Abstract
Hilbert space representations of cross product ∗-algebras of the Hopf ∗-algebras U q(gl 2) with the coordinate algebras O ̂ ( C q 2) and O( R q 3) of quantum vector spaces, and of U q(su 2) with the coordinate algebras O(SU q(2)) and O(S q 2) of the corresponding quantum spheres, are investigated and classified. Invariant states on the coordinate ∗-algebras are described by two variants of the quantum trace.
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