Ideal decompositions of a ternary ring of operators with predual

Abstract

We show that any TRO (ternary ring of operators) with predual can be decomposed into the direct sum of a two-sided ideal, a left ideal, and a right ideal in some von Neumann algebra using an extreme point of the unit ball of the TRO.Recall that an operator space X is called a triple system or a ternary ring of operators (TRO for short) if there exists a complete isometry ι from X into a C * -algebra such that ι(x)ι(y) * ι(z) ∈ ι(X ) for all x, y, z ∈ X .Our main result is that any TRO with predual can be decomposed into the direct sum of a two-sided ideal, a left ideal, and a right ideal in some von Neumann algebra:Theorem.Let X be a TRO which is also a dual Banach space.Then X can be decomposed into the direct sum of TROs X T , X L , and X R ,so that there is a complete isometry ι from X into a von Neumann algebra in which ι(X T ), ι(X L ), and ι(X R ) are a weak * -closed two-sided, left, and right ideal, respectively, andIn the special case that the TRO is finite-dimensional, the decomposition is into a direct sum of rectangular matrices, as first proved essentially by R. R. Smith [2000].In the Appendix we give a short proof of that result.The following lemma is a version of Kadison's theorem [1951, Theorem 1] as found in [Pedersen 1979, Proposition 1.4.8] or [Sakai 1971, Proposition 1.6.5].Together with the idea of embedding an off-diagonal corner into a diagonal corner developed in [Blecher and Kaneda 2004, Section 2] (see also [Kaneda 2003, Section 2.2]), it plays a key role in the proof of our theorem.