By Marco A. Pérez B.

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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.