Journal of Operator Theory

Volume 44, Issue 1, Summer 2000 pp. 207-222.

Left quotients of

$C^*$ -algebras, II: Atomic parts of left quotients

Authors: Lawrence G. Brown (1), and Ngai-Ching Wong (2)
Author institution: (1) Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA
(2) Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan, R.O.C.

Summary: Let

$A$ be a

$C^*$ -algebra. Let

$z$ be the maximal atomic projection in

$\A$ . By a theorem of Brown, an element

$x$ in

$\A$ has a continuous atomic part, \ie

$zx=za$ for some

$a$ in

$A$ , whenever

$x$ is uniformly continuous on the set of pure states of

$A$ . Let

$L$ be a closed left ideal of

$A$ . Under some additional conditions, we shall show that if

$x$ is uniformly continuous on the set of pure states of

$A$ killing

$L$ , or its weak* closure, then

$x$ has a continuous atomic part modulo

$L^{**}$ in an appropriate sense.

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