Subspace structure of some operator and Banach spaces

TL;DR

This paper constructs specific operator and Banach spaces to analyze the complexity of subspace relations, revealing complete isomorphism relations and unique isometry properties.

Contribution

It introduces a family of Hilbertian operator spaces with complex isomorphism relations and explores unique properties of unconditional bases and subspace isometries.

Findings

Complete isomorphism relation is complex (KS-complete) for subspaces of constructed operator spaces.

Existence of Banach space where subspace isometry corresponds to real number equality.

Analysis of unconditional bases in the constructed operator spaces.

Abstract

We construct a family of separable Hilbertian operator spaces, such that the relation of complete isomorphism between the subspaces of each member of this family is complete $\ks$. We also investigate some interesting properties of completely unconditional bases of the spaces from this family. In the Banach space setting, we construct a space for which the relation of isometry of subspaces is equivalent to equality of real numbers.