Quantum Eberlein compactifications and invariant means

TL;DR

This paper introduces a new class of $C^*$-bialgebras called $C^*$-Eberlein algebras, explores their properties, especially invariant means, and applies these to quantum group compactifications and splitting results.

Contribution

It defines $C^*$-Eberlein algebras, constructs their compactifications for quantum groups, and proves the existence of invariant means leading to splitting theorems.

Findings

$C^*$-Eberlein algebras always admit invariant means.

The $C^*$-Eberlein compactification splits into the quantum Bohr compactification and annihilated elements.

Results generalize classical and Kac algebra cases, extending previous work.

Abstract

We propose a definition of a "$C^*$-Eberlein" algebra, which is a weak form of a $C^*$-bialgebra with a sort of "unitary generator". Our definition is motivated to ensure that commutative examples arise exactly from semigroups of contractions on a Hilbert space, as extensively studied recently by Spronk and Stokke. The terminology arises as the Eberlein algebra, the uniform closure of the Fourier-Stieltjes algebra $B(G)$, has character space $G^{\mathcal E}$, which is the semigroup compactification given by considering all semigroups of contractions on a Hilbert space which contain a dense homomorphic image of $G$. We carry out a similar construction for locally compact quantum groups, leading to a maximal $C^*$-Eberlein compactification. We show that $C^*$-Eberlein algebras always admit invariant means, and we apply this to prove various "splitting" results, showing how the $C^*$-Eberlein compactification splits as the quantum Bohr compactification and elements which are annihilated by the mean. This holds for matrix coefficients, but for Kac algebras, we show it also holds at the algebra level, generalising (in a semigroup-free way) results of Godement.