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...ons in this case, for example x ∈ Xh iff ‖1 + itx‖2 = 1 + t2‖x‖2 for all t ∈ R, and iff ‖1 + itx‖ = 1 + o(t) (see e.g. [8, 20]). An abstract characterization of unital operator spaces may be found in =-=[6]-=-. The reader may also find metric characterizations of operator systems there, and the fact that the involution on a dual operator system is weak* continuous. The famous order theoretic characterizati...

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...linear-metric in our strict sense described in the paper). 1. Introduction The present paper is a sequel to our paper “Metric characterization of isometries and of unital operator spaces and systems” =-=[14]-=-. The goal of both papers is to characterize certain common objects in the theory of operator spaces (unitaries, unital operator spaces, operator systems, operator algebras, and so on), in terms which...

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...ork has finished, D. Blecher and M. Neal recently gave another abstract characterization for unital operator spaces (as well as an operator space characterization for operator systems) in their paper =-=[4]-=-. More precisely, they characterize the “identity” u of an operator space X by looking at the norms of some matrices defined by u together with an arbitrary elements in Mn(X) (n ∈ N). 2. We would like...

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...− ee∗)A(1− e∗e) = {0}. Proof. (a) ⇔ (b) If e is an extreme point of the closed unit ball of A, then trivially D A 1 (e) must be {0}. Conversely, if D A 1 (e) = {0}, and e = 12(x + y) with ‖x‖, ‖y‖ = 1, then taking z = 1 2 (x− y) we have ‖z‖ ≤ 1, ‖e + z‖ = ‖x‖ = 1 and ‖e− z‖ = ‖y‖ = 1. This implies that z ∈ DA1 (e) = {0}, and hence x = y = e. (b) ⇒ (c) If DA1 (e) = {0}, then D A 2 (e) = D A 1 (e) = {0}. Theorem 3.2 gives the desired statement. The implication (c) ⇒ (d) is clear. Finally, for (d) ⇒ (b), we observe that {e}⊥ ⊆ (1−ee∗)A(1−e∗e) = {0}, and hence Theorems 3.2 and 3.3 prove (b). In [9, 10] D.P. Blecher and M. Neal established a metric characterization of unitaries, isometries, and coisometries in terms of the operator space structure of C∗-algebras and TRO’s. We recall that a ternary ring of operators (a TRO in the terminology of D.P. Blecher and M. Neal in [8] and M. Neal and B. Russo in [54]) is a closed subspace Z of a C∗-algebra A such that ab∗c ∈ Z, for every a, b, c ∈ Z. For each complex Hilbert space H, and each natural number n, the symbol Hn stands for the direct sum of n copies of H. According to this notation, the space Mn(B(H)) of all n × n-matrices with entries in ...

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...es from X. If K is a field and X is a K -vector space, then Mn(X) will be identified with the K-vector space Mn (K)⊗X. The authors would like to thank David Blecher for referring them to the articles =-=[10,11]-=-. 2. Parametrizing operator systems and operator spaces We consider in this section several natural standard Borel parametrizations of the categories OSy and OSp of complete separable operator systems...

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...iged to give some justification. Following the general abstract characterization of an operator space due to Ruan (cf. [3]) abstract characterizations of a unital operator space have been provided in =-=[1]-=- and [2]. One knows that given any concretely represented unital operator space, its enveloping operator system is uniquely determined up to complete ∗-isometry regardless of the particular representa...

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... (1− δ)1 + (1 − 2δ)zi is in the weak* closure of FA for every i. Hence 1 + z is in this weak* closure too. Below we will also consider unital operator spaces: subspaces A of B(H) containing IH (see =-=[11]-=- for a matrix norm characterization of these). Here FA = {x ∈ A : ‖1A − x‖ ≤ 1}. One may define a cone in any operator algebra (or unital operator space) A by considering c = cA = R + FA. Probably 1 2...