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.

Keywords: C*-algebra, continuous fields of Hilbert spaces, left quotients, affine property.

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