Matthew Neal

Metric characterizations of isometries and of unital operator spaces and systems

We give some new characterizations of unitaries, isometries, unital operator spaces, unital function spaces, operator systems, C*-algebras, and related objects. These characterizations only employ the vector space and operator space structure.

Ordered Involutive Operator Spaces

This is a companion to recent papers of the authors; here we construct the ‘noncommutative Shilov boundary’ of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider ‘maximal’ and ‘minimal’ unitizations of such ordered operator spaces.

Contractive projections and operator spaces

We define a family of Hilbertian operator spaces Hnk, 1≤k≤n, containing the row and column Hilbert spaces Rn, Cn and show that an atomic subspace X⊂B(H) which is the range of a contractive projection on B(H) is isometrically completely contractive to an l∞-sum of the Hnk and Cartan factors of types 1 to 4. We also give a classification up to complete isometry of w∗-closed atomic JW∗-triples which have no infinite-dimensional rank 1 w∗-closed ideal.