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⊗opE is free, where “⊗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.
CITATION STYLE
Helemskii, A. Y. (2018). Projective quantum modules and projective ideals of C*-algebras. In Operator Theory: Advances and Applications (Vol. 262, pp. 223–241). Springer International Publishing. https://doi.org/10.1007/978-3-319-62527-0_6
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