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This paper is a sequel to our previous study of spherical representations inthe operator algebra setup. We first introduce possible analogs of dimensiongroups in the present context by utilizing the notion of operator systems andtheir relatives. We then apply our study to inductive limits of compact quantumgroups, and establish an analogue of Olshanski's notion of spherical unitaryrepresentations of infinite-dimensional Gelfand pairs of the form $G < G\timesG$ (via the diagonal embedding) in the quantum group setup. This, inparticular, justifies Ryosuke Sato's approach to asymptotic representationtheory for quantum groups.

Spherical Representations of $C^*$-Flows II: Representation System and Quantum Group Setup