Projective quantum modules and projective ideals of C*-algebras

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.