A survey of amenability theory for direct-limit groups

Dawson, Matthew; Ólafsson, Gestur

doi:10.1090/conm/650/13041

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“…We touch on that theme here again, proving, among other things, that holomorphy, although much weaker than linearity, is strong enough to imply for instance complete positivity of multiplicative homomorphisms of C * -algebras (see Proposition 3.2 below). The present approach to representation theory of infinite dimensional unitary groups is based on C * -algebras and their multiplicative representations and is thus complementary to other recent works on representation theory of infinite dimensional Lie groups, such as [Wo14] or [DwOl15]. This approach allows us to shed fresh light on, and to partially extend, some results on Schur-Weyl duality in infinite dimensions from [BN12], [Nes13], and [EnIz15].…”

Section: Introductionmentioning

confidence: 85%

Polynomial Representations of $$C^*$$ C ∗ -Algebras and their Applications

Beltiţă

Neeb

2016

Integr. Equ. Oper. Theory

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This is a sequel to our paper on nonlinear completely positive maps and dilation theory for real involutive algebras, where we have reduced all representation classification problems to the passage from a C * -algebra A to its symmetric powers S n (A), resp., to holomorphic representations of the multiplicative * -semigroup (A, ·). Here we study the correspondence between representations of A and of S n (A) in detail. As S n (A) is the fixed point algebra for the natural action of the symmetric group Sn on A ⊗n , this is done by relating representations of S n (A) to those of the crossed product A ⊗n ⋊ Sn in which it is a hereditary subalgebra. For C * -algebras of type I, we obtain a rather complete description of the equivalence classes of the irreducible representations of S n (A) and we relate this to the Schur-Weyl theory for C * -algebras. Finally we show that if A ⊆ B(H) is a factor of type II or III, then its corresponding multiplicative representation on H ⊗n is a factor representation of the same type, unlike the classical case A = B(H). Mathematics Subject Classification 2010: 46L06, 22E66, 46L45

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