The algebraic structure of quantum partial isometries

TL;DR

This paper explores the algebraic structures of quantum partial isometries, constructing their liberations and half-liberations, and provides detailed algebraic and probabilistic analyses of these quantum semigroups.

Contribution

It introduces new half-liberated and liberated quantum semigroups related to partial isometries and analyzes their algebraic and probabilistic properties.

Findings

Construction of half-liberations and liberations of quantum partial isometry semigroups

Detailed algebraic analysis of the new quantum semigroups

Probabilistic study supporting the liberation concepts

Abstract

The partial isometries of $\mathbb R^N,\mathbb C^N$ form compact semigroups $\widetilde{O}_N,\widetilde{U}_N$. We discuss here the liberation question for these semigroups, and for their discrete versions $\widetilde{H}_N,\widetilde{K}_N$. Our main results concern the construction of half-liberations $\widetilde{H}_N^\times,\widetilde{K}_N^\times,\widetilde{O}_N^\times,\widetilde{U}_N^\times$ and of liberations $\widetilde{H}_N^+,\widetilde{K}_N^+,\widetilde{O}_N^+,\widetilde{U}_N^+$. We include a detailed algebraic and probabilistic study of all these objects, justifying our "half-liberation" and "liberation" claims.