Journal of Operator Theory

Volume 39, Issue 2, Spring 1998 pp. 319-338.

Unique extension of pure states of $\scriptstyle C^*$-algebras

Authors: L.J. Bunce (1), and C.-H. Chu (2)
Author institution: (1) Department of Mathematics, University of Reading, Reading RG6 2AX, England
(2) Goldsmiths College, University of London, London SE14 6NW, England

Summary: Let $A$ be a $C^*$-subalgebra of a $C^*$-algebra $B$. We say that $A$ has the {\it pure extension property} in $B$ if every pure state of $A$ has a unique pure state extension to $B$. We show that $A$ has the pure extension property in $B$ if and only if there is a weak expectation on $B$ for the atomic representation of $A$, among several equivalent conditions, including the unique extension of type I factor states. If $A$ is separable and $B$ is a von Neumann algebra, we show that the pure extension property is equivalent to that every factor state of $A$ extends to a unique factor state of $B$ which is in turn equivalent to that $A$ is dual and the minimal projections of $A$ are minimal in $B$. If $A$ has the pure extension property in $B$, then there is a natural map $\widehat{\alpha}$ between their spectra $\widehat {A}$ and $\widehat {B}$. We study the relationship of $\widehat {A}$ and $\widehat {B}$ under $\widehat{\alpha}$ as well as the unique extension of atomic states.

Keywords: $C^*$-algebra, pure extension property, atomic extension property, weak expectation, hereditary subalgebra

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