# Polynomials in operator space theory: matrix ordering and algebraic aspects

## TL;DR

This paper extends the $bbbb-theory of operator spaces to matrix ordered spaces and Banach $*$-algebras, introducing cones related to $bbbb for tensor products and analyzing ideal structures.

## Contribution

It generalizes the $bbbb-theory to new algebraic and ordered contexts, including matrix regular operator spaces and operator systems.

## Key findings

- Introduces cones related to $bbbb that respect matricial structures.
- Discusses the ideal structure of $bbbb$-tensor products of $C^*$-algebras.
- Extends the tensor norm concepts to matrix ordered spaces and Banach $*$-algebras.

## Abstract

We extend the $\lambda$-theory of operator spaces given by Defant and Wiesner (2014), that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach $*$-algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to $\lambda$ for the algebraic tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of $\lambda$-tensor product of $C^*$-algebras has also been discussed.