We consider real Banach spaces X for which the quotient algebra (X)/ℐn(X) is finite-dimensional, where ℐn(X) stands for the ideal of inessential operators on X. We show that these spaces admit a decomposition as a finite direct sum of indecomposable subspaces for which is isomorphic as a real algebra to either the real numbers ℝ, the complex numbers ℂ, or the quaternion numbers ℍ. Moreover, the set of subspaces can be divided into subsets in such a way that if and are in different subsets, then ; and if they are in the same subset, then and are isomorphic, up to a finite-dimensional subspace. Moreover, denoting by X̂ the complexification of X, we show that (X)/ℐn(X) and (X̂)/ℐn(X̂) have the same dimension.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-1-1,
author = {Manuel Gonz\'alez and Jos\'e M. Herrera},
title = {Decompositions for real Banach spaces with small spaces of operators},
journal = {Studia Mathematica},
volume = {178},
year = {2007},
pages = {1-14},
zbl = {1134.47010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-1-1}
}
Manuel González; José M. Herrera. Decompositions for real Banach spaces with small spaces of operators. Studia Mathematica, Tome 178 (2007) pp. 1-14. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm183-1-1/