# Projective quantum modules and projective ideals of C*-algebras

## TL;DR

This paper introduces the concepts of quantum algebras and modules, defines projectivity and freeness in this context, and proves that all closed left ideals in a separable C*-algebra are projective quantum modules.

## Contribution

It develops the theory of quantum modules over quantum algebras, including projectivity and freeness, and applies it to show that closed left ideals in separable C*-algebras are projective modules.

## Key findings

- All closed left ideals in a separable C*-algebra are projective quantum modules.
- The paper establishes a connection between projectivity and freeness in quantum modules.
- It introduces the notion of quantum algebra and quantum module, expanding the framework of operator modules.

## Abstract

We introduce in non-coordinate presentation the notions of a quantum algebra and of a quantum module over such an algebra. Then we give the definition of a projective quantum module and of a free quantum module, the latter as a particular case of the notion of a free object in a rigged category. (Here we say "quantum" instead of frequently used protean adjective "operator"). After this we discuss the general connection between projectivity and freeness. Then we show that for a Banach quantum algebra A and a Banach quantum space E the Banach quantum A-module $A\widehat\otimes_{op}E$ is free, where " $\widehat\otimes_{op}$ " denotes the operator-projective tensor product of Banach quantum spaces. This is used in the proof of the following theorem: all closed left ideals in a separable C*-algebra, endowed with the standard quantization, are projective left quantum modules over this algebra.