Left quotients of a $C^*$-algebra, III: Operators on left quotients
Volume 218 / 2013
Studia Mathematica 218 (2013), 189-217
MSC: 46L05, 46L45.
DOI: 10.4064/sm218-3-1
Abstract
Let $L$ be a norm closed left ideal of a $C^*$-algebra $A$. Then the left quotient $A/L$ is a left $A$-module. In this paper, we shall implement Tomita's idea about representing elements of $A$ as left multiplications: $\pi _p(a)(b+L)=ab+L$. A complete characterization of bounded endomorphisms of the $A$-module $A/L$ is given. The double commutant $\pi _p(A)''$ of $\pi _p(A)$ in $B(A/L)$ is described. Density theorems of von Neumann and Kaplansky type are obtained. Finally, a comprehensive study of relative multipliers of $A$ is carried out.
