Matthew Daws

The Haagerup property for locally compact quantum groups

The Haagerup property for locally compact groups is generalised to the context of locally compact quantum groups, with several equivalent characterisations in terms of the unitary representations and positive-definite functions established. In particular it is shown that a locally compact quantum group G has the Haagerup property if and only if its mixing representations are dense in the space of all unitary representations. For discrete G we characterise the Haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group

COMPLETELY POSITIVE MULTIPLIERS OF QUANTUM GROUPS

\hat{G}

Multipliers of locally compact quantum groups via Hilbert C*-modules

; by the existence of a real proper cocycle on G, and further, if G is also unimodular we show that the Haagerup property is a von Neumann property of G. This extends results of Akemann, Walter, Bekka, Cherix, Valette, and Jolissaint to the quantum setting and provides a connection to the recent work of Brannan. We use these characterisations to show that the Haagerup property is preserved under free products of discrete quantum groups.

Arens Regularity of the Algebra of Operators on a Banach Space

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group 𝔾 (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of 𝔾. It follows that there is an order bijection between the completely positive multipliers of L1(𝔾) and the positive functionals on the universal quantum group . We provide a direct link between the Junge, Neufang, Ruan representation result and the representing element of a multiplier, and use this to show that their representation map is always weak*–weak*-continuous.

Representing Multipliers of the Fourier Algebra on Non-Commutative L-p Spaces

A result of Gilbert shows that every completely bounded multiplier

Closed ideals in the Banach algebra of operators on classical non-separable spaces

f

Categorical Aspects of Quantum Groups: Multipliers and Intrinsic Groups

of the Fourier algebra

Shift invariant preduals of ℓ 1(ℤ)

A(G)

Amenability of Ultrapowers of Banach Algebras

arises from a pair of bounded continuous maps

Corrigendum: amenability of ultrapowers of Banach algebras

\alpha,\beta:G \rightarrow K