A twisted inclusion between tensor products of operator spaces

TL;DR

The paper investigates a specific tensor product map between operator spaces, revealing a contractive extension that links projective and minimal tensor products, with implications for constructing antisymmetric cocycles.

Contribution

It introduces a new contractive map between tensor products of operator spaces involving opposite structures, expanding understanding of their interactions.

Findings

The identity map extends to a contractive linear map between tensor products.

This extension connects projective and minimal tensor products in operator space theory.

Potential applications include constructing antisymmetric 2-cocycles on Fourier algebras.

Abstract

Given operator spaces $V$ and $W$, let $\widetilde{W}$ denote the opposite operator space structure on the same underlying Banach space. Although the identity map $W\to \widetilde{W}$ is in general not completely bounded, we show that the identity map on $V\otimes W$ extends to a contractive linear map $V\widehat{\otimes} W \to V\otimes_{\min} \widetilde{W}$, where $\widehat{\otimes}$ and $\otimes_{\rm min}$ denote the projective and injective tensor products of operator spaces. In future work, this will be applied to construct antisymmetric $2$-cocycles on certain Fourier algebras.