K-theory for algebras of operators on Banach spaces

TL;DR

This paper constructs Banach spaces with prescribed K-theory groups for their operator algebras and computes K-groups of specific ideals, advancing understanding of operator algebra invariants.

Contribution

It demonstrates the existence of Banach spaces with specified K_0 and K_1 groups for their operator algebras, and computes K-groups of certain operator ideals.

Findings

Existence of Banach spaces with prescribed K-theory groups

Computed K-groups of all closed ideals within strictly singular operators

Results on splittings of certain short exact sequences

Abstract

It is proved that, for each pair (m,n) of non-negative integers, there is a Banach space X for which the group K_0(B(X)) is isomorphic to m copies of the integers and the group K_1(B(X)) is isomorphic to n copies of the integers. Along the way we compute the K-groups of all closed ideals of operators contained in the ideal of strictly singular operators, and we derive some results about the existence of splittings of certain short exact sequences.