Group representations in the Liouville representation and the algebraic approach

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Abstract
Group representations on the Liouville representation spaces are considered. It is shown that the state space I1(H) of trace class operators on Hilbert space H and the observable space L(H) of bounded operators are completely reducible under physically induced representations of compact Hausdorff groups when appropriate topologies are used. For state space I1(H) both the norm topology and the weak topology lead to complete reducibility, while for observable space L(H) the weak‐* topology—but not the norm topology—suffices. This leads to conservation laws, selection rules, and Wigner–Eckart theorems for the Liouville representation. It is shown that serious difficulties are encountered when a similar theory is attempted on the observable space and state space used in the algebraic approach.

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