Title: Quantum Eberlein compactifications and invariant means

Authors: Matthew Daws and Biswarup Das

Issue: Volume 65 (2016), Issue 1, 307-352


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 because the Eberlein algebra, the uniform closure of the Fourier-Stieltjes $B(G)$, has character space $G^{\mathcal{E}}$, which is the semigroup compactification given by considering all semigroups of contractions on a Hilbert space that 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 that 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.