A completely bounded characterization of operator algebras

Supported by a grant from the NSF     

The Ricci operator of completely solvable metric lie algebras

We study the Ricci curvature of completely solvablemetric Lie algebras. In particular,we prove that the Ricci operator of every completely solvable nonunimodular or every noncommutative nilpotent metric Lie algebra has at least two negative eigenvalues.     

On center-valued traces on real operator algebras

V. I. Romanovskii Institute of Mathematics of the Uzbek Academy of Sciences. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 2, pp. 1–9, April–June, 1992.     

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.     

Model theory of operator algebras II: model theory

We introduce a version of logic for metric structures suitable for applications to C*-algebras and tracial von Neumann algebras. We also prove a purely model-theoretic result to the effect that the theory of a separable metric structure is stable if and only if all of its ultrapowers associated with nonprincipal ultrafilters on ? are isomorphic even when the Continuum Hypothesis fails.     

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.     

Geometric representation theory for unitary groups of operator algebras

Geometric realizations for the restrictions of GNS representations to unitary groups of C^∗-algebras are constructed. These geometric realizations use an appropriate concept of reproducing kernels on vector bundles. To build such realizations in spaces of holomorphic sections, a class of complex coadjoint orbits of the corresponding real Banach-Lie groups is described and some homogeneous holomorphic Hermitian vector bundles that are naturally associated with the coadjoint orbits are constructed.     

On operator algebras and operator ranges

Integral Equations and Operator Theory -     

Jordan operator algebras

A survey of recent results in classification of JW-algebras (weakly closed Jordan algebras of self-adjoint operators in the Hilbert space) is given along with connections of JW-algebras with their enveloping W^*-algebras. It is shown how these results are applied in the proofs of analogs of many important results in the theory of W^*-algebras.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 67–98, 1985.     

Involutive operator algebras

Positivity - Examples of operator algebras with involution include the operator $$*$$-algebras occurring in noncommutative differential geometry studied recently by Mesland, Kaad, Lesch, and...     

Lax operator algebras

In this paper, we develop the general approach, introduced in [l], to Lax operators on algebraic curves. We observe that the space of Lax operators is closed with respect to their usual multiplication as matrix-valued functions. We construct orthogonal and symplectic analogs of Lax operators, prove that they form almost graded Lie algebras, and construct local central extensions of these Lie algebras.     

Similarity of an operator to an isometric operator

Leningrad State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 2, pp. 77–78, April–June, 1989.     

Commutative quantum operator algebras

A key notion bridging the gap between quantum operator algebras [26] and vertex operator algebras [4, 9] is the definition of the commutativity of a pair of quantum operators (see Section 2). This is not commutativity in any ordinary sense, but it is clearly the correct generalization to the quantum context. In [26] we give a definition of a commutative quantum operator algebra. We show in [26] that a vertex operator algebra gives rise to a special case of a CQOA. The main purpose of the current paper is to further develop the foundations for a complete mathematical theory of CQOAs. We give proofs of most of the relevant results announced in [26], and we carry out some calculations with sufficient detail to enable the interested reader to become proficient with the algebra of commuting quantum operators.     

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.     

Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.     

Vertex operator algebras and representations of affine Lie algebras

We announce the construction of an explicit basis for all integrable highest weight modules over the Lie algebra A ₁ ⁽¹⁾. The construction uses representations of vertex operator algebras and leads to combinatorial identities of Rogers-Ramanujan-type.