Example 4

Let X be a locally compact space and let $A =\mathscr{C}_b(X)$ be the commutative unital star algebra of continuous bounded functions on X (Example 2 of I, p. 102). Let $g\in A$; its spectrum S is the closure in $\mathbf{C}$ of the set $g(X)$ of values of $g$ (Example 3 of I, p. 17). The functional calculus mapping of $g$ is then the mapping $f\mapsto f\circ g$, for $f\in \mathscr{C}(S)$. In fact, this mapping is a unital morphism of star algebras such that the identical mapping on S has image $g$.

In the case where X is compact, one has $A =\mathscr{C}(X)$ and $S =g(X)$.

Let $\pi : A\rightarrow A'$ be a unital morphism of unital star algebras. The element $\pi (x)$ of $A'$ is normal and its spectrum relative to $A'$ is contained in Sp$_A(x)$. One then has: