Kenneth R. Davidson

Invariant Subspaces and Hyper-Reflexivity for Free Semigroup Algebras

In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces. Such algebras are called hyper-reflexive. This property is very strong, but it has been established in only a very few cases. Moreover the prototypes of this class of algebras are the natural candidate for a non-commutative analytic Toeplitz algebra on n variables. The case we make for this analogy is very compelling. In particular, in this paper, the key to the invariant subspace analysis is a good analogue of the Beurling theorem for invariant subspaces of the unilateral shift. This leads to a notion of inner–outer factorization in these algebras. In a sequel to this paper [13], we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in Cn. A free semigroup algebra is the weak operator topology closed algebra generated by a set S1, . . . , Sn of isometries with pairwise orthogonal ranges. These conditions are described algebraically by S∗ i Sj = δijI for 1 ≤ i, j ≤ n; (F)

Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras

The non-commutative analytic Toeplitz algebra is the WOT-closed algebra generated by the left regular representation of the free semigroup onn generators. We obtain a distance formula to an arbitrary WOT-closed right ideal and thereby show that the quotient is completely isometrically isomorphic to the compression of the algebra to the orthogonal complement of the range of the ideal. This is used to obtain Nevanlinna-Pick type interpolation theorems

Convexity and Optimization

Optimization is a central theme of applied mathematics that involves minimizing or maximizing various quantities. This is an important application of the derivative tests in calculus. In addition to the first and second derivative tests of one-variable calculus, there is the powerful technique of Lagrange multipliers in several variables. This chapter is concerned with analogues of these tests that are applicable to functions that are not differentiable. Of course, some different hypothesis must replace differentiability, and this is the notion of convexity. It turns out that many applications in economics, business, and related areas involve convex functions. As in other chapters of this book, we concentrate on the theoretical underpinnings of the subject. The important aspect of constructing algorithms to carry out our program is not addressed. However, the reader will be well placed to read that material. Results from both linear algebra and calculus appear regularly.

Periodicity in Rank 2 Graph Algebras

Kumjian and Pask introduced an aperiodicity condition for higher rank graphs. We present a detailed analysis of when this occurs in certain rank 2 graphs. When the algebra is aperiodic, we give another proof of the simplicity of C(F θ ). The periodic C∗-algebras are characterized, and it is shown that C(F θ ) ≃ C(T)⊗ A where A is a simple C∗-algebra. Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1 e-mail: krdavids@uwaterloo.ca Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4 e-mail: dyang@uwindsor.ca Received by the editors June 5, 2007. The first author partially supported by an NSERC grant. AMS subject classification: Primary: 47L55; secondary: 47L30, 47L75, 46L05.

On operators commuting with Toeplitz operators modulo the compact operators

Abstract We prove that an operator on H2 of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.

The Choquet boundary of an operator system

We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary representations to generate the C*-envelope.

Isometric Dilations of Non-Commuting Finite Rank n-Tuples

A contractive n-tuple A = (A1, . . . , An) has a minimal joint isometric dilation S = (S1, . . . , Sn) where the Sis are isometries with pairwise orthogonal ranges. This determines a rep- resentation of the Cuntz-Toeplitz algebra. When A acts on a finite dimensional space, the wot-closed nonself-adjoint algebra S generated by S is completely described in terms of the properties of A. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra S is always hyper-reflexive. In the last section, we describe similarity invariants. In partic- ular, an n-tuple B of d × d matrices is similar to an irreducible n-tuple A if and only if a certain finite set of polynomials vanish on B.

The isomorphism problem for some universal operator algebras

Abstract This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C ⁎ -envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot -closures of these algebras as well.

Isomorphisms between topological conjugacy algebras

Abstract A family of algebras, which we call topological conjugacy algebras, is associated with each proper continuous map on a locally compact Hausdorff space. Assume that is a continuous proper map on a locally compact Hausdorff space , for i= 1,2. We show that the dynamical systems and are conjugate if and only if some topological conjugacy algebra of is isomorphic as an algebra to some topological conjugacy algebra of . This implies as a corollary the complete classification of the semicrossed products , which was previously considered by Arveson and Josephson [W. Arveson, K. Josephson, Operator algebras and measure preserving automorphisms II, J. Funct. Anal. 4 (1969), 100–134.], Peters [J. Peters, Semicrossed products of C*-algebras, J. Funct. Anal. 59 (1984), 498–534.], Hadwin and Hoover [D. Hadwin, T. Hoover, Operator algebras and the conjugacy of transformations, J. Funct. Anal. 77 (1988), 112–122.] and Power [S. Power, Classification of analytic crossed product algebras, Bull. London Math. Soc. 24 (1992), 368–372.]. We also obtain a complete classification of all semicrossed products of the form , where denotes the disc algebra and a continuous map which is analytic on the interior. In this case, a surprising dichotomy appears in the classification scheme, which depends on the fixed point set of η. We also classify more general semicrossed products of uniform algebras.

Operator algebras for analytic varieties

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictionsMV of the multiplier algebraM of Drury-Arveson space to a holomorphic subvariety V of the unit ball Bd. We nd that MV is completely isometrically isomorphic toMW if and only if W is the image of V under a biholomorphic auto- morphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthend to show that, when d <1, every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (al- gebraically) isomorphic is also studied. When V and W are each a nite union of irreducible varieties and a discrete variety in Bd with d <1, then an isomorphism betweenMV andMW deter- mines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak- continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold|particularly, smooth curves and Blaschke sequences.