×

Direct sums of Cartan factors. (English) Zbl 0807.46076

In [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat. Rend., IX. Ser. 1, No. 3, 203-213 (1990; Zbl 0747.46040)] we envisaged the orbit of the origin in the unit ball of a direct sum of two complex Banach spaces (endowed with a suitable norm), with respect to the group of holomorphic automorphisms, and we obtained some general results. As a special case we considered the class of \(p\)-norms, and we proved that the most interesting case is when \(p\) equals 2. For \(p=2\) we succeeded in giving some information about the orbit of the origin when one of the spaces is either a Hilbert space or a commutative \(C^*\)-algebra with identity. In this paper we consider the case when one of the spaces is a Cartan factor. The reason for considering Cartan factors is that, as we proved in [loc. cit.], only spaces in which the orbit of the origin in the unit ball is non-trivial can give rise to a direct sum in which such an orbit is non-trivial: and the unit ball of a Cartan factor is homogeneous.
Our main result can be expressed in the following way: if \(F\) is a Cartan factor of type I, II, III or IV and \(F\) is not isometric to a Hilbert space, then, given a non-trivial complex Banach space \(G\), no point in the orbit of the origin in the unit ball of the 2-sum of \(G\) and \(F\) can have non-zero \(F\)-coordinate.
In the last section we prove some results concerning duality theory for Cartan factors.

MSC:

46L35 Classifications of \(C^*\)-algebras
46G20 Infinite-dimensional holomorphy

Citations:

Zbl 0747.46040
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] E. Calabi - E. Vesentini , On compact, locally symmetric Kähler manifolds , Ann. Math. , 71 ( 1960 ), pp. 472 - 507 . MR 111058 | Zbl 0100.36002 · Zbl 0100.36002 · doi:10.2307/1969939
[2] L.A. Harris , Bounded symmetric homogeneous domains in infinite dimensional spaces , Lectures Notes in Mathematics, 364 , Springer , Berlin - Heildelberg - New York ( 1973 ), pp. 13 - 40 . MR 407330 | Zbl 0293.46049 · Zbl 0293.46049
[3] C. Petronio , Holomorphic automorphism groups in certain compact operator spaces , Rend. Accad. Naz. Lincei , Serie IX , Vol. I , pp. 125 - 130 . MR 1081395 | Zbl 0738.47031 · Zbl 0738.47031
[4] C. Petronio , Holomorphic automorphisms of the unit ball of a direct sum , Rend. Accad. Naz. Lincei , Serie IX , Vol. I , pp. 203 - 213 . MR 1083249 | Zbl 0747.46040 · Zbl 0747.46040
[5] R. Schatten , A Theory of Cross-Spaces , Princeton University Press ( 1950 ). MR 36935 | Zbl 0041.43502 · Zbl 0041.43502
[6] R. Schatten , Norm ideals of completely continuous operators , Ergebn. der Math. , 27 , Springer-Verlag ( 1960 ). MR 119112 | Zbl 0090.09402 · Zbl 0090.09402
[7] L.L. Stachò , A short proof of the fact that biholomorphic automorphisms of the unit ball in certain LP spaces are linear , Acta Sci. Math. , 41 ( 1979 ), pp. 381 - 383 . MR 555432 | Zbl 0432.58006 · Zbl 0432.58006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.