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Transactions of the American Mathematical Society

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

 

 

Simplexes of extensions of states of $ C\sp{\ast} $-algebras


Author: C. J. K. Batty
Journal: Trans. Amer. Math. Soc. 272 (1982), 237-246
MSC: Primary 46L05; Secondary 46L55
DOI: https://doi.org/10.1090/S0002-9947-1982-0656488-8
MathSciNet review: 656488
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Abstract: Let $ B$ be a $ {C^\ast}$-subalgebra of a $ {C^\ast}$-algebra $ A$, $ F$ a compact face of the state space $ S(B)$ of $ B$, and $ {S_F}(A)$ the set of all states of $ A$ whose restrictions to $ B$ lie in $ F$. It is shown that $ {S_F}(A)$ is a Choquet simplex if and only if (a) $ F$ is a simplex, (b) pure states in $ {S_F}(A)$ restrict to pure states in $ F$, and (c) the states of $ A$ which restrict to any given pure state in $ F$ form a simplex. The properties (b) and (c) are also considered in isolation.

Sets of the form $ {S_F}(A)$ have been considered by various authors in special cases including those where $ B$ is a maximal abelian subalgebra of $ A$, or $ A$ is a $ {C^\ast}$-crossed product, or the Cuntz algebra $ {\mathcal{O}_n}$.


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DOI: https://doi.org/10.1090/S0002-9947-1982-0656488-8
Keywords: Pure state, extension, restriction, simplex, face, irreducible representation, invariant state, crossed product
Article copyright: © Copyright 1982 American Mathematical Society