Definition 29.3.3 (Index Group).label Let $A$ be a unital Banach algebra, then the index group $I(A)$ is the quotient $G(A)/G_{0}(A)$, which is a discrete group.
Proof. By Definition 29.3.1, the components of $G(A)$ are the cosets of $G_{0}(A)$, so the quotient group is discrete.$\square$
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