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# Journal of Operator Theory

Volume 32, Issue 1, Summer 1994  pp. 185-201.

Left quotients of a C*-algebra I: Representation via vector sections

Authors Ngai-Ching Wong
Author institution: Department of Applied Mathematics, National Sun Yat-sen University Kao-hsiung, 80424, Taiwan, R.O.C.

Summary:  Let A be a C*-algebra, L a closed left ideal of A and p the closed projection related to L. We show that for an xp in $A**p (\cong A**/L**)$ if $pAxp \subset pAp$ and $px*xp \in pAp$ then $xp \in Ap (\cong A/L)$. The proof goes by interpreting elements of A**p (resp. Ap) as admissible (resp. continuous admissible) vector sections over the base space $F(p) = \{\varphi \in A* : \varphi \ge 0, \varphi (p) = \left\| \varphi \right\| \le 1\}$ in the notions developed by Diximier and Douady, Fell, and Tomita. We consider that our results complement both Kadison function representation and Takesaki duality theorem.

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