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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. Akad. Nauk SSSR Ser. Mat., 1989 Volume 53, Issue 2, Pages 227–242 (Mi im1238)

Diameters of state spaces of Jordan Banach algebras

Sh. A. Ayupov, Sh. M. Usmanov


Abstract: The notion of diameter $D(A)$ of the state space of a Jordan Banach algebra ($JBW$-algebra $A$) is introduced. The diameters of the state spaces for $JBW$-factors of type $\mathrm I_n$ ($n<+\infty$), $\mathrm I_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, $\mathrm{III}_\lambda$ ($0<\lambda<1$) are computed.
It is proved that if $A$ is not a factor, or is a factor of type $\mathrm I_\infty$ or $\mathrm{II}_1$, then $D(A)=2$. If $A$ is a $JBW$-factor of type $\mathrm I_n$ ($n<+\infty$), then $D(A)=2(1-1/n)$, and if $A$ is a $JBW$-factor of type $\mathrm{III}_\lambda$ ($0<\lambda<1$), then $D(A)=2(1-\sqrt\lambda)/(1+\sqrt\lambda)$ or $D(A)=2(1-\sqrt[4]\lambda)/(1+\sqrt[4]\lambda)$.
Bibliography: 15 titles.

UDC: 517.986

MSC: Primary 46L30; Secondary 46L35

Received: 16.06.1987


 English version:
Mathematics of the USSR-Izvestiya, 1990, 34:2, 229–244

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