Proposition 24.7.4.label Let $X$ be a compact Hausdorff space, $J: C(X; \complex) \to C(X; \complex)^{**}$ be the natural embedding, and $B$ be the closed unit ball of $C(X; \complex)^{**}$, then
- (1)
There exists a unique weak*-continuous involution $C(X; \complex)^{**}$ such that $J(f^{*}) = J(f)^{*}$ for all $f \in C(X; \complex)$, given by
\[C(X; \complex)^{**}\to C(X; \complex)^{**}\quad \dpn{\mu, \phi^*}{C(X; \complex)^*}= \ol{\dpn{\mu, \phi}{C(X; \complex)^*}}\] - (2)
There exists a unique seperately weak*-continuous bilinear map on $C(X; \complex)^{**}$ such that $J(fg) = J(f)J(g)$ for all $f, g \in C(X; \complex)$.
- (3)
$C(X; \complex)^{**}$ equipped with the above involution and product is a commutative unital $C^{*}$-algebra.
Proof. It is sufficient to construct the maps in (1) and (2). Let $\seqi{\mu}$ be a maximal mutually singular family of Radon measures on $X$. Using Theorem 24.7.2 and the Riesz Representation Theorem, identify
Under the above, $C(X; \complex)$ may be identified as the diagonal
which is weak*-dense in $[l^{\infty}(I); L^{\infty}(\mu_{i}; \complex)]$ by Goldstine’s Theorem.
(1): For each $g \in [l^{\infty}(I); L^{\infty}(\mu_{i}; \complex)]$, let $g^{*} = \ol g$. For any $\mu \in [l^{1}(I); L^{1}(\mu_{i}; \complex)]$,
so the conjugation map is weak*-continuous.
(2): Let $f, g \in [l^{\infty}(I); L^{\infty}(\mu_{i}; \complex)]$ and $\mu \in [l^{1}(I); L^{1}(\mu_{i}; \complex)]$,
so the composition map is separately weak*-continuous.
(3): $[l^{\infty}(I); L^{\infty}(\mu_{i}; \complex)]$ is a commutative unital $C^{*}$-algebra.$\square$
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