Polynomial representations of C^*-algebras and their applications

2016/08/08 by Daniel Beltita, Beltita, Daniel, Karl-Hermann Neeb +1
Mathematics · #22E66 #46L06 #46L45 #FOS: Mathematics #Operator Algebras (math.OA) #Representation Theory (math.RT) #math.OA #math.RT #msc:22E66 #msc:46L06 #msc:46L45

paper · pdf · doi:10.48550/arxiv.1608.02445

32 pages

arxiv created 2016/08/08 · arxiv updated 2016/08/09

Abstract

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 \mathcal A to its symmetric powers Sⁿ(\mathcal A), resp., to holomorphic representations of the multiplicative *-semigroup (\mathcal A,⋅). Here we study the correspondence between representations of \mathcal A and of Sⁿ(\mathcal A) in detail. As Sⁿ(\mathcal A) is the fixed point algebra for the natural action of the symmetric group S_n on \mathcal A^{⊗ n}, this is done by relating representations of Sⁿ(\mathcal A) to those of the crossed product \mathcal A^{⊗ n} \rtimes S_n 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 Sⁿ(\mathcal A) and we relate this to the Schur--Weyl theory for C^*-algebras. Finally we show that if \mathcal A⊆ B(\mathcal H) is a factor of type II or III, then its corresponding multiplicative representation on \mathcal H^{⊗ n} is a factor representation of the same type, unlike the classical case \mathcal A=B(\mathcal H).

Related

• Schur--Weyl Theory for C^*-algebras
• C^*-algebras associated with the ax+b-semigroup over \mathbb N
• Holomorphic geometric models for representations of C^*-algebras
• Nonlinear completely positive maps and dilation theory for real involutive algebras
• Inclusions of C^*-algebras arising from fixed-point algebras