An operator-theoretic approach to invariant integrals on quantum homogeneous sl_{n+1}(R)-spaces

TL;DR

This paper introduces an operator-theoretic method for defining invariant integrals on complex quantum homogeneous spaces where traditional algebraic approaches are ineffective, expanding the toolkit for quantum geometry analysis.

Contribution

It develops a novel operator-theoretic framework for invariant integrals on quantum spaces with unbounded generators and continuous spectra, overcoming limitations of algebraic methods.

Findings

Invariant integrals are defined as trace functionals on trace class operator algebras.

The approach applies to quantum spaces lacking bounded Hilbert space representations.

The method generalizes the quantum trace to more complex quantum homogeneous spaces.

Abstract

We present other examples illustrating the operator-theoretic approach to invariant integrals on quantum homogeneous spaces developed by Kuersten and the second author. The quantum spaces are chosen such that their coordinate algebras do not admit bounded Hilbert space representations and their self-adjoint generators have continuous spectrum. Operator algebras of trace class operators are associated to the coordinate algebras which allow interpretations as rapidly decreasing functions and as finite functions. The invariant integral is defined as a trace functional which generalizes the well-known quantum trace. We argue that previous algebraic methods would fail for these examples.