Extreme points of convex fully symmetric sets of measurable operators. (English) Zbl 0787.46051
Let \(M\) be a semifinite non-atomic von Neumann algebra with a faithful semifinite normal trace \(\tau\). A subset \(W\) of \(L^ 1(M,\tau)+ M\) is called fully symmetric if for any \(x\in W\) there is a function \(f\) in \(L^ 1([0;\tau(1)[)+ L^ \infty([0;\tau(1)[)\) with the same non- increasing rearrangement. Let \(\widetilde W\) be the set of all functions that correspond to elements of \(W\) in the way described above. If \(W\) is convex, then \(x\) is an extreme point of \(W\) if and only if its rearrangement \(\mu(x)\) is an extreme point of \(\widetilde W\), and either \(\mu_ \infty(x)=0\) or the orthogonal complements of the left and right supports of \(x\) are centrally orthogonal and \(| x|\geq \mu_ \infty(x)r(x)\). This result can be applied to convex subsets of non- commutative Banach function spaces [see K. Watanabe, Hokkaido Math. J. 22, 349-364 (1993)].
Reviewer: S.Goldstein (Łódź)
MSC:
| 46L51 | Noncommutative measure and integration |
| 46L53 | Noncommutative probability and statistics |
| 46L54 | Free probability and free operator algebras |
Keywords:
measurable operator; fully symmetric set; semifinite non-atomic von Neumann algebra with a faithful semifinite normal; fully symmetric; non- increasing rearrangement; extreme pointReferences:
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