The Jordan socle and finitary Lie algebras

In this paper we introduce the notion of Jordan socle for nondegenerate Lie algebras, which extends the definition of socle given in [A. Fernández López et al., 3-Graded Lie algebras with Jordan finiteness conditions, Comm. Algebra, in press] for 3-graded Lie algebras. Any nondegenerate Lie algebra with essential Jordan socle is an essential subdirect product of strongly prime ones having nonzero Jordan socle. These last algebras are described, up to exceptional cases, in terms of simple Lie algebras of finite rank operators and their algebras of derivations. When working with Lie algebras which are infinite dimensional over an algebraically closed field of characteristic 0, the exceptions disappear and the algebras of derivations are computed.     

The socle of a nondegenerate Lie algebra

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.     

An Artinian theory for Lie algebras

Complemented Lie algebras are introduced in this paper (a notion similar to that studied by O. Loos and E. Neher in Jordan pairs). We prove that a Lie algebra is complemented if and only if it is a direct sum of simple nondegenerate Artinian Lie algebras. Moreover, we classify simple nondegenerate Artinian Lie algebras over a field of characteristic 0 or greater than 7, and describe the Lie inner ideal structure of simple Lie algebras arising from simple associative algebras with nonzero socle.     

Jordan radicals of u-algebras

In this paper we show that strong noncommutative Jordan algebras R over an arbitrary ring of scalars having the alternator mappings y,y,-1 as Jordon derivations are U-algebras, algebras such that U_ablpar;crpar; lies in the Jordan ideal generated by a. For any U-algebra R we relate the radical theories of R and R⁺. Our main result is that any radical property p′ of U-algebras such that P′-radR? p-radR⁺. If p is nondegenerate the P′ is nondegenerate and P′-radR=p-radR⁺. This applies in particular to the McCrimmon, locally nilpotent, nil, Jacobson and Brown-McCoy radicals of Jordan algebras     

Derivations of noncommutative Arens algebras

The present paper deals with derivations of noncommutative Arens algebras. We prove that every derivation of an Arens algebra associated with a von Neumann algebra and a faithful normal finite trace is inner. In particular, each derivation on such algebras is automatically continuous in the natural topology, and in the commutative case, even for semi-finite traces, all derivations are identically zero. At the same time, the existence of noninner derivations is proved for noncommutative Arens algebras with a semi-finite but nonfinite trace.     

Prime Quotients of Jordan Systems and Lie Algebras

We show that, unlike alternative algebras, prime quotients of a nondegenerate Jordan system or a Lie algebra need not be nondegenerate, even if the original Jordan system is primitive, or the Lie algebra is strongly prime, both with nonzero simple hearts. Nevertheless, for Jordan systems and Lie algebras directly linked to associative systems, we prove that even semiprime quotients are necessarily nondegenerate.     

Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

Using the fact that absolute zero divisors in Jordan pairs become Lie sandwiches of the corresponding Tits–Kantor–Koecher Lie algebras, we prove local nilpotency of the McCrimmon radical of a Jordan system (algebra, triple system, or pair) over an arbitrary ring of scalars. As an application, we show that simple Jordan systems are always nondegenerate.     

Realizing ultragraph Leavitt path algebras as Steinberg algebras

In this article, we realize the finite range ultragraph Leavitt path algebras as Steinberg algebras. This realization allows us to use the groupoid approach to obtain structural results about these algebras. Using the skew product of groupoids, we show that ultragraph Leavitt path algebras are graded von Neumann regular rings. We characterize strongly graded ultragraph Leavitt path algebras and show that every ultragraph Leavitt path algebra is semiprimitive. Moreover, we characterize irreducible representations of ultragraph Leavitt path algebras. We also show that ultragraph Leavitt path algebras can be realized as Cuntz-Pimsner rings.     

Locally von Neumann algebras II

In this paper, we will prove some properties of locally von Neumann algebras. In particular, we will show that every locally von Neumann algebra is the dual of a certain locally convex space and also, we will show the existence of a polar decomposition for every element in a locally von Neumann algebra.     

Symmetries in synaptic algebras

A synaptic algebra is a generalization of the Jordan algebra of self-adjoint elements of a von Neumann algebra. We study symmetries in synaptic algebras, i.e., elements whose square is the unit element, and we investigate the equivalence relation on the projection lattice of the algebra induced by finite sequences of symmetries. In case the projection lattice is complete, or even centrally orthocomplete, this equivalence relation is shown to possess many of the properties of a dimension equivalence relation on an orthomodular lattice.     

A Characterization of Homomorphisms Between Banach Algebras

We show that every unital invertibility preserving linear map from a yon Neumann algebra onto a semi-simple Banach algebra is a Jordan homomorphism; this gives an affirmative answer to a problem of Kaplansky for all yon Neumann algebras. For a unital linear map Ф from a semi-simple complex Banach algebra onto another, we also show that the following statements are equivalent: (1)Ф is an homomorphism; (2) Ф is completely invertibility preserving; (3) Ф is 2-invertibility preserving.     

Strong regularity and generalized inverses in jordan systems

A notion of generalized inverse extending that of Moore—Penrose inverse for continuous linear operators between Hilbert spaces and that of group inverse for elements of an associative algebra is defined in any Jordan triple system (J, P). An element a?J has a (unique) generalized inverse if and only if it is strongly regular, i.e., a?P(a)²J. A Jordan triple system J is strongly regular if and only if it is von Neumann regular and has no nonzero nilpotent elements. Generalized inverses have properties similar to those of the invertible elements in unital Jordan algebras. With a suitable notion of strong associativity, for a strongly regular element a?J with generalized inverse b the subtriple generated by {a, b} is strongly associative     

Regular elements in generalized hermitian algebras

A generalized Hermitian (GH) algebra is a special Jordan algebra that is at the same time a spectral order-unit space. In this paper we characterize the von Neumann regular elements in a GH-algebra, relate maximal pairwise commuting subsets of the algebra to blocks in its projection lattice, and prove a Gelfand-Naimark type representation theorem for commutative GH-algebras.     

Similarity degrees for the crossed product of von Neumann algebras

In this paper, we will estimate an upper bound for the similarity degree of the crossed product of a hyperfinite finite von Neumann algebra by weakly compact action of an infinite discrete group. We will also improve some upper bounds for similarity degrees of some finite von Neumann algebras.     

About the Connes embedding conjecture

In his celebrated paper in 1976, A. Connes casually remarked that any finite von Neumann algebra ought to be embedded into an ultraproduct of matrix algebras, which is now known as the Connes embedding conjecture or problem. This conjecture became one of the central open problems in the field of operator algebras since E. Kirchberg’s seminal work in 1993 that proves it is equivalent to a variety of other seemingly totally unrelated but important conjectures in the field. Since then, many more equivalents of the conjecture have been found, also in some other branches of mathematics such as noncommutative real algebraic geometry and quantum information theory. In this note, we present a survey of this conjecture with a focus on the algebraic aspects of it.     

Nest-subalgebras of von Neumann algebras

The structure of a certain class of separably acting reflexive operator algebras is investigated for which the nest algebras of J. Ringrose can be considered prototypes. To a fixed von Neumann algebra and a complete nest of projections contained therein one associates the algebra of all operators in the von Neumann algebra which leave every member of the nest invariant. A generalization of the Ringrose criterion for inclusion in the Jacobson radical of a nest algebra is given for this more general class of algebras. Further properties of the radical are studied.     

Algebras of quotients of Jordan algebras

We define a Jordan analogue of Lambek and Utumi's associative algebra of quotients and we construct the maximal algebra of quotients for nondegenerate Jordan algebras. We apply those results to other classes of algebras of quotients appearing in the literature.     

The Socle of a Metric n-Lie Algebra

We define the socle of an n-Lie algebra as the sum of all the minimal ideals. An n-Lie algebra is called metric if it is endowed with an invariant nondegenerate symmetric bilinear form. We characterize the socle of a metric n-Lie algebra, which is closely related to the radical and the center of the metric n-Lie algebra. In particular, the socle of a metric n-Lie algebra is reductive, and a metric n-Lie algebra is solvable if and only if the socle coincides with its center. We also calculate the metric dimensions of simple and reductive n-Lie algebras and give a lower bound in the nonreductive case.     

von Neumann代数中CSL子代数上的Jordan(α,β)-导子

设■是Hilbert空间H上的von Neumann代数的CSL子代数.本文证明了,在一定的条件下,■上的Jordan(α,β)-导子是(α,β)-导子,其中α,β是■上的两个自同构.还证明了在没有添加任何条件的情况之下,CSL代数上的任意Jordan(α,β)-导子是(α,β)-导子.另外,讨论了von Neumann代数中的CSL子代数上的n次幂(α,β)-映射.     

Representation type of Jordan algebras

The problem of classification of Jordan bimodules over (non-semisimple) finite dimensional Jordan algebras with respect to their representation type is considered. The notions of diagram of a Jordan algebra and of Jordan tensor algebra of a bimodule are introduced and a mapping Qui is constructed which associates to the diagram of a Jordan algebra J the quiver of its universal associative enveloping algebra S(J). The main results are concerned with Jordan algebras of semi-matrix type, that is, algebras whose semi-simple component is a direct sum of Jordan matrix algebras. In this case, criterion of finiteness and tameness for one-sided representations are obtained, in terms of diagram and mapping Qui, for Jordan tensor algebras and for algebras with radical square equals to 0.