Left quotients of a C~*-algebra, III: Operators on left quotients

摘要

Let L be a norm closed left ideal of a C~*-algebra A. Then the left quotient A/Lis a left A-module. In this paper, we shall implement Tomita'sidea about representing elements of A as left multiplications: π_p(a)(b + L) = ab + L. A complete characterization of bounded endomorphisms of the A-module A/L is given. The double commutant π_p(A)" of π_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.