Given a Hilbert space $\mathcal{H}$.
Consider normals: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$
Denote their algebra: $$\mathcal{N}^\infty(N):=\{\eta(N):\|\eta\|_\infty<\infty\}\subseteq\mathcal{B}(\mathcal{H})$$
Denote for shorthand: $$\mathcal{N}(\mathcal{H}):=\{N:N^*N=NN^*\}$$
Regard a C*algebra: $$\mathcal{A}\subseteq\mathcal{B}(\mathcal{H}):\quad A'A=AA'\quad(A,A'\in\mathcal{A})$$
Then it admits one: $$N_0\in\mathcal{N}(\mathcal{H}):\quad\mathcal{A}\subseteq\mathcal{N}^\infty(N_0)$$
Can I always find such?
