Joseph L. Taylor

Several complex variables with connections to algebraic geometry and Lie groups

Selected problems in one complex variable Holomorphic functions of several variables Local rings and varieties The Nullstellensatz Dimension Homological algebra Sheaves and sheaf cohomology Coherent algebraic sheaves Coherent analytic sheaves Stein spaces Frechet sheaves--Cartans theorems Projective varieties Algebraic vs. analytic--Serres theorems Lie groups and their representations Algebraic groups The Borel-Weil-Bott theorem Bibliography Index.

Topological invariants of the maximal ideal space of a Banach algebra

The theme of this paper is as old as the concept of Banach algebra. Given the functor that assigns to a commutative Banach algebra A its maximal ideal space d, , the urge to interpret topological data concerning AA in terms of the algebraic structure of A is irresistable. The first result in this direction is due to Shilov [29]; it says that each opencompact subset of A, is the support of the Gelfand transform of a unique idempotent in A. A corollary is that A, itself is compact if and only if A has an identity. The Shilov idempotent theorem can be viewed as a characterization of the zero-dimensional Cech cohomology group H”(AA , Z). In fact, it implies immediately that N”(AA , Z) is isomorphic to the additive subgroup of A generated by the idempotents in A. An early result of Brushlinsky [9] points out that if X is compact, then the first Cech group W(X, Z) can be identified with C(X)-l,‘exp(C(X)), where for any commutative Banach algebra A with identity, A-l denotes the group of invertible elements of A and exp(A) denotes the subgroup consisting of elements of the form ea for a E A. Arens [l] and Royden [26] proved that the analogous result holds in general. That is, there is a natural isomorphism

Hochschild cohomology and perturbations of Banach algebras

Abstract Let A and B be Banach algebras with identity and let π : A → B be a continuous homomorphism. We obtain conditions on the Hochschild cohomology of A under which perturbations of π are similar to π. We also show that if A is a Banach algebra such that H 2 ( A , A ) = H 3 ( A , A ) = 0, then perturbations of the multiplication of A give algebras isomorphic to A . We use our techniques to partially answer some problems of Kadison and Kastler on perturbations of operator algebras.

A Remark on Casselman’s Comparison Theorem

This paper is an outgrowth of our attempt to understand a comparison theorem of Casselman which asserts that Lie algebra homology groups, with respect to certain nilpotent algebras of a Harish-Chandra module and its C ∞ completion, coincide. Let us start with a precise statement of this theorem.

Measures which are convolution exponentials

Let M(R) denote the measure algebra on the additive group of the reals. R. G. Douglas recently pointed out to us the importance of the following question in the study of Wiener-Hopf integral equations: if fxÇ:M(R) is invertible, then under what conditions does jit = exp(j>) for some vGM(R)? The relevance of the above question in integral equations stems from the fact that if JJLÇÏM(R) is invertible, then // is an exponential if and only if /x has a factorization of the form M=MI * M2, where jUi and ju2 are invertible elements of M[0, 00) and M(— <*>, O] respectively. In fact, if/x = exp(j>) and^i = ^| [0,00), 2̂ =v\ (-00,0), thenjUi = exp(j>i) and M2 = exp(^2) yields such a factorization. Now if Wp is the Wiener-Hopf operator on L [0, 00 ) (p ̂ > 1) given by

An analytic Riemann-Hilbert correspondence for semi-simple Lie groups

Geometric Representation Theory for semi-simple Lie groups has two main sheaf theoretic models. Namely, through Beilinson-Bernstein localization theory, Harish-Chandra modules are related to holonomic sheaves of D modules on the flag variety. Then the (algebraic) Riemann-Hilbert correspondence relates these sheaves to constructible sheaves of complex vector spaces. On the other hand, there is a parallel localization theory for globalized Harish-Chandra modules—i.e., modules over the full semi-simple group which are completions of Harish-Chandra modules. In particular, Hecht-Taylor and Smithies have developed a localization theory relating minimal globalizations of Harish-Chandra modules to group equivariant sheaves of D modules on the flag variety. The main purpose of this paper is to develop an analytic Riemann-Hilbert correspondence relating these sheaves to constructible sheaves of complex vector spaces and to discuss the relationship between this “analytic” study of global modules and the preceding “algebraic” study of the underlying Harish-Chandra modules.