Operator algebras of countable range multiplicity

Abstract

In this paper, the spectrum of operators commuting with operator algebras of countable range multiplicity is studied. It is shown that if the commutant of a set which does not contain any scalar operator has countable range multiplicity then it has a non trivial invariant subspace. If the range multiplicity of an operator algebra is one then it is shown that the strong and uniform topologies coincide on the commutant of the algebra and also each collection of mutually orthogonal projections in the commutant is finite. In addition, if the operator algebra is self adjoint also then it is shown that the underline Hilbert space has a finite orthogonal decomposition such that each of its components reduces the algebra.