Pure states on some group-invariant -algebras

Author:
Geoffrey L. Price

Journal:
Trans. Amer. Math. Soc. **283** (1984), 533-562

MSC:
Primary 46L30; Secondary 22D25

DOI:
https://doi.org/10.1090/S0002-9947-1984-0737883-7

MathSciNet review:
737883

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Abstract: Let be a UHF algebra of Glimm type , i.e., , where are matrix algebras. We define an AF-subalgebra of , consisting of those elements of invariant under a group of automorphisms of product type. is shown to be generated by an embedding of , the discrete group of finite permutations on countably many symbols. Let be a pure product state on , its restriction to . Let be a one-dimensional projection with corresponding projections . Then if both (i) , and (ii) hold, is not pure. is shown to be pure if there exist orthogonal one-dimensional projections of with corresponding projections such that or , , and for at most one .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1984-0737883-7

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