On quantum semigroup actions on finite quantum spaces

TL;DR

The paper demonstrates that continuous actions of quantum semigroups on finite quantum spaces can be derived from actions of their quantum Bohr compactification, providing a complete classification for actions on the 2x2 matrix algebra.

Contribution

It establishes a link between quantum semigroup actions and their Bohr compactifications, and classifies all such actions on M_2 preserving a faithful state.

Findings

Actions of quantum semigroups on finite quantum spaces originate from their Bohr compactification.

Complete classification of quantum semigroup actions on M_2 preserving a faithful state.

Identification of the structure of actions on finite-dimensional C*-algebras.

Abstract

We show that a continuous action of a quantum semigroup $\mathcal{S}$ on a finite quantum space (finite dimensional $\mathrm{C}^*$-algebra) preserving a faithful state comes from a continuous action of the quantum Bohr compactification $\mathfrak{b}\mathcal{S}$ of $\mathcal{S}$. Using the classification of continuous compact quantum group actions on $M_2$ we give a complete description of all continuous quantum semigroup actions on this quantum space preserving a faithful state.