Extensions, restrictions, and representations of states on 𝐶*-algebras

Anderson, Joel

doi:10.1090/S0002-9947-1979-0525675-1

Extensions, restrictions, and representations of states on $C^{\ast }$-algebras
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Trans. Amer. Math. Soc. 249 (1979), 303-329 Request permission

Abstract:

In the first three sections the question of when a pure state g on a ${C^{\ast }}$-subalgebra B of a ${C^{\ast }}$-algebra A has a unique state extension is studied. It is shown that an extension f is unique if and only if inf$\left \| {b\left ( {a - f\left ( a \right )1} \right )b} \right \| = 0$ for each a in A, where the inf is taken over those b in B such that $0 \leqslant b \leqslant 1$ and $g(b) = 1$. The special cases where B is maximal abelian and/or $A = B\left ( H \right )$ are treated in more detail. In the remaining sections states of the form $T \mapsto \lim \limits _{\mathcal {u}} \left ( {T{x_\alpha }, {x_\alpha }} \right )$, where $\left \{ {{x_\alpha }} \right \}{ _{\alpha \in \kappa }}$ is a set of unit vectors in H and $\mathcal {u}$ is an ultrafilter are studied.

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