Polynomial Representations of $C^*$-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 classification problems to the passage from a $C^*$-algebra $\mathcal {A}$ to its symmetric powers $S^n(\mathcal {A})$, resp., to holomorphic representations of the multiplicative $*$-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 $\mathcal {A}^{\otimes n} \rtimes S_n$ 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 $S^n(\mathcal {A})$ and we relate this to the Schur–Weyl theory for $C^*$-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 $\mathcal {A}=B(\mathcal {H})$.