Polynomial representations of $C^{\ast }$-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 $C^{\ast }$-algebra $\mathcal{A}$ to its symmetric powers $S^n(\mathcal{A})$, resp., to holomorphic representations of the multiplicative $\ast$-semigroup $({\mathcal A},\cdot)$\mathrm{.}Herewestudythecorrespondencebetweenrepresentationsof${\mathcal A}$andof$S^n({\mathcal A})$indetail\mathrm{.}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 ${\mathcal A}^{\otimes n} \rtimes S_n$inwhichitisahereditarysubalgebra\mathrm{.}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 $C^{\ast }$-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 ${\mathcal A}=B({\mathcal H})$.