Banach spaces whose algebra of bounded operators has the integers as their $K_0$-group

Abstract

Let $X$ and $Y$ be Banach spaces such that the ideal of operators which factor through $Y$ has codimension one in the Banach algebra $\mathcal{B}(X)$ of all bounded operators on $X$, and suppose that $Y$ contains a complemented subspace which is isomorphic to $Y\oplus Y$ and that $X$ is isomorphic to $X\oplus Z$ for every complemented subspace $Z$ of $Y$. Then the $K_0$-group of $\mathcal{B}(X)$ is isomorphic to the additive group $\mathbb{Z}$ of integers. A number of Banach spaces which satisfy the above conditions are identified. Notably, it follows that $K_0(\mathcal{B}(C([0,{\omega }_1])))\cong \mathbb{Z}$, where $C([0,{\omega }_1])$ denotes the Banach space of scalar-valued, continuous functions defined on the compact Hausdorff space of ordinals not exceeding the first uncountable ordinal ${\omega }_1$, endowed with the order topology.

References in corpus (5)

• Uniqueness of the maximal ideal of the Banach algebra of bounded operators on $C([0,{\omega }_1])$
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• A weak*-topological dichotomy with applications in operator theory
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Cited by in corpus (2)

• A weak*-topological dichotomy with applications in operator theory
• Separable Spaces of Continuous Functions as Calkin Algebras