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Journal of Operator Theory

Volume 76, Issue 2, Fall 2016  pp. 219-248.

Quasi-multipliers and algebrizations of an operator space. II. Extreme points and quasi-identities

Authors:  Masayoshi Kaneda
Author institution:Department of Mathematics and Natural Sciences, College of Arts and Sciences, American University of Kuwait, P.O. Box 3323, Safat 13034, Kuwait

Summary: We study extreme points of the unit ball of an operator space by introducing the new notion ``(approximate) quasi-identities''. More specifically, we characterize an operator algebra having a contractive (approximate) quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points of the unit ball (of the weak$^*$-closure) of the underlying operator space. Furthermore, we give a necessary and sufficient condition for a given operator space to be qualified to become a $C^*$-algebra or a one-sided ideal in a $C^*$-algebra in terms of quasi-multipliers.

Keywords: Quasi-multipliers, extreme points, quasi-identities, abstract operator algebras, approximate identities, injective operator spaces, dual operator spaces, $C^*$-algebras, ideals, ternary rings of operators

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