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.

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