Amalgamated direct sums of operator spaces

TL;DR

This paper investigates the properties of amalgamated direct sums of operator spaces, especially how they behave under different quantizations, providing new examples that answer previously open questions.

Contribution

It introduces the study of amalgamated direct sums of operator spaces under various quantizations and provides counterexamples to prior assumptions.

Findings

Amalgamated direct sums of two $L^{ abla}$-spaces over a common subspace are not necessarily minimal.

The paper demonstrates differences between minimal and maximal quantizations in amalgamated sums.

Answers a question posed by Vern Paulsen regarding the structure of these sums.

Abstract

We consider amalgamated direct sums (and their dual counterparts -- fibre products) of operator spaces and study their behaviour with respect to different quantisations (minimal and maximal). We show examples of amalgamated direct sums of two $L^{\infty}$-spaces over a common subspace that are not minimal themselves, thus answering a question posed by Vern Paulsen.