The left quotient of a C('*)-algebra and its representation through a continuous field of Hilbert spaces

Ngai-ching Wong, Purdue University

Abstract

In the late 50's M. Tomita began the discussion of the structure of the left quotient A/L of a C$\sp*$-algebra A by a closed left ideal L of A. In this dissertation, elements of A/L (resp. $A\sp{**}$/$L\sp{**}$) are represented as continuous admissible vector sections (resp. admissible vector sections) of a continuous field of Hilbert spaces over a closed face F of the quasi-state space of A. In this context we study the left regular representation of A (resp. $A\sp{**})$ on the Banach space A/L (resp. $A\sp{**}$/$L\sp{**})$. Elements of A and $A\sp{**}$ are represented as admissible operator fields over F. Problems related to various relative multipliers and operator topologies in B(A/L) are discussed, variants of classical density theorems are given and basic examples are presented. When the ideal L is zero our theory resembles and generalizes the Takesaki-Bichteler duality theorem and a similar construction of Akemann and Shultz.

Degree

Ph.D.

Advisors

Brown, Purdue University.

Subject Area

Mathematics

COinS