Let $A$ be a commutative complex Banach algebra, and let $\Delta(A)$ denote the character space of $A$, equipped with the weak-* topology inherited from $A^*$. Define the Gelfand transform
for every $a\in A$ and every $\tau\in\Delta(A)$, where $C_b(\Delta(A))$ is equipped with pointwise algebra operations and the supremum norm. Then $\Gamma$ is a contractive algebra homomorphism: