Projective quantum modules and projective ideals of \(C^\ast\)-algebras

DOI10.1007/978-3-319-62527-0_6zbMATH Open1465.46056arXiv1705.07123OpenAlexW2619862134MaRDI QIDQ1728892

Publication date: 26 February 2019

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 w i d e h a t o t i m e s_{o p} E

is free, where "

w i d e h a t o t i m e s_{o p}

" 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.

Full work available at URL: https://arxiv.org/abs/1705.07123

zbMATH Keywords

ideal freeness quantum algebra projective object \(C^\ast\)-algebras quantum module

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Operator spaces and completely bounded maps (46L07) Projective and injective objects in functional analysis (46M10) Ring-theoretic aspects of quantum groups (16T20)