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By T.Y. Lam

By way of aiming the extent of writing on the beginner instead of the gourmet and by way of stressing the position of examples and motivation, the writer has produced a textual content that's appropriate for a one-semester graduate path or for self-study.

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Example text

41 Let π be a representation of G and let ξ ∈ H(π ). The following are equivalent: (i) ξ is a cyclic vector for π. (ii) T ∈ π(G) and T ξ = 0 imply T = 0. 41 is often referred to as a separating vector for the algebra π (G) . When ξ is a separating vector for π(G) , the map T → T ξ is an injective linear map from π(G) into H(π ). So π(G) cannot be too large if π is a cyclic representation. A general representation can be decomposed into a sum of cyclic representations. 42 Let π be a representation of G.

In fact, the map P → P H(π ) is a bijection between the set of projections in π (G) and the set of closed π-invariant subspaces of H(π ). If P is a projection in π(G) for some representation π of G, let π P denote the subrepresentation formed by restricting each π(x) to P H(π). If P and Q are projections in π (G) , we can form the linear space Qπ(G) P = {QAP : A ∈ π (G) }. The map T → T |P H(π) identifies Qπ(G) P with HomG (π P , π Q ). This is formalized as the following proposition which allows us to view spaces of intertwining operators as a “part” of a commutant algebra.

Then ϕ is called a function of positive type associated with π. If S is a set of representations of G and ϕ is a function of positive type, we may say ϕ is associated with S if ϕ is associated with σ for some σ ∈ S. Note that equivalent representations have the same functions of positive type associated with them. Thus, we can unambiguously refer to the functions of positive type associated with π ∈ G. For a locally compact group G, the set P (G) of all continuous functions of positive type on G carries much of the representation theory of G in its structure.

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