On linear isometries of Cartan factors in infinite dimensions (Q1085405)

The open unit ball B(\({\mathfrak A})\) of any \(J^*\)-algebra \({\mathfrak A}\) is a bounded circular homogeneous domain. In order to know the group of its analytic automorphisms, Aut B(\({\mathfrak A})\), it therefore suffices to know the group of surjective linear isometries of \({\mathfrak A}\). In infinite dimensions, \textit{T. Franzoni} [ibid. 127, 51-66 (1981; Zbl 0483.46051)] has solved the problem when \({\mathfrak A}\) is a Cartan factor of type I. In the present article the solution when \({\mathfrak A}\) is a Cartan factor of type II or III is presented. The results obtained can in fact be extended to a wider class of cases including that in which \({\mathfrak A}={\mathcal L}_ c(H,K)\) is the space of compact operators from H to K, which has been solved by \textit{J. Arazy} [Isr. J. Math. 22(1975), 247-256 (1976; Zbl 0319.47020)] when H and K are separable Hilbert spaces.