Local completeness of operator algebras

Abstract

A normed ∗ \ast -algebra A \mathcal {A} is called a local C ∗ {C^\ast } -algebra, if all its maximal commutative ∗ \ast -subalgebras are C ∗ {C^\ast } -algebras. It is shown that any local C ∗ {C^\ast } -algebra dense in K ( H ) \mathcal {K}(\mathcal {H}) , the algebra of compact operators on the Hilbert space H \mathcal {H} equals K ( H ) \mathcal {K}(\mathcal {H}) . The same result holds also for local C ∗ {C^\ast } -algebras dense in A W ∗ A{W^\ast } -algebras without a II 1 {\text {II}_1} summand.