Extreme points in duals of complex operator spaces. (English) Zbl 0581.47029
It is proved that for complex Banach spaces X and Y, we have that the extreme points in the unit ball of the dual space of K(X,Y), the compact operators from X to Y, are given by
\[
ext B(K(X,Y)^*)=ext B(X^{**})\otimes ext B(Y^*).
\]
This result was first proved for real Banach spaces by W. M. Ruess and C. P. Stegall, Math. Ann. 261, 535-546 (1982; Zbl 0501.47015). It follows from this characterization that if K(X,Y) contains a proper M-summand, then Y contains a proper M-summand or X contains a proper L-summand.
MSC:
| 47L07 | Convex sets and cones of operators |
| 46B20 | Geometry and structure of normed linear spaces |
| 47B06 | Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators |
Keywords:
extreme points in the unit ball of the dual space of K(X,Y); compact operators; proper M-summand; proper L-summandCitations:
Zbl 0501.47015References:
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