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

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

2011-07-25

arxiv.org/abs/1107.4941v4

Mathematics

Functional Analysis

Some generalisation of main results in the original paper was obtained in this new one. 17pages

Scientific paper

The operator space $\text{OUMD}$ property was introduced by Pisier in the context of vector-valued noncommutative $L_p$-spaces. In this paper, we prove that the column Hilbert space $C$ is $\text{OUMD}_p$ for all $1 < p < \infty$. This answers positively a question asked by Zhong-Jin Ruan. We also prove that some iterated noncommutative $L_p$-spaces, e.g. $S_{p_1}[S_{p_2}[... [S_{p_n}] ...]]$ are $\text{OUMD}_p$, for all $1 < p, p_1, p_2, ..., p_n < \infty$. This gives a full generalisation of Musat's result in \cite{Musat1}. It is well known that the Banach space $\text{UMD}$ property is independent of $p$, but it is still unknown whether the operator space $\text{OUMD}$ property is independent of $p$.

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