Index map

Summary

The index map is the connecting homomorphism $\partial \colon K_1(B) \to K_0(I)$ attached to a short exact sequence $0 \to I \to A \to B \to 0$ of C*-algebras. It measures the obstruction to lifting a unitary of $B$ to a unitary of $A$. For the extension of the compact operators by the bounded operators, it is the Fredholm index of an operator. It is one of the two connecting maps of the six-term exact sequence in operator K-theory.