Operator space structure of JC*-triples and TROs, I

Abstract

We embark upon a systematic investigation of operator space structure of J C * -triples via a study of the TROs (ternary rings of operators) they generate. Our approach is to introduce and develop a variety of universal objects, including universal TROs, by which means we are able to describe all possible operator space structures of a J C * -triple. Via the concept of reversibility we obtain characterisations of universal TROs over a wide range of examples. We apply our results to obtain explicit descriptions of operator space structures of Cartan factors regardless of dimension Keywords CAR algebra · Cartan factor · JC-operator space · Operator space ideal · Reversibility · Ternary ring of operators · Universal TRO Mathematics Subject Classification (2000) 47L25 · 17C65 · 46L70

On the operator space structure of Hilbert spaces

“…The latter may be thought of as Jordan TROs. Our main aim in this paper is to complement and extend results of these authors and at the same time to bring to a completion a strand of our earlier work [3] on Cartan factors. (See § 1 for any undefined terms here and below.…”

“…A JC-operator space is a JC * -triple E together with an operator space structure on E induced by a linear isometry onto a JC * -subtriple of a C * -algebra (or TRO). We recall [3] that, associated with a JC * -triple E, there is a pair (T * (E), α E ), where T * (E) is a TRO and α E : E → T * (E) is a linear isometry onto a JC * -subtriple, such that if π : E → T is a triple homomorphism into a TRO, there is a unique TRO homomorphism,π : T * (E) → T , withπ • α E = π; moreover,π maps onto TRO(π(E)). The ideals kerπ of T * (E) that arise in this way when π is injective are called operator space ideals.…”

“…Equivalently, these are the norm closed ideals I of T * (E) having zero intersection with α E (E). For each such I we associate the JC-operator space E I induced by the embedding E → T * (E)/I (x → α E (x) + I), and every JC-operator space structure of E arises in this way [3,Proposition 6.2]. Given operator space ideals I and J , the formal identity map is a complete contraction from E I to E J whenever I ⊆ J , and we remark that distinct E I and E J can be completely isometric.…”

“…We remark that T (X) is a TRO and that references to 'triple homomorphisms' in [2,6] should be read as 'TRO homomorphisms'. We shall make frequent use of the following straightforward consequence of Theorem 6.5 and the proof of Theorem 6.11(a) from [3], the conditions of which are satisfied for every non-Hilbertian Cartan factor [3, Theorem 6.8]. Consider the anti-symmetric Fock space ∞ n=0 Λ n E, denoted by F − (E), associated with a Hilbert space E of dimension at least 2.…”

“…The operator space structures on a JC * -triple E induced by linear isometries from E onto JC * -subtriples of a TRO are the JC-operator spaces of E, a systematic study of which was started in [3], where we established the existence of a universal TRO, T * (E), of E. Each member I of a certain class of ideals of T * (E), referred to as operator space ideals, gives rise to a JC-operator space E I of E and these exhaust all possibilities. This new vehicle enables us to approach Hilbertian JC-operator spaces from a different point of view and to settle a number of outstanding questions.…”