# Structures on the way from classical to quantum spaces and their tensor products

## TL;DR

This paper explores tensor products of novel mathematical structures called Lambert and proto-Lambert spaces, revealing their unique properties and applications, especially in relation to $L_1$-spaces and Hilbert spaces.

## Contribution

It introduces and analyzes tensor products for Lambert and proto-Lambert spaces, bridging classical and quantum space theories with new norm properties.

## Key findings

- Proto-Lambert tensor product is well-behaved for spaces with maximal proto-Lambert norm.
- Lambert tensor product is suitable for Hilbert spaces with minimal Lambert norm.
- Different tensor products induce distinct norms, highlighting structural differences.

## Abstract

We study tensor products of two structures situated, in a sense, between normed spaces and (abstract) operator spaces. We call them Lambert and proto-Lambert spaces and pay more attention to the latter ones. The considered two tensor products lead to essentially different norms in the respective spaces. Moreover, the proto-Lambert tensor product is especially nice for spaces with the maximal proto-Lambert norm and in particular, for $L_1$-spaces. At the same time the Lambert tensor product is nice for Hilbert spaces with the minimal Lambert norm.