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Journal of the London Mathematical Society 1995 51(1):148-160; doi:10.1112/jlms/51.1.148
© 1995 by London Mathematical Society
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© The London Mathematical Society

Decomposability of Reflexive Cycle Algebras

K.J. Harrison and U.A. Mueller

Mathematics Department, Murdoch University Murdoch, Western Australia 6150, Australia E-mail: harrison{at}csuvaxl.murdoch.edu.au
Department of Mathematics, Edith Cowan University Perth, Western Australia 6050, Australia E-mail: u.mueller{at}cowan.edu.au

Received 19 January 1993. Revision received 14 June 1993.

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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