On the $\text{OUMD}$ property for the column Hilbert space $C$

TL;DR

This paper investigates whether the column Hilbert space possesses the operator space OUMD property, linking it to the study of OUMD_q for non-commutative L_p spaces using complex interpolation techniques.

Contribution

It establishes a connection between the OUMD property of the column Hilbert space and the OUMD_q property for non-commutative L_p spaces, advancing understanding of operator space properties.

Findings

Relates the OUMD property of the column Hilbert space to OUMD_q for non-commutative L_p spaces.

Uses complex interpolation techniques to analyze the problem.

Provides a framework for future research on operator space properties.

Abstract

The operator space $\text{OUMD}$ property was introduced by Pisier in the context of vector-valued noncommutative $L_p$-spaces. It is an open problem whether the column Hilbert space has this property. Based on some complex interpolation techniques, we are able to relate this problem to the study of the $\text{OUMD}_q$ property for non-commutative $L_p$-spaces.