Gerard J. Murphy

Co-Amenability of Compact Quantum Groups

Abstract We study the concept of co-amenability for a compact quantum group. Several conditions are derived that are shown to be equivalent to it. Some consequences of co-amenability that we obtain are faithfulness of the Haar integral and automatic norm-boundedness of positive linear functionals on the quantum group’s Hopf ∗ -algebra (neither of these properties necessarily holds without co-amenability).

Differential calculi over quantum groups and twisted cyclic cocycles

Abstract We study some aspects of the theory of non-commutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the theory of twisted graded traces and their associated twisted cyclic cocycles. One of our principal results is a new method of constructing differential calculi, using twisted graded traces.

Crossed products of C*-algebras by endomorphisms

The concept of a twisted crossed product associated to a non-classical C*-dynamical system is introduced and studied. The relationship between a covariant projective representation of the system and the corresponding induced representation of the twisted crossed product is investigated, particularly from the point of view of determining when the induced representation is faithful. Conditions are given on the C*-dynamical system that ensure nuclearity, simplicity or primeness of the twisted crossed product.

Positive definite kernels and Hilbert C*-modules

A theory of positive definite kernels in the context of Hilbert C*-modules is presented. Applications are given, including a representation of a Hilbert C*-module as a concrete space of operators and a construction of the exterior tensor product of two Hilbert C*-modules.

AMENABILITY AND COAMENABILITY OF ALGEBRAIC QUANTUM GROUPS

We define concepts of amenability and coamenability for algebraic quantum groups in the sense of Van Daele (1998). We show that coamenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or coamenability are obtained. Coamenability is shown to have interesting consequences for the modular theory in the case that the algebraic quantum group is of compact type.

Diagonality inC*-algebras

Soit A une C*-algebre et A Sa les elements hermitiens. Soit I un ideal ferme separable dans une C*-algebre A et on suppose que les elements de A Sa de spectres finis sont denses dans A Sa . Si a∈A Sa et e>0 alors il existe b∈A Sa diagonal relativement a I et il existe x∈I sa tel que ∥x∥

Unitarily-invariant linear spaces in C*-algebras

Characterisations and containment results are given for linear subspaces of a unital C*-algebra that are invariant under conjugation by sets of unitary elements of the algebra. The (unitarily-invariant) linear span of the projections in a simple, unital C*-algebra having non-scalar projections is shown to contain all additive commutators of the algebra and, in certain cases, to be equal to the algebra.

Amenability and co-amenability of algebraic quantum groups II

Abstract We continue our study of the concepts of amenability and co-amenability for algebraic quantum groups in the sense of A. Van Daele and our investigation of their relationship with nuclearity and injectivity. One major tool for our analysis is that every non-degenerate ∗ -representation of the universal C ∗ -algebra associated to an algebraic quantum group has a unitary generator which may be described in a concrete way.

Continuity of the spectrum and spectral radius

Let A be a Banach algebra containing an element x. Topological conditions on the spectrum of x are given which are necessary and sufficient to ensure the continuity of the spectrum or spectral radius at x.

Toeplitz operators on generalised H2 spaces

Some aspects are developed of the theory of Toeplitz operators on generalised HardyH2 spaces associated to function algebras. It is shown that a substantial number of results of the classical theory of Toeplitz operators on the circle extend to this situation, although counterexamples are given which show that there are also important differences. Spectral connectedness results are obtained, and a characterisation of invertibility for Toeplitz operators. The Fredholm theory is also studied.