Simplexes of extensions of states of 𝐶*-algebras

Batty, C. J. K.

doi:10.1090/S0002-9947-1982-0656488-8

Simplexes of extensions of states of $C^{\ast }$-algebras
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by C. J. K. Batty PDF

Trans. Amer. Math. Soc. 272 (1982), 237-246 Request permission

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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