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

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.