Decompositions for real Banach spaces with small spaces of operators

Manuel González, and José M. Herrera. "Decompositions for real Banach spaces with small spaces of operators." Studia Mathematica 183.1 (2007): 1-14. <http://eudml.org/doc/284505>.

@article{ManuelGonzález2007,
abstract = {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 $X_\{i\}$ for which $(X_\{i\})/ℐn(X_\{i\})$ 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 $X_\{i\}$ can be divided into subsets in such a way that if $X_\{i\}$ and $X_\{j\}$ are in different subsets, then $(X_\{i\},X_\{j\}) = ℐn(X_\{i\},X_\{j\})$; and if they are in the same subset, then $X_\{i\}$ and $X_\{j\}$ 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.},
author = {Manuel González, José M. Herrera},
journal = {Studia Mathematica},
keywords = {real hereditarily indecomposable Banach spaces; Fredholm ideals; complexification},
language = {eng},
number = {1},
pages = {1-14},
title = {Decompositions for real Banach spaces with small spaces of operators},
url = {http://eudml.org/doc/284505},
volume = {183},
year = {2007},
}

TY - JOUR
AU - Manuel González
AU - José M. Herrera
TI - Decompositions for real Banach spaces with small spaces of operators
JO - Studia Mathematica
PY - 2007
VL - 183
IS - 1
SP - 1
EP - 14
AB - 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 $X_{i}$ for which $(X_{i})/ℐn(X_{i})$ 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 $X_{i}$ can be divided into subsets in such a way that if $X_{i}$ and $X_{j}$ are in different subsets, then $(X_{i},X_{j}) = ℐn(X_{i},X_{j})$; and if they are in the same subset, then $X_{i}$ and $X_{j}$ 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.
LA - eng
KW - real hereditarily indecomposable Banach spaces; Fredholm ideals; complexification
UR - http://eudml.org/doc/284505
ER -