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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.