38.8 $L^{\infty}$
Proposition 38.8.1.label Let $X$ be a compact Hausdorff space and $\mathscr{M}\subset M_{R}(X; \complex)$ be a closed subspace such that:
- (A)
For each $\mu \in \mathscr{M}$ and $\nu \in M_{R}(X; \complex)$ with $\nu \ll \mu$, $\nu \in \mathscr{M}$.
and
then
- (1)
$J(C(X; \complex))$ is weak*-dense in $\mathscr{M}^{*}$.
- (2)
There exists a unique weak*-continuous involution on $\mathscr{M}^{*}$ such that $J(f^{*}) = J(f)^{*}$ for all $f \in C(X; \complex)$, given by
\[\dpn{\mu, \phi^*}{C(X; \complex)^*}= \ol{\dpn{\mu, \phi}{C(X; \complex)^*}}\] - (3)
There exists a unique seperately weak*-continuous bilinear map on $\mathscr{M}^{*}$ such that $J(fg) = J(f)J(g)$ for all $f, g \in C(X; \complex)$.
- (4)
$\mathscr{M}^{*}$ equipped with the above involution and product is a commutative unital $C^{*}$-algebra.
Proof. 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
(1): Under the above, $C(X; \complex)$ may be identified as the diagonal
By Goldstine’s Theorem, $C(X; \complex)$ is weak*-dense in $C(X; \complex)^{**}$. As a result, $J(C(X; \complex))$ is weak*-dense in $\mathscr{M}^{*}$.
(2): 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.
(3): 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.
(4): $[l^{\infty}(I); L^{\infty}(\mu_{i}; \complex)]$ is a commutative unital $C^{*}$-algebra.$\square$
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