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Facial structure in operator algebra theory. (English) Zbl 0759.46050

A closed non-empty subset \(F\) of a convex set \(C\) is called a face if \(F\) is convex, and if \(\lambda x+(1-\lambda)y\) in \(F\) for \(0<\lambda<1\) implies that \(x\) and \(y\) in \(F\) for all elements \(x\), \(y\) in \(C\). The authors give a complete description of the closed faces of convex sets that appear in operator algebra theory. These consist of positive parts of unit balls, of selfadjoint parts of unit balls and of general unit balls for \(C^*\)-algebras and for their dual spaces and for von Neumann algebras and their preduals [though stated in the authors’ unified methods, results for von Neumann algebras and their preduals are known by C. M. Edwards and G. T. Rüttimann, Math. Proc. Cambridge Philos. Soc. 98, 305-322 (1985; Zbl 0577.46007); J. London Math. Soc., II. Ser. 38, No. 2, 317-332 (1988; Zbl 0621.46043)]. The results have intrinsic forms for faces in algebras such as Theorem 3.10 and 4.10, and have forms of exposed faces in case for duals and preduals such as Theorem 2.10 and 3.6.
Defining facear and pre-facear (polar operation) for corresponding pairs of convex sets in their relevant operator spaces, the authors give a duality theory for faces in §5 proving the density theorem 5.6 (bipolar theorem). The final §6 is devoted to the discussion of facial structure in ideals and quotients.
Reviewer: J.Tomiyama (Tokyo)

MSC:

46L05 General theory of \(C^*\)-algebras
46L10 General theory of von Neumann algebras
46L55 Noncommutative dynamical systems
46L30 States of selfadjoint operator algebras
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