Representation of certain homogeneous Hilbertian operator spaces and applications

TL;DR

This paper characterizes homogeneous Hilbertian operator spaces within a specific class, providing explicit formulas for their fundamental constants and maps, thus advancing the understanding of their structure and applications.

Contribution

It introduces a canonical representation for homogeneous spaces in QS(C⊕R) based on fundamental sequences, enabling explicit computation of key operator space constants.

Findings

Representation theorem for homogeneous spaces in QS(C⊕R)

Explicit formulas for exactness and projection constants

Characterization of completely 1-summing maps between spaces

Abstract

Following Grothendieck's characterization of Hilbert spaces we consider operator spaces $F$ such that both $F$ and $F^*$ completely embed into the dual of a C*-algebra. Due to Haagerup/Musat's improved version of Pisier/Shlyakhtenko's Grothendieck inequality for operator spaces, these spaces are quotients of subspaces of the direct sum $C\oplus R$ of the column and row spaces (the corresponding class being denoted by $QS(C\oplus R)$). We first prove a representation theorem for homogeneous $F\in QS(C\oplus R)$ starting from the fundamental sequences defined by column and row norms of unit vectors. Under a mild regularity assumption on these sequences we show that they completely determine the operator space structure of $F$ and find a canonical representation of this important class of homogeneous Hilbertian operator spaces in terms of weighted row and column spaces. This canonical representation allows us to get an explicit formula for the exactness constant of an $n$-dimensional subspace $F_n$ of $F$ involving the fundamental sequences. Similarly, we have formulas for the the projection (=injectivity) constant of $F_n$. They also permit us to determine the completely 1-summing maps in Effros and Ruan's sense between two homogeneous spaces $E$ and $F$ in $QS(C\oplus R)$.