To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

We study aspects of noncommutative geometry on hyperbolic dynamical systems known as Smale spaces. In particular, there are two C*-algebras, defined on the stable and unstable groupoids arising from the hyperbolic dynamics. These give rise to two additional crossed product C*-algebras known as the stable and unstable Ruelle algebras. We show that the Ruelle algebras exhibit noncommutative Poincaré duality. As a consequence we obtain isomorphisms between the K-theory and K-homology groups of the stable and unstable Ruelle algebras. A second result defines spectral triples on these C*-algebras and we show that the spectral dimension recovers the topological entropy of the Smale space itself. Finally we define a natural Fredholm module on the Ruelle algebras in the special case that the Smale space is a shift of finite type. Using unitary operators arising from the Pimsner-Voiculescu sequence we compute the index pairing with our Fredholm module for specific examples.

Given a C*-algebra $A$ endowed with an action $\alpha$ of $\R$ and an $\alpha$-invariant trace $\tau$, there is a canonical dual trace $\widehat \tau$ on the crossed product $A \rtimes_\alpha \R$. This dual trace induces (as would any suitable trace) a real-valued homomorphism $\widehat \tau_* : K_0(A \rtimes_\alpha \R) \to \R$ on the even $K$-theory group. Recall there is a natural isomorphism $\phi_\alpha^i : K_i(A) \to K_{i+1}(A \rtimes_\alpha \R)$, the Connes-Thom isomorphism. The attraction of describing $\widehat \tau_* \circ \phi_\alpha^1$ directly in terms of the generators of $K_1(A)$ is clear. Indeed, the paper where the isomorphisms $\{\phi_\alpha^0,\phi_\alpha^1\}$ first appear sees Connes show that $\widehat \tau_* \phi_\alpha^1[u] = \frac{1}{2 \pi i} \tau(\delta(u) u^*)$, where $\delta = \frac{d}{dt} \big|_{t=0} \alpha_t(\cdot)$ and $u$ is any appropriate unitary. A careful proof of the aforementioned result occupies a central place in this thesis. To place the result in its proper context, the right-hand side is first considered in its own right, i.e., in isolation from mention of the crossed-product. A study of 1-parameter dynamical systems and exterior equivalence is undertaken, with several useful technical results being proven. A connection is drawn between a lemma of Connes on exterior equivalence and projections, and a quantum-mechanical theorem of Bargmann-Wigner. An introduction to the Connes-Thom isomorphism is supplied and, in the course of this introduction, a refined version of suspension isomorphism $K_1(A) \to K_0(\susp A)$ is formulated and proven. Finally, we embark on a survey of unbounded traces on C*-algebras; when traces are allowed to be unbounded, there is inevitably a certain amount of hard, technical work needed to resolve various domain issues and justify various manipulations. / Graduate / 0280

An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric n-tuple V. The structure of a free semigroup algebra contains a great deal of information about V. Thus it is natural to study this algebra in order to study V. A free semigroup algebra is said to be analytic if it is isomorphic to the noncommutative analytic Toeplitz algebra, which is a higher-dimensional generalization of the classical algebra of bounded analytic functions on the complex unit disk. This notion of analyticity is of central importance in the general theory of free semigroup algebras. A vector x in H is said to be wandering for an isometric n-tuple V if the set of words in the entries of V map x to an orthonormal set. As in the classical case, the analytic structure of the noncommutative analytic Toeplitz algebra is determined by the existence of wandering vectors for the generators of the algebra. In the first part of this thesis, we prove the following dichotomy: either an isometric n-tuple V has a wandering vector, or the free semigroup algebra it generates is a von Neumann algebra. This implies the existence of wandering vectors for every analytic free semigroup algebra. As a consequence, it follows that every free semigroup algebra is reflexive, in the sense that it is completely determined by its invariant subspace lattice. In the second part of this thesis we prove a decomposition for an isometric tuple of operators which generalizes the classical Lebesgue-von Neumann-Wold decomposition of an isometry into the direct sum of a unilateral shift, an absolutely continuous unitary and a singular unitary. The key result is an operator-algebraic characterization of an absolutely continuous isometric tuple in terms of analyticity. We show that, as in the classical case, this decomposition determines the weakly closed algebra and the von Neumann algebra generated by the tuple.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2000. / Includes bibliographical references. Also available on the Internet.

In a foundational paper “Operators Possesing an Open Set of Eigenvalues” written several decades ago, Cowen and Douglas showed that an operator T on a Hilbert space ‘H possessing an open set Ω C of eigenvalues determines a holomorphic Hermitian vector bundle ET . One of the basic theorems they prove states that the unitary equivalence class of the operator T and the equivalence class of the holomorphic Hermitian vector bundle ET are in one to one correspondence. This correspondence appears somewhat mysterious until one detects the invariants for the vector bundle ET in the operator T and vice-versa. Fortunately, this is possible in some cases. Thus they point out that if the operator T possesses the additional property that the dimension of the eigenspace at ω is 1 for all ω Ω then the map ω ker(T - ω) admits a non-zero holomorphic section, say γ, and therefore defines a line bundle on Ω. As is well known, the curvature defined by the formula is a complete invariant for the line bundle . On the other hand, define and note that NT (ω)2 = 0. It follows that if T is unitarily equivalent to T˜, then the corresponding operators NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω. However, Cowen and Douglas prove the non-trivial converse, namely that if NT (ω) and NT˜(ω) are unitarily equivalent for all ω Ω then T and T˜ are unitarily equivalent. What does this have to do with the line bundles and .To answer this question, we must ask what is a complete invariant for the unitary equivalence class of the operator NT (ω). To find such a complete invariant we represent NT (ω) with respect to the orthonormal basis obtained from the two linearly independent vectors γ(ω),∂γ(ω) by Gram-Schmidt orthonormalization process. Then an easy computation shows that It then follows that is a complete invariant for NT (ω), ω Ω. This explains the relationship between the line bundle and the operator T in an explicit manner. Subsequently, in the paper ”Operators Possesing an Open Set of Eigenvalues”, Cowen and Douglas define a class of commuting operators possessing an open set of eigenvalues and attempt to provide similar computations as above. However, they give the details only for a pair of commuting operators. While the results of that paper remain true in the case of an arbitrary n tuple of commuting operators, it requires additional effort which we explain in this thesis.

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

We introduce the notion of an abstract projection in an operator system and when a finite number of positive contractions in an operator system are all simultaneously abstract projections in that operator system. We extend this notion to Archimedean order unit spaces where we prove when a positive contraction is an abstract projection in some operator system, and furthermore when a finite number of positive contractions in an Archimedean order unit space are all simultaneously abstract projections in a single operator system. These methods are then used to provide new characterizations of both nonsignalling and quantum commuting correlations. In particular, we construct a universal Archimedean order unit space such that every quantum commuting correlation may be realized as the image of a unital linear positive map acting on the generators of that Archimedean order unit space. We also construct an Archimedean order unit space which is universal (in the same way) to nonsignalling correlations. We conclude with results concerning weak dual matrix ordered *-vector spaces and the operator systems they induce.

Two main objects of the research in this thesis are countable discrete groups and their operator algebras (C*-algebras and von Neumann algebras). Discrete groups are often succesfully studied using geometric and ergodic-theoretic methods, the corresponding areas of mathematics being called geometric resp. measured group theory. This thesis has a cumulative form: each chapter is a research article, and therefore has its own abstract and bibliography. The majority of these publications have been peer-reviewed and published in various journals.