Linearizing holomorphic functions on operator spaces

TL;DR

This paper develops a framework for completely bounded holomorphic functions on operator spaces, establishing their operator space structure and identifying a noncommutative predual with key properties, advancing the understanding of noncommutative function theory.

Contribution

It introduces a new notion of completely bounded holomorphic functions on operator spaces and constructs their operator space predual with properties analogous to classical cases.

Findings

Identified an operator space predual for scalar-valued completely bounded holomorphic functions.

Established a linearization property for vector-valued functions.

Showed transfer of approximation properties between the predual and the domain.

Abstract

We introduce a notion of completely bounded holomorphic functions defined on the open unit ball of an operator space. We endow the set of these functions with an operator space structure, and in the scalar-valued case we identify an operator space predual for it which is a noncommutative version of Mujica's predual for the space of bounded holomorphic functions and satisfies similar properties. In particular, our predual is a free holomorphic operator space in the sense that it satisfies a linearization property for vector-valued completely bounded holomorphic functions. Additionally, several different operator space approximation properties transfer between the predual and the domain.