On the operator space structure of Hilbert spaces

Abstract

Operator spaces of Hilbertian JC*‐triples E are considered in the light of the universal ternary ring of operators (TRO) introduced in recent work. For these operator spaces, it is shown that their triple envelope (in the sense of Hamana) is the TRO they generate, that a complete isometry between any two of them is always the restriction of a TRO isomorphism and that distinct operator space structures on a fixed E are never completely isometric. In the infinite‐dimensional cases, operator space structure is shown to be characterized by severe and definite restrictions upon finite‐dimensional subspaces. Injective envelopes are explicitly computed.

Derivations and projections on Jordan triples: An introduction to nonassociative algebra, continuous cohomology, and quantum functional analysis

“…The paper [48], together with its sequel [49], initiates a systematic investigation of the operator space structure of JC * -triples via a study of the TROs they generate. The approach is through the introduction and development of a variety of universal objects (TROs and C * -algebras).…”

“…It follows from this that the triple envelope, in the sense of [109], of a JC * -triple E which is isometric to a Cartan factor is identified with the TRO generated by E. The following theorem shows that distinct JC-operator space structures on a Hilbert space cannot be completely isometric. It is also shown in [49,Theorem 3.4] how infinite-dimensional Hilbertian JC-operator spaces are determined explicitly by their finite-dimensional subspaces, and that, in turn, they impose very rigid constraints upon the operator space structure of their finite-dimensional subspaces. Finally, the operator space ideals of the universal TRO of a Hilbert space are identified [49,Theorem 3.7], as well as the corresponding injective envelopes [49,Theorem 4.4].…”

“…In turn, the results of [48,49,50] are used in [51] to extend Theorem 10.4 to general JC * -triples ([51, Theorem 4.7]), provide the structure of universally reversible JW * -triples and W * -TROs ([51, Theorem 4.14]), and show under mild conditions that a triple homomorphism on a TRO decomposes into a sum of a TRO homomorphism and a TRO antihomorphism ( [51,Proposition 5.14]).…”

“…The remainder of Part III discusses three topics, two very recent, which involve the interplay between Jordan theory and operator space theory (quantum functional analysis). The first one, a joint work of the author [187], discusses the structure theory of contractively complemented Hilbertian operator spaces, and is instrumental to the third topic, which is concerned with some recent work on enveloping TROs and K-theory for JB*-triples [48], [49], [39]. The second topic presents some very recent joint work by the author concerning quantum operator algebras which leans very heavily on contractive projection theory [191].…”

Universally reversible JC *-triples and operator spaces

“…Operator space structure of reflexive JC * -triples has been investigated in important articles by Neal, Ricard and Russo [25] and by Neal and Russo [26,28], who also proved [27] that an operator space X is completely isometric to a TRO if and only if M n (X) is a JC *triple (in the abstract sense defined above) for all n ≥ 2. In [3] B. Feely and the authors began a general study of JC-operator spaces (the operator spaces induced by linear isometries onto concrete JC * -triples) which was continued in [4,5]. The JC-operator spaces of all Cartan factors (see §2) were described and enumerated in the process via instrumental use of the notions (inaugurated in [3]) of the universal TRO of a JC * -triple and of a universally reversible JC * -triple, conceived by analogy with companion notions in Jordan operator algebras [15,16] which they precisely generalise [3, §4].…”