Open partial isometries and positivity in operator spaces

Abstract

Abstract. We first study positivity in C * -modules using tripotents (= partial isometries) which are what we call open. This is then used to study ordered operator spaces via an "ordered noncommutative Shilov boundary" which we introduce. This boundary satisfies the usual universal diagram/property of the noncommutative Shilov boundary, but with all the arrows completely positive. Because of their independent interest, we also systematically study open tripotents and their properties.1. Introduction. We are interested here in cones of positive operators X + = {x ∈ X : x ≥ 0}, for a space X of bounded linear operators on a Hilbert space, where ≥ denotes the usual ordering of such operators. Besides the intrinsic interest of such objects (for example, operator positivity plays a central role in many areas of mathematical physics today), our work is a sequel to [13], which was a first step in a new approach to positivity in an operator space X, namely studying it in terms of the "noncommutative Shilov boundary" of X (see [6,26,12]). The latter object is a Hilbert C * -module, or, equivalently, a ternary ring of operators (or TRO for short), by which we will mean a closed subspace Z of a C * -algebra A such that ZZ * Z ⊂ Z. If X contains positive operators, then so will any containing TRO. The starting point of the present investigation and [13] was the question of whether, in this case, all morphisms in the universal property of the noncommutative Shilov boundary can also be chosen to be positive (allowing this boundary to be used as a new tool in the study of ordered operator spaces). To answer this, one is led immediately to study positivity in TROs, and we

Metric characterizations of isometries and of unital operator spaces and systems

“…We will not need this until much later, but it is worth mentioning that the set of positive elements Δ u + in a unital operator space (X, u) is precisely d u ∩ X, in the notation of [8]. We leave it to the reader to check this useful alternative description of the positive elements in X.…”

“…p. 238 in [8] for an exposition of the TRO case of this), which shows that u is a partial isometry, and so uu …”

“…Proof. We use notation and facts from [8]. Consider the ordered ternary envelope of X whose positive cone is a natural cone…”

A decomposition theorem for Lie ideals in nest algebras

“…The diagonal of a nest algebra provides a concrete example of a ternary ring of operators (TRO), i.e., a subspace T of bounded linear operators on Hilbert space satisfying T S * R ∈ T for all elements T, S and R of the subspace T . A subspace S of a TRO T is called a sub-TRO if SS * S ⊆ S and is called an inner ideal if ST * S ⊆ S (see [1,10] and the references therein). In a nest algebra T (N ), the intersection Vol.…”

On range tripotents in JBW*-triples

“…[2,5]) and, more recently, the suprema of increasing nets of range tripotents lying in the bidual of a ternary ring of operators have come to play an important rôle in the study of positivity in operator spaces (cf. [3]). The present work is concerned with the subset of tripotents in a JBW * -triple B formed by the range tripotents of the elements of a JB * -subtriple A of B endowed with the partial ordering inherited from the set of all tripotents.…”