In classical geometry the calculation of the Chern classes of a vector bundle using a connection is independent of the choice of connection. Does any such result hold for projective modules in noncommutative geometry?
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Yes. You can see the construction in detail, for example, in Max Karoubi's ‘Homologie cyclique et $K$theorie’ (Asterisque 149, SMF; you can get this from his web page), where he constructs the Chern classes $K_0(A)\to H(A)$ using connections much à ChernWeyl (Here $H(A)$ is the noncommutative de Rham theory, or one of the various cyclic homologies of $A$) He also constructs higher Chern classes on the higher algebraic $K$theory by a similar procedure. The book by Loday on cyclic homology also covers this. 

