Duality and operator algebras II: operator algebras as Banach algebras

We answer, by counterexample, several questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the ‘nonselfadjoint analogue’ of a w*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors, is sharp.     

Morita equivalence of dual operator algebras

We consider notions of Morita equivalence appropriate to weak* closed algebras of Hilbert space operators. We obtain new variants, appropriate to the dual algebra setting, of the basic theory of strong Morita equivalence, and new nonselfadjoint analogues of aspects of Rieffel’s W^∗-algebraic Morita equivalence.     

Duality and normal parts of operator modules

For an operator bimodule X over von Neumann algebras A⊆B(H) and B⊆B(K), the space of all completely bounded A,B-bimodule maps from X into B(K,H), is the bimodule dual of X. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To X a normal operator bimodule X_n is associated so that completely bounded A,B-bimodule maps from X into normal operator bimodules factorize uniquely through X_n. A construction of X_n in terms of biduals of X, A and B is presented. Various operator bimodule structures are considered on a Banach bimodule admitting a normal such structure.     

The weak* similarity property on dual operator algebras

This paper is devoted to dual operator algebras, that isw ^*-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebrasA with the following weak^* similarity property: for every Hilbert spaceH, anyw ^*-continuous unital homomorphism fromA intoB(H) is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras, in analogy with some recent work of Pisier on the similarity problem on operator algebras.     

Fully symmetric operator spaces

It is shown that certain interpolation theorems for non-commutative symmetric operator spaces can be deduced from their commutative versions. A principal tool is a refinement of the notion of Schmidt decomposition of a measurable operator affiliated with a given semi-finite von Neumann algebra.Research supported by A.R.C.     

RUC-decompositions in symmetric operator spaces

We show that ifX is a Banach space of type 2 andG is a compact Abelian group, then any system of eigenvectors {x }_G (with respect to a strongly continuous representation ofG onX) is an RUC-system. As an application, we exhibit new examples of RUC-bases in certain symmetric spaces of measurable operators.Research supported by the Australian Research Council     

Jordan operator algebras: basic theory

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras.     

Lipschitz continuity of the absolute value in preduals of semifinite factors

We prove a weak-type estimate for the absolute value mapping in the preduals of semifinite factors which extends an earlier result of Kosaki for the trace class.Research supported by the Australian Research Council (ARC) and the Netherlands Organization for Scientific Research (NWO)     

Jordan operator algebras revisited

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . In two recent papers by the authors and Neal, a theory for these spaces was developed. It was shown there that much of the theory of associative operator algebras, in particular, surprisingly much of the associative theory from several recent papers of the first author and coauthors, generalizes to Jordan operator algebras. In the present paper we complete this task, giving several results which generalize the associative case in these papers, relating to unitizations, real positivity, hereditary subalgebras, and a couple of other topics. We also solve one of the three open problems stated at the end of our earlier joint paper on Jordan operator algebras.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

Multipliers on matrix ordered operator spaces and some K-groups

Matrix ordered operator spaces are ‘non-commutative Banach spaces equipped with a non-commutative order’. Examples include C*-algebras as well as their duals. In this article, we define and intrinsically characterize the multiplier algebra for this class of spaces and briefly tackle the problem of extending K-theory to this context.     

On some norm equalities in pre-Hilbert C*-modules

We characterize when the norm of the sum of two elements in a pre-Hilbert C*-module equals the sum of their norms. We also give the necessary and sufficient conditions for two orthogonal elements of a pre-Hilbert C*-module to satisfy Pythagoras’ equality.     

Embedding of C q and R q into noncommutative L p -spaces, 1≤p<q≤2

We prove that a quotient of a subspace of C_p⊕_pR_p (1≤p<2) embeds completely isomorphically into a noncommutative L_p -space, where C_p and R_p are respectively the p-column and p-row Hilbertian operator spaces. We also represent C_q and R_q (p<q≤2) as quotients of subspaces of C_p⊕_pR_p. Consequently, C_q and R_q embed completely isomorphically into a noncommutative L_p (M). We further show that the underlying von Neumann algebra M cannot be semifinite.     

Noncommutative topology and Jordan operator algebras

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space with for all . We study noncommutative topology, noncommutative peak sets and peak interpolation, and hereditary subalgebras of Jordan operator algebras. We show that Jordan operator algebras present perhaps the most general setting for a “full” noncommutative topology in the ‐algebraic sense of Akemann, L. G. Brown, Pedersen, etc, and as modified for not necessarily selfadjoint algebras by the authors with Read, Hay and other coauthors. Our breakthrough relies in part on establishing several strong variants of ‐algebraic results of Brown relating to hereditary subalgebras, proximinality, deeper facts about for a left ideal L in a ‐algebra, noncommutative Urysohn lemmas, etc. We also prove several other approximation results in ‐algebras and various subspaces of ‐algebras, related to open and closed projections and technical ‐algebraic results of Brown.     

Matrix-valued interpolation and hyperconvex sets

We prove that it is possible for two uniform algebras to have the same scalar interpolating sets, yet still have different matrix-valued interpolating sets.We prove a result for tensor products of uniform algebras that extends Agler's interpolation formula for the bidisk to more general product domains. This is accomplished by introducing a dual object for interpolation problems, which we call a Schur ideal, and proving that the Schur ideal for a tensor product is the intersection of the corresponding Schur ideals.Research supported in part by a grant from the NSF     

Quasi-multipliers and algebrizations of an operator space

Let X be an operator space, let φ be a product on X, and let (X,φ) denote the algebra that one obtains. We give necessary and sufficient conditions on the bilinear mapping φ for the algebra (X,φ) to have a completely isometric representation as an algebra of operators on some Hilbert space. In particular, we give an elegant geometrical characterization of such products by using the Haagerup tensor product. Our result makes no assumptions about identities or approximate identities. Our proof is independent of the earlier result of Blecher, Ruan and Sinclair [D.P. Blecher, Z.-J. Ruan, A.M. Sinclair, A characterization of operator algebras, J. Funct. Anal. 89 (1) (1990) 188-201] which solved the case when the bilinear mapping has an identity of norm one, and our result is used to give a simple direct proof of this earlier result. We also develop further the connections between quasi-multipliers of operator spaces and their representations on a Hilbert space or their embeddings in the second dual, and show that the quasi-multipliers of operator spaces defined in [M. Kaneda, V.I. Paulsen, Quasi-multipliers of operator spaces, J. Funct. Anal. 217 (2) (2004) 347-365] coincide with their C^∗-algebraic counterparts.     

Reflexive *-derivations and lattices of invariant subspaces of operator algebras associated with them

The paper studies unbounded reflexive *-derivations δ of C*-algebras of bounded operators on Hilbert spaces H whose domains D(δ) are weekly dense in B(H and contain compact operators. It describes a one-to-one correspondence between these derivations and pairs S,L, where S are symmetric densely operators on H and L are J-orthogonal π-reflexive lattices of subspaces in the deficiency spaces of S. The domains D(δ) of these *-derivations are associated with some non-selfadjoint reflexive algebras A_δ of bounded operators on H⊕H. The paper analyzes the structure of the lattices of invariant subspaces of A_δ and of the normalizers of A_δ-the largest Lie subalgebras of B(H⊕H) such that A_δ are their Lie ideals.     

Representations of Certain Banach C*-Modules

The possibility of extending the well known Gelfand–Naimark–Segal representation of *-algebras to certain Banach C*-modules is studied. For this aim the notion of modular biweight on a Banach C*-module is introduced. For the particular class of strict pre CQ*-algebras, two different types of representations are investigated.     

A Morita type equivalence for dual operator algebras

We generalize the main theorem of Rieffel for Morita equivalence of W^∗-algebras to the case of unital dual operator algebras: two unital dual operator algebras A,B have completely isometric normal representations α,β such that α(A)=[M^∗β(B)M]^−w∗ and β(B)=[Mα(A)M^∗]^−w∗ for a ternary ring of operators M (i.e. a linear space M such that MM^∗M⊂M) if and only if there exists an equivalence functor which “extends” to a ∗-functor implementing an equivalence between the categories and . By we denote the category of normal representations of A and by the category with the same objects as and Δ(A)-module maps as morphisms (Δ(A)=A∩A^∗). We prove that this functor is equivalent to a functor “generated” by a B,A bimodule, and that it is normal and completely isometric.