Peter Fillmore

Triviality Theorems for Hilbert Modules

In a recent paper of Kasparov [K] the theory of Hilbert modules over noncommutative C* -algebras is used to establish a general theory of extensions of C*-algebras that extends results of Brown, Douglas, and Fillmore [BDF], Fillmore [F], and Pimsner, Popa, and Voiculescu [PPV]. Since the category of Hilbert C (X) -modules is equivalent to the category of Hilbert bundles over X [DD;DG], many questions of topological interest can be recast in terms of Hilbert C(X)-modules which then give rise to questions about general Hilbert modules. In particular, Kasparov’s stability theorem [K] (which plays an essential part in the proof that inverses exist in the general theory of EXT) is the noncommutative extension of a triviality theorem of Dixmier and Douady [DD, Th.4] (which itself provides the existence of classifying maps for arbitrary separable Hilbert bundles over paracompact spaces).

Lectures on operator theory

C*-algebras: C*-algebras: Definitions and examples C*-algebras: Constructions Positivity in C*-algebras K-theory I Tensor products of C*-algebras Crossed products I Crossed products II: Examples Free products K-theory II: Roots in topology and index theory C*-algebraic K-theory made concrete, or trick or treat with

Operator Algebras and Their Applications

2 \times 2

Entire Cyclic Cohomology of Banach Algebras

matrix algebras Dilation theory C*-algebras and mathematical physics C*-algebras and several complex variables von Neumann algebras: Basic structure of von Neumann algebras von Neumann algebras (Type

The Shift Operator

II_1

K-theory I

factors) The equivalence between injectivity and hyperfiniteness, part I The equivalence between injectivity and hyperfiniteness, part II On the Jones index Introductory topics on subfactors The Tomita-Takesaki theory explained Free products of von Neumann algebras Semigroups of endomorphisms of

C*-algebras: Constructions

\mathcal{B}(H)

The range of the invariant

Classification of C*-algebras AF-algebras and Bratteli diagrams Classification of amenable C*-algebras I Classification of amenable C*-algebras II Simple AI-algebras and the range of the invariant Classification of simple purely infinite C*-algebras I Hereditary subalgebras of certain simple non real rank zero C*-algebras: Preface Introduction The isomorphism theorem The range of the invariant Bibliography Paths on Coxeter diagrams: From platonic solids and singularities to minimal models and subfactors: Preface/Acknowledgements The Kauffman-Lins recoupling theory Graphs and connections An extension of the recoupling model Relations to minimal models and subfactors Bibliography.

Relations to minimal models and subfactors

A generalized intertwining lifting theorem by B. V. Rajarama Bhat On the classification of C*-algebras of real rank zero, III: The infinite case by O. Bratteli, G. A. Elliott, D. E. Evans, and A. Kishimoto On the classification of C*-algebras of real rank zero, IV: Reduction to local spectrum of dimension two by G. A. Elliott, G. Gong, and H. Su Simple approximate circle algebras by I. Stevens The classification of certain non-simple approximate interval algebras by K. H. Stevens Right inverse of the module of approximately finite dimensional factors of type III and approximately finite ergodic principal measured groupoids by C. E. Sutherland and M. Takesaki.

C*-algebraic K-theory made concrete, or trick or treat with 2×2 matrix algebras

Connes has introduced the notion of cyclic cohomology as a replacement for de Rham cohomology in the non-commutative setting. Entire cyclic cohomology is an infinite dimensional version of cyclic cohomology. The technical complications of this subject are severe and consequently some of the fundamental properties that one would expect have until now been conjectural. We report on some recent results in this area, notably Morita and homotopy invariance. The details will appear elsewhere [10].