Polynomial representations of $C^{\ast }$-algebras and their applications

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 $C^{\ast }$-algebra $\mathcal{A}$ to its symmetric powers $S^n(\mathcal{A})$, resp., to holomorphic representations of the multiplicative $\ast$-semigroup $(\mathcal{A},\cdot )$. Here we study the correspondence between representations of $\mathcal{A}$ and of $S^n(\mathcal{A})$ in detail. As $S^n(\mathcal{A})$ is the fixed point algebra for the natural action of the symmetric group $S_n$ on ${\mathcal{A}}^{\otimes n}$, this is done by relating representations of $S^n(\mathcal{A})$ to those of the crossed product ${\mathcal{A}}^{\otimes n}⋊S_n$ in which it is a hereditary subalgebra. For $C^{\ast }$-algebras of type I, we obtain a rather complete description of the equivalence classes of the irreducible representations of $S^n(\mathcal{A})$ and we relate this to the Schur--Weyl theory for $C^{\ast }$-algebras. Finally we show that if $\mathcal{A}\subseteq B(\mathcal{H})$ is a factor of type II or III, then its corresponding multiplicative representation on ${\mathcal{H}}^{\otimes n}$ is a factor representation of the same type, unlike the classical case $\mathcal{A}=B(\mathcal{H})$.

32 pages

References in corpus (2)

• A survey of equivariant map algebras with open problems
• Schur-Weyl duality for the unitary groups of $I{I}_1$-factors