Tro Equivalent Algebras

In this work we study a new equivalence relation between w∗ closed algebras of operators on Hilbert spaces. The algebras A and B are called TRO equivalent if there exists a ternary ring of operatorsM (i.e. MM∗M ⊂ M) such that A = [M∗BM]−w ∗ and B = [MAM∗]−w ∗ . We prove that two reflexive algebras are TRO equivalent if and only if there exists a ∗ isomorphism between the commutants of their diagonals mapping the invariant projection lattice of the first algebra onto the lattice of the second one. In the case of separably acting CSL algebras the criterion is simpler: they are TRO equivalent if and only if there exists an isomorphism between their lattices which “respects continuity”. We explore some consequences of TRO equivalence for CSL algebras. We also prove that TRO equivalence is stronger than “spatial Morita equivalence”. Two CSL algebras are “spatially Morita equivalent” if and only if their lattices are isomorphic. In this case if one of them is synthetic then so is the other.