Matricial Banach spaces

TL;DR

This paper explores the structure and properties of matricial Banach spaces, extending classical Banach space constructions to the matrix-normed setting and introducing a Haagerup tensor algebra with a universal property.

Contribution

It characterizes the maximal matrix-norm on Banach spaces as dual operator spaces and constructs a Haagerup tensor algebra with a universal property, extending the theory of matrix-normed spaces.

Findings

Many Banach space constructions extend naturally to matricial Banach spaces

The maximal matrix-norm is characterized as a dual operator space

A Haagerup tensor algebra with a universal property is introduced

Abstract

This work performs a study of the category of complete matrix-normed spaces, called matricial Banach spaces. Many of the usual constructions of Banach spaces extend in a natural way to matricial Banach spaces, including products, direct sums, and completions. Also, while the minimal matrix-norm on a Banach space is well-known, this work characterizes the maximal matrix-norm on a Banach space from the work of Effros and Ruan as a dual operator space. Moreover, building from the work of Blecher, Ruan, and Sinclair, the Haagerup tensor product is merged with the direct sum to form a Haagerup tensor algebra, which shares the analogous universal property of the Banach tensor algebra from the work of Leptin.