Polynomial representations of $C^*$-algebras and their applications

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 $S^n({\mathcal A})$, 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 ${\mathcal A}=B({\mathcal H})$.