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Diameters of state spaces of Jordan Banach algebras. (English. Russian original) Zbl 0716.46049
Math. USSR, Izv. 34, No. 2, 229-244 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 2, 227-242 (1989).
Given a von-Neumann algebra M and a normal state \(\omega\) on M, [\(\omega\) ] stands for the closure of the orbit of \(\omega\) under all inner automorphisms of M. Consider the metric: \[ d([\omega],[\psi])=\inf \{\| \omega '-\psi '\|:\;\omega '\in [\omega],\quad \psi '\in [\psi]\} \] and set \(D(M)=\sup \{d([\omega],[\psi])\}\) for all normal states \(\omega\), \(\psi\) on M. This number is called the diameter of the state space of M. Such diameters were studied by A. Connes, U. Haagerup and E. Störmer [Lect. Notes Math. 1132, 91-116 (1985; Zbl 0566.46028)]; and A. Connes and E. Störmer [J. Funct. Anal. 28, 187-196 (1978; Zbl 0408.46048)]. It was shown that:
(a) \(D(M)=2(1-1/n)\) if M is a factor of type \(I_ n\) \((n<\infty),\)
(b) \(D(M)=2\) if M is not a factor, or M is a factor of type \(I_{\infty}\), \(II_ 1\), or \(II_{\infty},\)
(c) \(D(M)=2(1-\lambda^{1/2})/(1+\lambda^{1/2})\) if M is a \(\sigma\)- finite factor of type \(III_{\lambda}\) (0\(\leq \lambda \leq 1).\)
In the present paper the authors obtain similar results for the diameters of the state spaces of JBW-algebras, which are the real non- associative analogs of von-Neumann algebras. Namely, if A is a JBW- algebra, then (a) and (b) remain true in the same form. The diversity comes in(c). If A is a JBW factor of type \(III_{\lambda}\) \((0<\lambda <1)\), then D(A) can be \(2(1-\lambda^{1/2})/(1+\lambda^{1/2})\) or \(2(1-\lambda^{1/4})/(1+\lambda^{1/4})\) depending on the structure of the enveloping von-Neumann algebra M(A). The cases \(\lambda =0\) and \(\lambda =1\) in (c) remain open.
Reviewer: Kh.N.Boyadzhiev
MSC:
46L70 Nonassociative selfadjoint operator algebras
17C65 Jordan structures on Banach spaces and algebras
46H70 Nonassociative topological algebras
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