Projective Hilbert Modules and Sequential Approximation

We show that, when $A$ is a separable C*-algebra, every countably generated Hilbert $A$-module is projective (with bounded module maps as morphisms). We also study the approximate extensions of bounded module maps. In the case that $A$ is a $σ$-unital simple C*-algebra with strict comparison and every strictly positive lower semicontinuous affine function on quasitraces can be realized as the rank of an element in Cuntz semigroup, we show that the Cuntz semigroup is the same as unitarily equivalent class of countably generated Hilbert $A$-modules if and only if $A$ has stable rank one.

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