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

TL;DR

This paper characterizes Banach spaces whose algebra of bounded operators has a $K_0$-group isomorphic to the integers, identifying conditions and specific examples including $C([0,])$, and explores their algebraic structure.

Contribution

It establishes sufficient conditions for the $K_0$-group of the algebra of bounded operators on a Banach space to be isomorphic to the integers, and identifies several spaces satisfying these conditions.

Findings

$K_0( ext{B}(X)) \\cong \\mathbb{Z}$ for certain Banach spaces

Identifies $C([0,\omega_1])$ as an example with $K_0$-group isomorphic to $\\mathbb{Z}$

Provides conditions on Banach spaces related to complemented subspaces and ideal codimension

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 $\mathscr{B}(X)$ 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 $\mathscr{B}(X)$ 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 $K_0(\mathscr{B}(C([0,\omega_1])))\cong\mathbb{Z}$, 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.