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. Several characterizations of these objects are already known; see e.g. [15, Theorem 9.5.16], [1], the discussion on p. 316 of [4], [14], and [13]. One difference between our paper and these cited references, is that our results only use the vector space structure of the space and its matrix norms, in the spirit of Ruan’s matrix norm characterization of operator spaces [18], whereas the other cited references use criteria involving maps or functionals on the space. Our first main result characterizes unital operator spaces, that is, subspaces of a unital C-algebra containing the identity. More abstractly, a unital operator space is a pair (X,u) consisting of an operator space X containing a fixed element u such that there exists a Hilbert space H and a complete isometry T : X → B(H) with T (u) = IH . Such spaces have played a significant role since the birth of operator space theory in [2]. Indeed, although the latter paper is mostly concerned with unital operator algebras, it was remarked in several places there that many of the results are valid for unital operator spaces. The text [6] also greatly emphasizes unital operator spaces. The abstract characterization of these objects has been missing until recently, and we had wondered about this over the years; the following is our answer to this question. Our result complements Ruan’s characterization of operator spaces [18], the Blecher-Ruan-Sinclair abstract characterization of operator algebras [8], and a host of other theorems of this type (see e.g. [6, 16]). To state it, we will write un for the diagonal matrix in Mn(X) with u in each diagonal entry.