Beltita D, Neeb KH (2016)
Publication Type: Journal article
Publication year: 2016
Book Volume: 86
Pages Range: 545-578
Journal Issue: 4
DOI: 10.1007/s00020-016-2335-9
This is a sequel to our paper on nonlinear completely positive maps and dilation theory for real involutive algebras, where we have reduced all classification problems to the passage from a C ∗-algebra A to its symmetric powers Sn(A), resp., to holomorphic representations of the multiplicative ∗-semigroup (A, ·). Here we study the correspondence between representations of A and of Sn(A) in detail. As Sn(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 Sn(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 Sn(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).
APA:
Beltita, D., & Neeb, K.H. (2016). Polynomial representations of C*-algebras and their applications. Integral Equations and Operator Theory, 86(4), 545-578. https://dx.doi.org/10.1007/s00020-016-2335-9
MLA:
Beltita, Daniel, and Karl Hermann Neeb. "Polynomial representations of C*-algebras and their applications." Integral Equations and Operator Theory 86.4 (2016): 545-578.
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