Diameters of state spaces of Jordan Banach algebras

Sh. A. Ayupov ; Sh. M. Usmanov

Izvestiya. Mathematics , Tome 34 (1990) no. 2, pp. 229-244

Résumé

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 $\mathrm I_n$ ($n+\infty$), $\mathrm I_\infty$, $\mathrm{II}_1$, $\mathrm{II}_\infty$, $\mathrm{III}_\lambda$ ($0\lambda1$) 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 $\mathrm{III}_\lambda$ ($0\lambda1$), 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.