# Type, cotype and twisted sums induced by complex interpolation

## TL;DR

This paper explores how complex interpolation induces twisted sums of Banach spaces, analyzing the relationship between type, cotype, and the properties of these extensions, with applications to Schatten classes.

## Contribution

It introduces new results linking type and cotype of interpolated spaces to the triviality or singularity of induced extensions, especially in Schatten classes.

## Key findings

- Extensions induced by complex interpolation can be nontrivial and singular.
- The type and cotype of spaces influence the triviality of the induced extensions.
- Nontrivial extensions of spaces without the CAP are constructed.

## Abstract

This paper deals with extensions or twisted sums of Banach spaces that come induced by complex interpolation and the relation between the type and cotype of the spaces in the interpolation scale and the nontriviality and singularity of the induced extension. The results are presented in the context of interpolation of families of Banach spaces, and are applied to the study of submodules of Schatten classes. We also obtain nontrivial extensions of spaces without the CAP which also fail the CAP.