Decompositions for real Banach spaces with small spaces of operators
Volume 183 / 2007
Abstract
We consider real Banach spaces $X$ for which the quotient algebra $\mathcal{L}(X)/\mathcal{I}n(X)$ is finite-dimensional, where $\mathcal{I}n(X)$ 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 $\mathcal{L}(X_i)/\mathcal{I}n(X_i)$ is isomorphic as a real algebra to either the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, or the quaternion numbers $\mathbb{H}$. 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 $\mathcal{L}(X_i,X_j)=\mathcal{I}n(X_i,X_j)$; 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 $\widehat X$ the complexification of $X$, we show that $\mathcal{L}(X)/\mathcal{I}n(X)$ and $\mathcal{L}(\widehat X)/\mathcal{I}n(\widehat X)$ have the same dimension.
