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arXiv · 1705.07123

Projective quantum modules and projective ideals of C*-algebras

A. Ya. Helemskii
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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.

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A. Ya. Helemskii. 2017-05-19. Projective quantum modules and projective ideals of C*-algebras. https://arxiv.org/abs/1705.07123

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