Operators in finite distributive subspace lattices II
In a previous paper we gave an example of a finite distributive subspace lattice ℒ on a Hilbert space and a rank two operator of Algℒ that cannot be written as a finite sum of rank one operators from Algℒ. The lattice ℒ was a specific realization of the free distributive lattice on three generators. In the present paper, which is a sequel to the aforementioned one, we study Algℒ for the general free distributive lattice with three generators (on a normed space). Necessary and sufficient conditions are given for 1) a finite rank operator of Algℒ to be written as a finite sum of rank ones from Algℒ, and 2) a realization of ℒ to contain a finite rank operator of Algℒ with the preceding property. These results are then used to show the curiosity that the product of two finite rank operators of Algℒ always has the above property.