The equivariant index of $C^*$-elliptic operators

Let $G$ be a compact Lie group, and $A$ a $C^*$-algebra with identity. A $K$-theory of $G$-equivariant $A$-vector bundles is developed along with a corresponding theory of Fredholm operators, and the analytic and topological indices of an elliptic equivariant pseudodifferential operator over a $C^*$-algebra $A$ are defined. An index theorem generalizing the Mishchenko–Fomenko theorem is proved.
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