Decomposability of Reflexive Cycle Algebras

We give, for each n 3, an example of a reflexive operator algebra_n with the following properties: (i) each finite rank operatorwith rank less than n – 1 is the sum of rank-one operatorsin _n, and (ii) there is an operator of rank n – 1 in _nwhich is not the sum of rank-one operators in _n. The invariantsubspace lattice of _n is finite and distributive with 2n join-irreducibleelements. We show also that the indecomposability of _n is relatedto the existence of a chordless cycle in a bipartite graph associatedwith _n.