It is well known that for any unital C*-algebra $A$, $K_0(A)\cong \pi_0(\text{Fred}(\mathcal{H}_A))$ (the non-commutative Atiyah-Jänich theorem). In the monograph "Lectures on Operator Theory", it's stated that $K_1(A)$ is isomorphic to the group of path components of the space of self-adjoint Fredholm operators on the standard Hilbert $A$-module. Does anyone know where to find a proof of it?
2026-02-23 03:01:43.1771815703
Non commutative Atiyah-Singer theorem
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