Isometries between quantum convolution algebras

TL;DR

This paper characterizes isometric algebra isomorphisms between quantum convolution algebras, showing they imply isomorphisms or commutant relations between the underlying quantum groups, extending classical and Kac algebra results.

Contribution

It extends known results on isometries of quantum convolution algebras to general locally compact quantum groups, including measure algebras, with new proof techniques.

Findings

Isometric algebra isomorphisms imply quantum group isomorphisms or commutants.

The adjoint of such isomorphisms are *-isomorphisms up to a twist.

Results generalize classical theorems of Wendel and Walter.

Abstract

Given locally compact quantum groups $\G_1$ and $\G_2$, we show that if the convolution algebras $L^1(\G_1)$ and $L^1(\G_2)$ are isometrically isomorphic as algebras, then $\G_1$ is isomorphic either to $\G_2$ or the commutant $\G_2'$. Furthermore, given an isometric algebra isomorphism $\theta:L^1(\G_2) \rightarrow L^1(\G_1)$, the adjoint is a *-isomorphism between $L^\infty(\G_1)$ and either $L^\infty(\G_2)$ or its commutant, composed with a twist given by a member of the intrinsic group of $L^\infty(\G_2)$. This extends known results for Kac algebras (although our proofs are somewhat different) which in turn generalised classical results of Wendel and Walter. We show that the same result holds for isometric algebra homomorphisms between quantum measure algebras (either reduced or universal). We make some remarks about the intrinsic groups of the enveloping von Neumann algebras of C$^*$-algebraic quantum groups.