Geodesic Mappings of Vn(K)-Spaces and Concircular Vector Fields

In the present paper, we study geodesic mappings of special pseudo-Riemannian manifolds called V n ( K ) -spaces. We prove that the set of solutions of the system of equations of geodesic mappings on V n ( K ) -spaces forms a special Jordan algebra and the set of solutions generated by concircular fields is an ideal of this algebra. We show that pseudo-Riemannian manifolds admitting a concircular field of the basic type form the class of manifolds closed with respect to the geodesic mappings.

A note on ideals in synaptic algebras

The notion of a synaptic algebra was introduced by David Foulis. Synaptic algebras unite the notions of an order-unit normed space, a special Jordan algebra, a convex effect algebra and an orthomodular lattice. In this note we study quadratic ideals in synaptic algebras which reflect its Jordan algebra structure. We show that projections contained in a quadratic ideal from a p-ideal in the orthomodular lattice of projections in the synaptic algebra and we find a characterization of those quadratic ideals which are generated by their projections.

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.

Synaptic algebras

A synaptic algebra is both a special Jordan algebra and a spectral order-unit normed space satisfying certain natural conditions suggested by the partially ordered Jordan algebra of bounded Hermitian operators on a Hilbert space. The adjective “synaptic”, borrowed from biology, is meant to suggest that such an algebra coherently “ties together” the notions of a Jordan algebra, a spectral order-unit normed space, a convex effect algebra, and an orthomodular lattice.

Tensor Products of Jordan Algebras

Let J1 and J2 be two Jordan algebras with unit elements. We define various tensor products of J1 and J2. The first, which we call the Kronecker product, is the most obvious and is based on the tensor product of the vector spaces. We find conditions sufficient for its existence and for its non-existence. Motivated by the universal mapping property for the tensor product of associative algebras we define, in Section 2, tensor products of J1 and J2 by means of a universal mapping property. The tensor products always exist for special Jordan algebras and need not coincide with the Kronecker product when the latter exists. In Section 3 we construct a more concrete tensor product for special Jordan algebras. Here the tensor product of a special Jordan algebra and an associative Jordan algebra coincides with the Kronecker product of these algebras. We show that this "special" tensor product is the natural tensor product for some Jordan matrix algebras.

On Homomorphic Images of Special Jordan Algebras

A linear algebra is called a Jordan algebra if it satisfies the identities(1) ab = ba, (a2b) a = a2(ba).It is well known that a linear algebra S over a field of characteristic different from two is a Jordan algebra if there is an isomorphism a → a of the vector-space underlying S into the vector-space of some associative algebra A such that1,where the dot denotes the multiplication in A. Such an algebra S is called a special Jordan algebra.