An Operator-theoretic Approach to Invariant Integrals on Quantum Homogeneous ${\mathrm{SU}}_{n,1}$-spaces

Abstract

An operator-theoretic approach to invariant integrals on non-compact quantum spaces is introduced on the examples of quantum ball algebras. In order to describe an invariant integral, operator algebras are associated to the quantum space which allow an interpretation as “rapidly decreasing” functions and as functions with compact support. 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. The important feature of the approach is that these operator algebras are topological spaces in a natural way. For suitable representations and with respect to the bounded and weak operator topologies, it is shown that the algebra of functions with compact support is dense in the algebra of closeable operators used to define these algebras of functions and that the infinitesimal action of the quantum symmetry group is continuous.