Invariant integration theory on non-compact quantum spaces: Quantum (n,1)-matrix ball

TL;DR

This paper develops an operator algebra framework for invariant integration on the non-compact quantum (n,1)-matrix ball, generalizing the quantum trace and exploring Hilbert space representations.

Contribution

It introduces an operator theoretic approach to invariant integration on non-compact quantum spaces, specifically for the quantum (n,1)-matrix ball, and classifies related Hilbert space representations.

Findings

Invariant integral is a generalization of the quantum trace.

Operator algebras interpret functions with rapid decay or compact support.

Hilbert space representations are classified and topologically analyzed.

Abstract

An operator theoretic approach to invariant integration theory on non-compact quantum spaces is introduced on the example of the quantum (n,1)-matrix ball O_q(Mat_{n,1}). In order to prove the existence of an invariant integral, operator algebras are associated to O_q(Mat_{n,1}) which allow an interpretation as ``rapidly decreasing'' functions and as functions with compact support on the quantum (n,1)-matrix ball. It is shown that the invariant integral is given by a generalization of the quantum trace. If an operator representation of a first order differential calculus over the quantum space is known, then it can be extended to the operator algebras of integrable functions. Hilbert space representations of O_q(Mat_{n,1}) are investigated and classified. Some topological aspects concerning Hilbert space representations are discussed.