A Banach space with an infinite dimensional reflexive quotient algebra L(X)/SS(X)

TL;DR

This paper constructs a Banach space where the algebra of bounded operators decomposes into a reflexive part and a strictly singular ideal, revealing new structural insights into operator algebras on Banach spaces.

Contribution

It introduces a Banach space with a unique decomposition of its operator algebra into a reflexive quotient and strictly singular operators, a novel structural example.

Findings

The operator algebra L(X) splits into a reflexive quotient and strictly singular operators.

The constructed space is of Gowers-Maurey type with specific algebraic properties.

Provides a new example of Banach space with a complex operator algebra structure.

Abstract

We construct a Banach space X of Gowers-Maurey type such that the algebra of bounded operators L(X) is a direct sum of an infinite dimensional reflexive Banach space and the operator ideal of strictly singular operators SS(X).