Local completeness of operator algebras

A normed $\ast$-algebra $\mathcal{A}$ is called a local ${C^\ast}$-algebra, if all its maximal commutative $\ast$-subalgebras are ${C^\ast}$-algebras. It is shown that any local ${C^\ast}$-algebra dense in $\mathcal{K}(\mathcal{H})$, the algebra of compact operators on the Hilbert space $\mathcal{H}$ equals $\mathcal{K}(\mathcal{H})$. The same result holds also for local ${C^\ast}$-algebras dense in $A{W^\ast}$-algebras without a ${\text{II}_1}$ summand.