# Projective tensor product of protoquantum spaces

## TL;DR

This paper introduces a new projective tensor product for proto-quantum spaces, exploring its properties, computations for specific cases like L1 spaces, and comparing it with existing tensor products in operator space theory.

## Contribution

It defines a universal projective tensor product for proto-quantum spaces and analyzes its properties, including a proto-quantum analogue of Grothendieck's theorem.

## Key findings

- The tensor product has a universal property for completely bounded bilinear operators.
- Computed the tensor product for L1 spaces, extending classical results.
- Compared the new tensor product with existing operator space tensor products, highlighting differences.

## Abstract

A proto-quantum space is a (general) matricially normed space in the sense of Effros and Ruan presented in a `matrix-free' language. We show that these spaces have a special (projective) tensor product possessing the universal property with respect to completely bounded bilinear operators. We study some general properties of this tensor product (among them a kind of adjoint associativity), and compute it for some tensor factors, notably for $L_1$ spaces. In particular, we obtain what could be called the proto-quantum version of the Grothendieck theorem about classical projective tensor products by $L_1$ spaces. At the end, we compare the new tensor product with the known projective tensor product of operator spaces, and show that the standard construction of the latter is not fit for general proto-quantum spaces.