# Twisting Operator Spaces

## TL;DR

This paper investigates a specific three space problem in operator space theory, demonstrating that certain operator spaces containing a Hilbert space and an OH copy do not necessarily have to be completely isomorphic to OH, through analysis of complex interpolation sequences.

## Contribution

It provides the first negative solutions to the three space problem for operator spaces, using complex interpolation and exact sequence analysis.

## Key findings

- The three space problem has a negative answer.
- Two different counterexamples are constructed.
- Analysis of complex interpolation sequences is key.

## Abstract

In this work we study the following three space problem for operator spaces: if X is an operator space with base space isomorphic to a Hilbert space and X contains a completely isomorphic copy of the operator Hilbert space OH with respective quotient also completely isomorphic to OH, must X be completely isomorphic to OH? This problem leads us to the study of short exact sequences of operator spaces, more specifically those induced by complex interpolation, and their splitting. We show that the answer to the three space problem is negative, giving two different solutions.