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 operators M (i.e. MM*M\subset M) such that A
is the w*-closed span of M*BM and B is the w*-closed span of MAM*. 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.