Central operators on complex Banach lattices

Abstract

We show that the centre of a Dedekind complete complex Banach lattice is a commutative $\mathrm{C}^\ast$-algebra in the order unit norm. This implies that the order unit norm and the operator norm coincide. As an application of the latter, a Fuglede--Putnam--Rosenblum-type theorem is established. Under an extra condition on the underlying real Banach lattice, which is satisfied when it is order continuous, a spectral theorem is given for the centre of the complex Banach lattice as a whole and for an individual central operator. The ensuing functional calculus for an individual operator is applied to show that central operators have many spectral properties similar to those of normal operators on complex Hilbert spaces. For example, when the spectrum is countable every element is the sum of an order convergent series of eigenvectors.