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By Eberhard Kaniuth

The twin area of a in the community compact team G involves the equivalence periods of irreducible unitary representations of G. This publication offers a entire consultant to the idea of brought about representations and explains its use in describing the twin areas for vital sessions of teams. It introduces a variety of induction structures and proves the middle theorems on precipitated representations, together with the basic imprimitivity theorem of Mackey and Blattner. an intensive advent to Mackey research is utilized to compute twin areas for a wide selection of examples. Fell's contributions to realizing the average topology at the twin also are provided. within the ultimate chapters, the idea is utilized in various settings together with topological Frobenius houses and non-stop wavelet transforms. This publication may be beneficial to graduate scholars trying to input the realm in addition to specialists who desire the speculation of unitary staff representations of their learn.

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Induced Representations of Locally Compact Groups

The twin house of a in the neighborhood compact workforce G includes the equivalence sessions of irreducible unitary representations of G. This ebook presents a accomplished advisor to the idea of brought on representations and explains its use in describing the twin areas for vital sessions of teams. It introduces a number of induction structures and proves the middle theorems on brought about representations, together with the basic imprimitivity theorem of Mackey and Blattner.

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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.

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