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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Geoffrey L. Price
Journal: Trans. Amer. Math. Soc. 283 (1984), 533-562
MSC: Primary 46L30; Secondary 22D25
MathSciNet review: 737883
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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$.

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Keywords: Pure state, factor state, product state, UHF, AF-algebras, lattice algebra, invariant algebra, symmetric group, GNS representation
Article copyright: © Copyright 1984 American Mathematical Society