| Publications and Preprints |
| The tame symbol and determinants of Toeplitz operators |
| preprint |
| Suppose that $\phi$ and $\psi$ are smooth complex-valued functions on the circle that are invertible, have winding number zero with respect to the origin, and have meromorphic extensions to an open neighborhood of the closed unit disk. Let $T_\phi$ and $T_\psi$ denote the Toeplitz operators with symbols $\phi$ and $\psi$ respectively. We give an explicit formula for the determinant of $T_\phi T_\psi T_\phi^{-1} T_\psi^{-1}$ in terms of the products of the tame symbols of $\phi$ and $\psi$ on the open unit disk. |
| K_0 theory of n-potents in rings and algebras |
| (with Jody Trout) |
| preprint |
| Let $n \geq 2$ be an integer. An \emph{$n$-potent} is an element $e$ of a ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a complex algebra, there is a group isomorphism $K_0^n(A) \cong \bigl(K_0(A)\bigr)^{n-1}$ for all $n \geq 2$. However, for algebras over cyclotomic fields, this is not true in general. We consider $K_0^n$ as a covariant functor, and show that it is also functorial for a generalization of homomorphism called an \emph{$n$-homomorphism}. |
| On the nonexistence of nontrivial involutive n-homomorphisms of C*-algebras |
| (with Jody Trout) |
| Transactions of the American Mathematical Society, October 22, 2008 |
| An n-homomorphism is a linear map f from an algebra A to an algebra B with the property that f(a_1a_2...a_n) = f(a_1)f(a_2)...f(a_n) for all a_1, a_2, ... a_n in A. Every homomorphism is an n-homomorphism for all n, but the converse is false, in general. A question posed by earlier researchers is this: is every *-preserving n-homomorphism between *-algebras continuous? We answer their question in the affirmative, but the even and odd n arguments are surprisingly disjoint. We then use these results to prove stronger ones: if n is even, then f is just an ordinary *-homomorphism, and if n is odd, then f is a difference of two orthogonal *-homomorphisms. Thus, there are no nontrivial *-linear n-homomorphisms between C*-algebras. |
| Complex Topological K-Theory |
| Cambridge Studies in Advanced Mathemtics, Cambridge University Press |
| Topological K-theory is a key tool in topology, differential geometry and index theory, yet this is the first contemporary introduction for graduate students new to the subject. No background in algebraic topology is assumed; the reader need only have taken the standard first courses in real analysis, abstract algebra, and point-set topology. The book begins with a detailed discussion of vector bundles and related algebraic notions, followed by the definition of K-theory and proofs of the most important theorems in the subject, such as the Bott periodicity theorem and the Thom isomorphism theorem. The multiplicative structure of K-theory and the Adams operations are also discussed and the final chapter details the construction and computation of characteristic classes. With every important aspect of the topic covered, and exercises at the end of each chapter, this is the definitive book for a first course in topological K-theory. |
| Index theory of Toeplitz operators associated to transformation group C*-algebras |
| Pacific Journal of Mathematics, Volume 223. Number 1 (2006), 159--166 |
| In this paper, we extend the results of our paper "The index of Toeplitz operators on free transformation group C*-algebras" to the case of arbitrary smooth actions of finite groups. |
| Toeplitz algebras and extensions of irrational rotation algebras |
| Canadian Mathematical Bulletin, Volume 48, Number 4 (2005), 607--613 |
| We define Toeplitz operators with symbols in the irrational rotation algebras and show that several of the properties of classical Toeplitz operators carry over to this situation. In particular, we prove an index theorem for these Toeplitz operators that generalizes the winding number theorem for Toeplitz operators on the circle. |
| A Hopf index theorem for foliations |
| (with Victor Belfi and Ken Richardson) |
| Differential Geometry and its Applications, Volume 18 (2003), 319--341 |
| We formulate and prove an analog of the Hopf Index Theorem for Riemannian foliations. We compute the basic Euler characteristic of a closed Riemannian manifold as a sum of indices of a non-degenerate basic vector field at critical leaf closures. The primary tool used to establish this result is an adaptation to foliations of the Witten deformation method. |
| The index of Toeplitz operators on free transformation group C*-algebras |
| Bulletin of the London Mathematics Society, Volume 34 (2002), 84--90 |
| Let $\Gamma$ be a discrete group acting on a compact manifold $X$, let $V$ be a $\Gamma$-equivariant Hermitian vector bundle over $X$, and let $D$ be a first-order elliptic self-adjoint $\Gamma$-equivariant differential operator acting on sections of $V$. We use this data to define Toeplitz operators with symbols in the transformation group $C^*$-algebra $C(X) \rtimes \Gamma$, and we show that if the symbol of such a Toeplitz operator is invertible, then the operator is Fredholm. In the case where $\Gamma$ is finite and acts freely on $X$, we prove a geometric-topological formula for the index. |
| Representable E-theory for $C_0(X)$-algebras (with Jody Trout) |
| Journal of Functional Analysis, Volume 177 (2000), 178--202 |
| Let $X$ be a locally compact space, and let $A$ and $B$ be $C_0(X)$-algebras. We define the notion of an asymptotic $C_0(X)$-morphism from $A$ to $B$ and use this to define representable $E$-theory groups $RE(X;A,B)$. We show that these are the universal groups on the category of separable $C_0(X)$-algebras that are $C_0(X)$-stable, $C_0(X)$-homotopy-invariant, and half-exact. If $A$ is $RKK(X)$-nuclear, these groups are naturally isomorphic to Kasparov's representable $KK$-theory groups $RKK(X;A,B)$. Applications and examples are also discussed. |
| The mathematics of apportionment |
| The University of Chicago Law School Roundtable, Volume 7 (2000), 227--237 |
| We consider various methods of apportionment that have been used for the U.S. House of Representatives. In particular, we illustrate some of the unavoidable paradoxes that arise in apportionment. |
| Isometry groups of unbounded Fredholm modules over manifolds |
| Houston Journal of Mathematics, Volume 26, Number 1 (2000), 131--144 |
| A self-adjoint first-order elliptic differential operator $D$ acting on sections of a Hermitian vector bundle over a compact Riemannian manifold $M$ naturally determines an unbounded Fredholm module over $M_n(C(M))$ for each positive integer $n$. We show that the group of automorphisms of $M_n(C(M))$ that respect the unbounded Fredholm module is a compact topological group in the topology of pointwise convergence. If $D$ is an operator of Dirac type and we restrict to scalar functions, then this group is also a Lie group. |
| Toeplitz algebras associated to isometric flows |
| Illinois Journal of Mathematics, Volume 41, Number 1 (1997), 93--102 |
| Let $M$ be a compact Riemannian manifold, and let $\Phi = \{\phi_t\}$ be a smooth one-parameter group of isometries of $M$. The group $\Phi$ is called an isometric flow on $M$. In this paper, we associate a Toeplitz $C^*$-algebra $T(\Phi)$ to an isometric flow $\Phi$ on $M$, and study how the $C^*$-algebraic properties of $T(\Phi)$ are related to the geometric and topological properties of $\Phi$. |
| The basic Laplacian of a Riemannian foliation (with Ken Richardson) |
| American Journal of Mathematics, Volume 118 (1996), 1249--1275 |
| We study the basic Laplacian on Riemannian foliations by writing the basic Laplacian in terms of the orthogonal projection from square--integrable forms to basic square--integrable forms. Using a geometric interpretation of this projection, we relate the ordinary Laplacian to the basic Laplacian. Among other results, we show the existence of the basic heat kernel and establish estimates for the eigenvalues of the basic Laplacian. |
| Isometries of unbounded Fredholm modules over reduced group C^*-algebras |
| Proceedings of the American Mathematical Society, Volume 123, Number 6 (1995), 1839--1843 |
| In this paper, the author studies a class of unbounded Fredholm modules over a reduced group $C^*$-algebra, and he shows that the isometry groups of these unbounded Fredholm modules are always compact Lie groups. The author also proves a result about the fixed point algebra of such an isometry. |
| Isometries of noncommutative metric spaces |
| Proceedings of the American Mathematical Society, Volume 123, Number 1 (1995), 97--105 |
| Alain Connes has shown that a unital $C^*$-algebra equipped with an unbounded Fredholm module can be viewed as a "noncommutative" metric space. In this paper, the author defines a notion of an isometry of a noncommutative metric space, and computes several examples. |
| Index theory and Toeplitz algebras on one-parameter subgroups of Lie groups |
| Pacific Journal of Mathematics, Volume 158, Number 1 (1993), 189--199 |
| We form the Toeplitz algebra $T(G;X)$ associated to the one-parameter subgroup $exp(tX)$ defined by a left-invariant vector field $X$ on a compact Lie group $G$. We compute the $K$-theory of $T(G;X)$ and its commutator ideal $C(G;X)$. We also define an abstract analytical index for $T(G;X)$ and show that this analytical index can be computed in terms of topological data. |
| On the K-theory of quarter-plane Toeplitz algebras (with Claude Schochet) |
| International Journal of Mathematics, Volume 2, Number 2 (1991), 195--204 |
| Given a $C^$-algebra $A$ that is filtered by a collection of closed ideals $A_i$, there is a spectral sequence that relates the $K$-theory of $A$ to the $K$-theory of the various quotient algebras $A_i/A_{i-1}$. The $d_1$ differentials in this spectral sequence are familiar index invariants, but the higher differentials are not well-understood. By considering the case of Toeplitz $C^*$-algebras associated with cones in $Z^2$, it is shown that a $d_2$ differential is nontrivial. This differential turns out to be an obstruction to a classical lifting problem in operator theory. Analysis of this obstruction leads to necessary and sufficient conditions for the lifting problem. |
| Index theory and Toeplitz algebras on certain cones in Z^2 |
| Journal of Operator Theory, Volume 23 (1990), 125--146 |
| We associate a Toeplitz algebra $T^{\alpha,\beta}$ to the cone in $Z^2$ defined by a quarter-plane determined by lines through the origin of slopes $\alpha$ and $\beta$. The compute the $K$-theory of these algebras, as well as the $K$-theory of various related algebras. We then use cyclic cohomology to produce an index theorem for Fredholm operators in $T^{\alpha,\beta}$. |