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Operator Algebras and Their Modules$
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David P. Blecher and Christian Le Merdy

Print publication date: 2004

Print ISBN-13: 9780198526599

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198526599.001.0001

C*-modules and operator spaces

Chapter:
(p.296) 8 C*-modules and operator spaces
Source:
Operator Algebras and Their Modules
Author(s):

David P. Blecher

Christian Le Merdy

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198526599.003.0008

This chapter has three main goals. First, to examine Hilbert C*-modules (and their W*-algebra variant, W *-modules) as operator modules. It aims to show that the theory of C *-modules fits comfortably into the operator module framework. Second, to consider space X. In particular, it will discuss the noncommutative Shilov boundary ☐(X) of X. TRO methods and this Shilov boundary provide important insights into the structure of X. Third, to illustrate how C*-module and TRO methods can lead to interesting results about operator spaces. Notes and historical remarks are presented at the end of the chapter.

Keywords:   Hilbert C*-modules, noncommutative Shilov boundary, W*-modules, C*-module maps, operator space multipliers, operator modules

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