈ I. This implies that I is a two sided ideal in A. The next theorem is cited without proof and will be quoted when we show that Morita Equivalence is an equivalence relation on a C * -algebra.

16]
Suppose A 0 is a dense *-subalgebra of a C * -algebra A, and that X 0 is a right A 0 -module. We suppose that X 0 is a pre-inner product A 0 -module with preinner product ·, · 0 . Then there is a Hilbert A-module X and a linear map for all x ∈ X 0 , a ∈ A 0 , and q(x), q(y) A = x, y 0 . We call X the completion of the pre-inner product module X 0 . q : X 0 → X such that q(X 0 ) is dense and for all x, y ∈ X, a ∈ A 0 , and b ∈ B.
In the following theorem we mention completion of an imprimitivity bimodule, this can be understood in terms of Lemmas 3.2.11 and 3.2.12.
Theorem 3.2.13. [17, p. 48] Let A, B and C be C * -algebras. Suppose that X is an A − B-imprimitivity bimodule and Y is a B − C-imprimitivity bimodule. Then Z := X B Y is an A − C-bimodule, and there are unique A and C valued pre-inner products A ·, · and ·, · C respectively on Z satisfying

MORITA EQUIVALENCE
With respect to these pre-inner products, Z is an A − C-pre-imprimitivity bimodule. The completion of Z is an A − C-imprimitivity bimodule, which we also denote by X ⊗ B Y and call the internal tensor product.
Proof. To prove transitivity we use the internal tensor product. If A X B and B Y C are Morita equivalences, then according to Theorem 3.2.13 we see that A (X ⊗ B Y ) C implements a Morita equivalence between A and C. It is routine to show that for any C * -algebra A, A A A is an A − A-imprimitivity bimodule [17,Example 3.5] and so the relation is reflexive. To observe the symmetric nature of the relation we introduce the dual module. If X is an A − B-imprimitivity bimodule, let X be the conjugate vector space, so that there is by definition an additive bijection .
ϕ(x), ϕ(y) A = A x, y for x, y ∈ X, a ∈ A and b ∈ B. Now we will look at a particular case which will be of of interest in the next chapter. Consider the vector space for a, b and c in F (A) and λ, µ in C. We have linearity in the first variable, so property (1) is satisfied. Let a, b ∈ F (A) and c ∈ A, then But the span of elements of the form a i b * i is dense in A. This can easily be seen for both the unital and non-unital case. In the former we can choose b = I and in the latter we can choose b = u λ where u λ is an approximate identity as defined in [14, p. 77]. Then we can write any element of A as a linear combination of elements of the form ab * . From [8, p. 73] we know that finite dimensional normed spaces are complete. We know that A ·, · induces a norm on F (A). Furthermore F (A) is complete in this norm. This shows that F (A) is a full left Hilbert A-module.  We have linearity in the second variable, so property (1) is satisfied. Moreover where a ij is the entry of the matrix in the (i, j) position. Since π is faithful by assumption this implies that ϕ must also be faithful. Furthermore, let h ∈ H n be such that So if we can show that h, ϕ( a, a Mn(A) )h ≥ 0 then ϕ( a, a Mn(A) ) is positive, in other words we make use of the standard characterization of positive operators on Hilbert spaces, which in turn implies that a, a Mn(A) ≥ 0.

CHAPTER 3. K-THEORY AND MORITA EQUIVALENCE
Consider the following a 1 ), . . . , ϕ(a n )) If a, a Mn(A) = 0 (the n × n matrix with only zero entries) then a = 0. Conversely, if a = 0, then clearly a * a = 0 = a, a Mn(A) . So the five properties of Definition 3.2.2 are satisfied. We know that finite dimensional normed spaces are complete, so the norm induced by ·, · Mn(A) is complete. It only remains to verify that F (A) is full. Consider the ideal Let b j be an approximate identity [14, p. 77] of the C * -algebra A and choose two vectors in F (A), a and b with entries a i and b j at positions i and j respectively and zero's everywhere else. Using these vectors we can create matrices with a single entry of a i b * j at position (i, j) and zero's everywhere. It is then clear that the span of such matrices is dense in M n (A). This concludes the proof. Proof. This is clear from the above two lemmas.

THE CONNECTION BETWEEN MORITA EQUIVALENCE AND K 0 -GROUPS
The result of Lemma 3.2.18 will be used in chapter 4 when we consider homomorphisms from a particular quantum torus to the matrix algebra with entries in another quantum torus.

The Connection Between Morita
Equivalence and K 0 -groups The next theorem is of great importance since it establishes the fact that when two C * -algebras are Morita equivalent they will have the same K 0groups which we will use in proving the existence of *-homomorphims between different quantum tori. Proof. As usual, let K denote the algebra of compact operators on some separable infinite dimensional Hilbert space and let p be a rank-one projection in K. We can construct the mappings They are both *-homomorphisms and we clearly have ψφ =Id and similarly φψ =Id. So B is isomorphic to B ⊗ p. Now we define P := I ⊗ p.
So P is a projection in B ⊗ p, and we find that for any b ∈ B and k ∈ K we have where ϕ is the state on K defined by ϕ(k) = ζ, kζ with ζ a unit vector in the range of p. Then ϕ(k)p = pkp. Now since ϕ(k)p is simply a scalar multiple of p it follows that P (B ⊗ * K) P = B ⊗ p. This implies that B ⊗ p is a corner of B ⊗ K. We will now show that it is a full corner which in turn implies that B is Morita equivalent to B ⊗ K [17, p. 50]. It is enough to show that span{( The theory developed in the second chapter made use of the existence of certain maps between the different quantum tori. Up until now we have only been concerned about the M n (C) case and in this chapter we extend the theory to the quantum torus. In this chapter we state theorems and give proofs of the existence of *-homomorphisms between different quantum tori. We also investigate under which circumstances these *-homomorphisms become *-isomorphisms. These existence proofs are of fundamental importance since without the existence of such *-homomorphisms we cannot create a noncommutative σ-model. This section forms the most technical part of this thesis and combines results from K-theory and Morita equivalence to prove the existence of the *-homomorphisms. Proof. Suppose g is of the same form as in equation (4.1.1), furthermore suppose that aθ + b cθ + d = x y for some integers x and y. Then we can write for ay − cx = 0. So then θ is rational and we have a contradiction. When ay − cx = 0 it implies that dx − by = 0 and we can write Hence ad − bc = 0. But this is a contradiction since Hence for any irrational θ we find that gθ is also irrational. Hence k (lθ) = (kl) θ. This concludes the proof.
These results will be used when we consider homomorphisms of the form It is a familiar result that the matrices

Classification of the Homomorphisms
Now we are ready to look at the homomorphisms between quantum tori and also under which circumstances these homomorphisms are in fact isomorphisms. The following subsection is the first stepping stone in the theory.

CLASSIFICATION OF THE HOMOMORPHISMS
4.2.1. Unital *-Homomorphism Between Noncommutative Tori. The first case we consider is the simplest and most natural. Here we will show under what circumstances unital *-homomorphisms exist between different noncommutative tori. This case has a very clear physical interpretation from the point of view of string theory as explained in the introduction and is the most important result of this chapter. Proof. Suppose there exists a unital *-homomorphism By Lemma 3.1.19, K 0 (A Θ ) is mapped isomorphically to Z + ΘZ. From Lemma 3.1.20 we know the range of the trace on projections from the quantum torus to itself is (Z + ΘZ) [0, 1] and Lemma 3.1.26 shows that ϕ induces an order preserving map ϕ * of the K 0 -groups sending the class of the identity to the class of the identity. Remark 3.1.22 shows us that we can associate the number 1 ∈ Z + ΘZ with the 1 × 1 identity matrix I (1) A Θ . By Proposition 3.1.26 we know that we can view the map ϕ * as the inclusion of the subgroup K 0 (A Θ ) into K 0 (A θ ) with 1 going to 1. So we can now view ϕ * as an inclusion and write So if we regard Θ as being the generator of the group K 0 (A Θ ) ∼ = Z + ΘZ we see by the inclusion that we can write Conversely by Lemma 1.4.21 we see that A cθ+d ∼ = A cθ , so we can say that A cθ+d is generated by the unitaries U and V satisfying the commutation relation U V = e 2πicθ V U.
While A θ is generated by unitariesŨ andṼ satisfying

CHAPTER 4. EXISTENCE THEOREMS
Now we define the homomorphism ϕ to act on the generating unitaries in the following way: we see that ϕ is indeed a unital *-homomorphism. Furthermore we can calculate Here two special cases arise naturally for us to evaluate. They are respectively when c = +1, −1. The equation above reduces to for the respective cases. From the universal property of the irrational rotation algebra we can see that in the case where c = ±1 the images of U and V generate the quantum torus A θ and the *-homomorphism is bijective. We know that if ϕ is surjective then ϕ * will also be surjective. Consider c = ±1. Since ϕ * is the inclusion map we have ϕ * (Z + ΘZ) = Z + ΘZ = Z + (cθ + d) Z = Z + cθZ which is strictly included in Z+θZ. So if |c| = 1 then ϕ cannot be surjective. So by a contrapositive argument we can see when ϕ * is not surjective it implies that ϕ cannot be surjective. We conclude that indeed the constructed homomorphism can be chosen to be an isomorphism if and only if c = ±1.

Unital *-Homomorphisms to Matrix Algebras over A θ .
We would like to generalize the results of Theorem 4.2.1 to the case where

CLASSIFICATION OF THE HOMOMORPHISMS
the target space becomes the set of all n × n matrices with entries in A θ . This case has a more distorted physical interpretation in the sense that in the classical limit we can think of the parameter space as being replaced by T 2 × {1, · · · , n} since M n (A θ ) ∼ = A θ ⊗ M n (C). However, the application to string theory is not clear. These results are of a more mathematical nature with applications in the general theory of σ-models and therefore from the point of view of physics does not form such a cardinal part of this thesis.
We are interested in unital *-homomorphisms of the form  Let I A Θ be the unit of A Θ then we have or equivalently ϕ(I A Θ ) = ϕ(I A Θ ) * = ϕ(I A Θ ) 2 .
So we may write ϕ(I A Θ ) = p where p is some projection in M n (A θ ). Clearly ϕ (A Θ ) ⊂ pM n (A θ ) p and it can be shown that pM n (A θ )p is a unital C * -algebra for any projection p in M n (A θ ). From [19, Corollary 2.6] we see that since pM n (A θ )p is unital and Morita equivalent to A θ we may write for some integer k. Then putting everything together we find that the *homomorphism ϕ reduces to and is unital. So we are then once again reduced to the unital case.  if and only if n cθ+d = x 1 cθ+d + y for some integers x and y such that x = 0. By setting x = n and y = 0 the conditions of Theorem 4.2.1 are satisfied and we embed A n cθ+d unitally into A 1 cθ+d , which in turn is Morita equivalent to A cθ+d . By Lemma 1.4. 21 we can see that A cθ+d ∼ = A cθ , which once CHAPTER 4. EXISTENCE THEOREMS again according to Theorem 4.2.1 can be embedded unitally into A θ . If we denote Morita equivalence by ∼ and let ∼ = denote that two C * -algebras are *-isomorphic, then schematically we can write the procedure as follows: Since all the C * -algebras we are dealing with above are unital we can embed Remark 4.2.4. In the theorem that follows we will be investigating unital homomorphisms of the form where we will find that Θ = cθ + d n .
With c, d and n integers. We want to investigate the respective K-theories involved. For that purpose we give a schematic representation of the argument. where the η j mappings implement the Morita equivalence by embedding into full corners of the corresponding matrix algebras. If we define to be the composition map then, on the level of the K 0 -groups, we can write ϕ * = η 1 * • η 2 * • η 3 * • η 4 * and here ϕ * is just multiplication by n.
Lemma 4.2.5. Let θ be some irrational number. On the level of the K 0groups K 0 (A θ n ) and K 0 (A n θ ), the Morita equivalence of A θ n and A n θ is implemented through multiplication by n θ .
Proof. We know that when two quantum tori are Morita equivalent their K 0 -groups are isomorphic and these groups are induced by the unique Z.
This concludes the proof. So the class of the identity is represented by the number n when we are dealing with K 0 (M n (A θ )). By Lemma 3.1.24 we can regard ϕ * as multiplication by n. If we again regard Θ as the generator of the group K 0 (A Θ ) and apply the map ϕ * to Θ we find that nΘ = cθ + d for some c, d ∈ Z. By Lemma 4.2.3 we know that we can embed A Θ into a matrix algebra over A θ . In the end we have a non-zero *-homomorphism This *-homomorphism will in general not be unital and l = n. By Lemma 4.2.5 and Remark 4.2.4 we see that the Morita equivalence is implemented by multiplication by n on the level of the K 0 -groups. By [19, Corollary 2.5] we know that if we have two projections, say p and q in M n (A θ ) with θ irrational that have the same trace then they are unitarily equivalent which means we can write p = UqU * for some unitary matrix U in M n (A θ ). But we know that projections in M n (A θ ) that have the same trace will be mapped to the same element in K 0 (A θ ). Hence, projections are determined up to unitary equivalence by their classes in K 0 (A θ ) which implies that we can conjugate by a unitary and arrange for ϕ to map A Θ unitally to M n (A θ ). Finally in [18,Theorem 3] we see that if we have two matrix algebras M m (A α ) and M n (A β ) with α and β irrational then the only way M m (A α ) can be isomorphic to M n (A β ) is when n = m and α = β. So we conclude that A Θ can only be isomorphic to M n (A θ ) when n = 1. commutation rule u(z)v(z) = Θ(z)v(z)u(z).
The theory we have developed in this thesis can be seen as a noncommutative free field theory. To include interactions we have to modify the Polyakov action with another well known correction called the Wess-Zumino term. Lots of work has been done on noncommutative Wess-Zumino theory also known as Wess-Zumino-Witten theory, however as of yet there is no established theory where parameter space and world-time are replaced by noncommutative C * -algebras. Vargese Mathai and Jonathan Rosenberg have started to develop the theory [13].
In their paper Vargese Mathai and Johnathan Rosenberg explore a physical model using the theory that this thesis is based on. They determine the partition function However this integral is much too difficult to evaluate even in the commutative case. To gain some ground they over-simplify by considering a semi-classical approximation. The partition function is then approximated by a sum of the form Z ≈ e −iS(ϕ) .
The study of noncommutative path integrals is still in its infancy with the first steps only recently taking place to give a mathematically rigorous description of these objects which are "taken for granted" in quantum field theory and string theory. We conclude this thesis by stating that there is great room for improving the mathematical foundations of quantum field theory and string theory. Such endeavors can be seen as a course for future study.
-The End -APPENDIX A

REGARDING EXACT SEQUENCES
Exact and short exact sequences play an essential role in the K-theory of C * -algebras and was used in connection with K-theory in Chapter 3.
Definition A.0.8. (Exact Sequence of C * -algebras) Let J, A and B be C * -algebras. Suppose that j : J → A and π : A → B are *-homomorphisms such that Im(j) = ker(π) then the sequence is exact.
Definition A.0.9. (Short Exact Sequence of C * -algebras) [14, p. 211] Let J, A and B be C * -algebras. Suppose that j : J → A is an injective *-homomorphism and that π : A → B is a surjective *-homomorphism, and that Im(j) = ker(π). Then is a short exact sequence of C * -algebras.
Most of the time we will be working with a surjective *-homomorphism π : A → B of C * -algebras. We define J := ker(π) and let j : J → A be the inclusion map. This clearly satisfies the conditions mentioned in definition A.0.9. Where I B denotes the identity in B.
We sometimes write equation A.0.4 as to emphasize that we are dealing with a split short exact sequence. The next lemma is important in determining the K-theory of the quantum torus: Lemma A.0.11. If defines a split short exact sequence of C * -algebras then A is isomorphic to J ⊕ B.