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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