This is an announcement for the paper "Real hereditarily indecomposable Banach spaces and uniqueness of complex structure" by Valentin Ferenczi.

Abstract: There exists a real hereditarily indecomposable Banach space $X$ such that the quotient space $L(X)/S(X)$ by strictly singular operators is isomorphic to the complex field (resp. to the quaternionic division algebra). Up to isomorphism, the example with complex quotient space has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable; this extends results of J. Bourgain and S. Szarek from 1986. The quaternionic example, on the other hand, has unique complex structure up to isomorphism; there also exists a space with an unconditional basis, non isomorphic to $l_2$, which admits a unique complex structure. These examples answer a question of Szarek.

Archive classification: Functional Analysis

Mathematics Subject Classification: 46B03; 46B04; 47B99

Remarks: 29 pages

The source file(s), cplexstructure_ferenczi.tex: 70811 bytes, is(are) stored in gzipped form as 0511166.gz with size 22kb. The corresponding postcript file has gzipped size 87kb.

Submitted from: ferenczi@ccr.jussieu.fr

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