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• # Artykuł - szczegóły

## Studia Mathematica

2007 | 183 | 1 | 1-14

## Decompositions for real Banach spaces with small spaces of operators

EN

### Abstrakty

EN
We consider real Banach spaces X for which the quotient algebra 𝓛(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 $𝓛(X_{i})/ℐn(X_{i})$ 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 $𝓛(X_{i},X_{j}) = ℐn(X_{i},X_{j})$; 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 𝓛(X)/ℐn(X) and 𝓛(X̂)/ℐn(X̂) have the same dimension.

1-14

wydano
2007

### Twórcy

autor
• Departamento de Matemáticas, Universidad de Cantabria, E-39071 Santander, Spain
autor
• Departamento de Matemáticas, Universidad de Cantabria, E-39071 Santander, Spain