Completely bounded representations of convolution algebras of locally compact quantum groups

Details Completely bounded representations of convolution algebras of locally compact quantum groups Completely bounded representations of convolution algebras of locally compact quantum groups

2011-07-11

arxiv.org/abs/1107.2094v1

Mathematics

Operator Algebras

30 pages

Scientific paper

Given a locally compact quantum group $\G$, we study the question of when completely bounded homomorphisms $\pi:L^1(\mathbb G)\rightarrow\mathcal B(H)$ are similar to *-homomorphisms. By analogy with the cocommutative case (representations of the Fourier algebra $A(G)$) we are lead to consider the associated map $\pi^*:L^1_\sharp(\mathbb G) \rightarrow \mathcal B(H)$ given by $\pi^*(\omega) = \pi(\omega^\sharp)^*$. Completely bounded homomorphisms $\pi$ of $L^1(\mathbb G)$ are in a one-to-one correspondence with corepresentations $V_\pi$ of $L^\infty(\mathbb G)$. Our main technical result is that $\pi$ and $\pi^*$ are both completely bounded if and only if $V_\pi$ is invertible (if $\pi$ is non-degenerate). An averaging argument then shows that when $\mathbb G$ is amenable, $\pi$ is similar to a *-homomorphism if and only if $\pi^*$ is completely bounded. In the cocommutative case, we show that the assumption of complete boundedness is essential. We also give some further cases when completely bounded homomorphisms $\pi$ of $A(G)$ are similar to *-homomorphisms without further assumption on $\pi^*$.

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