Return to Revisions

There is a difference between algebraically finitely generated and topological finitely generated Hilbert C-modules. Consider C_0((0,1]) - the C-algebra of all continuous functions on the left-open unit interval with the usual metric topology. It is a Hilbert C([0,1])-module as a maximal ideal of that C-algebra. By the Theorem of Weierstrass the finite polynomials in one variable are norm-dense in C([0,1]), in a modular sense the generating set {x(t)=t} suffices to generated C_0((0,1]) as a C([0,1])-module topologically. However, C_0((0,1]) is not projective and not algebraically finitely generated. Kasparov subsummized this kind of Hilbert C-modules among countably generated ones.