Extreme points of the unit ball of an operator space

Details Extreme points of the unit ball of an operator space Extreme points of the unit ball of an operator space

2004-08-18

arxiv.org/abs/math/0408235v2

Mathematics

Operator Algebras

27 pages, http://www.math.uci.edu/~mkaneda/

Scientific paper

This paper is a revision and an enlargement of the previous version titled "Extreme points of the unit ball of a quasi-multiplier space" which had been circulated since 2004. We study extreme points of the unit ball of an operator space by introducing the new notion (approximate) "quasi-identities". Then we characterize an operator algebra with a contractive approximate quasi- (respectively, left, right, two-sided) identity in terms of quasi-multipliers and extreme points. Furthermore, we give a very neat necessary and sufficient condition for a given operator space to become a $C^*$-algebra or a one-sided ideal in a $C^*$-algebra in terms of quasi-multipliers. An extreme point is also used to show that any TRO with predual can be decomposed to the direct sum of a two-sided ideal, a left ideal, and a right ideal in some von Neumann algebra.

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