A+ CATEGORY SCIENTIFIC UNIT

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

Volume 218 / 2013

Lawrence G. Brown, Ngai-Ching Wong 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.

Authors

  • Lawrence G. BrownDepartment of Mathematics
    Purdue University
    West Lafayette, IN 47907, U.S.A.
    e-mail
  • Ngai-Ching WongDepartment of Applied Mathematics
    National Sun Yat-sen University
    Kaohsiung, 80424, Taiwan, R.O.C.
    e-mail

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