Abstract
We give, for each n ⩾ 3, an example of a reflexive operator algebra 𝒜n with the following properties: (i) each finite rank operator with rank less than n − 1 is the sum of rank-one operators in 𝒜n, and (ii) there is an operator of rank n − 1 in 𝒜n which is not the sum of rank-one operators in 𝒜n. The invariant subspace lattice of 𝒜n is finite and distributive with 2n join-irreducible elements. We show also that the indecomposability of 𝒜n is related to the existence of a chordless cycle in a bipartite graph associated with 𝒜n.
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