For any normal element $a \in A$ for a $C^*$-algebra $A$, we can easily define a continuous functional calculus using $\varphi:C(\sigma(a))\to C^*[1,a]$. We can also define a spectral measure using $a = \int_{\sigma(a)} \lambda \ dE(\lambda)$ by considering $C^*[1,a] \subset B(H)$ where $H$ is the universal representation of $A$. Most of the books I've sorted through only really care about the the spectral measure for normal elements in $B(H)$, but I'm wondering how this matches back up with $A$?
In particular, let's call $\overline A$ the weak closure of $A \subset B(H)$. Let's also explicitly call $E_a$ the spectral measure associated to $a \in A$. It's clear that for $f \in C_0(\mathbb C)$ we have that $\int_{\sigma(a)} f(\lambda) \ dE_a(\lambda) \in A$ for all normal $a \in A$.
Is it the case that for borel measurable $f$ that $\int_{\sigma(a)} f(\lambda) \ dE_a(\lambda) \in \overline{C^*[1,a]}$ for all normal $a \in A$? Moreover when is it the case that for all borel measurable $f$ and normal $a \in A$
that $\int_{\sigma(a)} f(\lambda) \ dE_a(\lambda) \in A$? From this answer, Is a von Neumann algebra just a C*-algebra which is generated by its projections? I'm guessing that this is not strong enough to characterize von neumann algebras, but what sorts of spaces is this true for? I think this last condition should be equivalent to $E_a(S) = \chi_S (a) \in A$ for all normal $a$, but I'm not sure. If the first question is true, then in this case we would have $\overline{C^*[1,a]} \subset A$ for all normal $a \in A$, but I don't know what sort of algebra that is!
Addendum: For the first question it seems like one could use the fact that borel sets appear at stage $\omega_1$ in the hierarchy? And so taking all $\omega_1$ length nets of functions it should be enough that $\langle \varphi(f_n)x,y \rangle \to \langle \varphi(f)x,y \rangle$. I'm not so sure about this $\omega_1$ net argument though.
Best Answer
Regarding your second paragraph, you can only have a meaningful spectral measure for normal elements.
The integral over a spectral measure is a norm limit, so it will stay in any C$^*$-algebra that contains the values of $E_a$. One can also show, via the Double Commutant Theorem, that any von Neumann algebra that contains the normal element $a$, also contains the range of the spectral measure $E_a$. So if $a\in A$, with $a$ normal and $A$ a von Neumann algebra, then $\int_{\sigma(a)}f(\lambda)\,dE_a(\lambda)\in A$ for all bounded Borel functions $f$.
The algebra you denote by $\overline{C^*(1,a)}$ is the von Neumann subalgebra generated by $a$, and the above applies. Said algebra contains, sometimes strictly the algebra $$ \Big\{\int_{\sigma(a)}f(\lambda)\,dE_a(\lambda):\ f\ \text{bounded Borel}\Big\}. $$