A Hilbert space approach to approximate diagonals for locally compact quantum groups

TL;DR

This paper introduces a Hilbert space method to construct approximate diagonals for locally compact quantum groups, linking operator amenability of quantum group algebras with properties of nets in their L^2 spaces.

Contribution

It develops a novel approach using nets in L^2 spaces and multiplicative unitaries to generate approximate diagonals, advancing the understanding of operator amenability in quantum groups.

Findings

Constructs approximate diagonals via nets in L^2()

Establishes conditions under which these nets produce bounded approximate diagonals

Provides a new method connecting operator amenability of L^1() and Fourier algebra A(G)

Abstract

For a locally compact quantum group $\mathbb{G}$, the quantum group algebra $L^1(\mathbb{G})$ is operator amenable if and only if it has an operator bounded approximate diagonal. It is known that if $L^1(\mathbb{G})$ is operator biflat and has a bounded approximate identity then it is operator amenable. In this paper, we consider nets in $L^2(\mathbb{G})$ which suffice to show these two conditions and combine them to make an approximate diagonal of the form $\omega_{{W'}^*\xi\otimes\eta}$ where $W$ is the multiplicative unitary and $\xi\otimes\eta$ are simple tensors in $L^2(\mathbb{G})\otimes L^2(\mathbb{G})$. Indeed, if $\mathbb{G}$ and $\hat{\mathbb{G}}$ both have a bounded approximate identity and either of the corresponding nets in $L^2(\mathbb{G})$ satisfies a condition generalizing quasicentrality then this construction generates an operator bounded approximate diagonal. In the classical group case, this provides a new method for constructing approximate diagonals emphasizing the relation between the operator amenability of the group algebra $L^1(G)$ and the Fourier algebra $A(G)$.