7.2 Abstract Index

Definition 7.2.1 (Index Group and Abstract Index).label Let $A$ be a unital Banach algebra. The principal component $G_{0}(A)$ of $G(A)$ is the connected component containing the identity which is a clopen normal subset of $G(A)$. The cosets of $G_{0}(A)$ are the connected components of $G(A)$. The quotient group $I(A)\define G(A)/G_{0}(A)$ is a discrete group called the index group of $A$ and the quotient mapping $\pi:G(A)\to I(A)$ is the abstract index of $A$ which is an invariant of Banach algebras in the following sense:

• Any homomorphism $\Phi:A\to B$ between unital Banach algebras induces a group homomorphism from $I(A)$ to $I(B)$ which is an isomorphism if $A\iso B$.