Pure states on some group-invariant $C^{\ast }$-algebras

Abstract:

Let $\mathfrak {A}$ be a UHF algebra of Glimm type ${n^\infty }$, i.e., $\mathfrak {A} = \otimes _{k \geqslant 1}^{\ast }{N_k}$, where $N = {N_1} = {N_2} = \cdots$ are $n \times n$ matrix algebras. We define an AF-subalgebra ${\mathfrak {A}^G}$ of $\mathfrak {A}$, consisting of those elements of $\mathfrak {A}$ invariant under a group of automorphisms $\{ {\alpha _g}:g \in G = \operatorname {SU} (n)\}$ of product type. ${\mathfrak {A}^G}$ is shown to be generated by an embedding of $S(\infty )$, the discrete group of finite permutations on countably many symbols. Let $\omega$ be a pure product state on $\mathfrak {A}$, ${\omega ^G}$ its restriction to ${\mathfrak {A}^G}$. Let $e \in N$ be a one-dimensional projection with corresponding projections ${e^k} \in {N_k}$. Then if both (i) ${\Sigma _{k \geqslant 1}}\omega ({e^k}) = \infty$, and (ii) $0 < {\Sigma _{k \geqslant 1}}\omega ({e^k})[1 - \omega ({e^k})] < \infty$ hold, ${\omega ^G}$ is not pure. ${\omega ^G}$ is shown to be pure if there exist orthogonal one-dimensional projections $\{ {p_i}:1 \leqslant i \leqslant n\}$ of $N$ with corresponding projections $p_i^k \in {N_k}$ such that $\omega (p_i^k) = 0$ or $1$, $1 \leqslant i \leqslant n, k \geqslant 1$, and $0 < {\Sigma _{k \geqslant 1}}\omega (p_i^k) < \infty$ for at most one $i$.