Crossed products of operator spaces

TL;DR

This paper introduces the concept of crossed products for operator spaces, generalizing the notion from $C^*$-algebras, but the main result was withdrawn due to a gap in the proof.

Contribution

It defines full and reduced operator space crossed products and relates them to $C^*$-algebra crossed products, extending the theory to operator spaces.

Findings

Defined operator space crossed products for group actions

Established equivalence with $C^*$-algebra crossed products in certain cases

The main theorem was withdrawn due to a proof gap.

Abstract

Let $V$ be an operator space and $\iso(V)$ be the group of all completely isometric bijective linear mappings on $V$. Let $G$ act on $V$ via a strongly continuous group homomorphism $\alpha:G \to \iso (V)$. We define the full (and reduced) operator space crossed product $V\rtimes^{\op}_{\alpha,(r)}G$ and show that for a $C^*$-algebra with its canonical operator space structure, it coincides with the corresponding $C^*$-algebra crossed product. Unfortunately, the proof of Theorem 4.3 of the paper contains a serious gap, which leads to the withdrawal of the paper.