Spectral theorem for a commutative operator algebra
= Spectral theorem for a commutative operator algebra
If $A$ is a commutative <C-star algebra> of bounded operators on a <Hilbert space> containing its identity, there is a unique regular <projection-valued measure> $E$ on the <character space> $\Phi_A$ such that $E(\Phi_A)=I$ and $a=\int_{\Phi_A}\widehat a\,dE$ for every $a\in A$, where $\widehat a$ is the <Gelfand transform>.