G-sequentially connectedness for topological groups with operations

The topological fundamental group and free topological groups

It is well known that the Lie algebra of derivations of the exceptional Jordan algebra M 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional Lie algebra F 4 of Killing-Cartan and that the Lie algebra of linear transformations in M 3 8 leaving a certain cubic norm form defined on M 3 8 invariant is the exceptional Lie algebra E 6 (Chevalley-Schafer [7], Freudenthal [11]). The program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple Jordan algebras. The present paper gives the definition of the groups and Lie algebras in the general case and gives their determination for special Jordan algebras with some restrictions on the base field. In a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple Jordan algebras and in a third paper we hope to study the generalized groups of types E 6 which arise from these Jordan algebras.

Let M be a von Neumann algebras, α — involutive *-anti-automorhism on M. Consider Mα(±1) = {x ∈ M : α(x) = ±x} the spectral subspaces of α. It follows that M α (+1) is a Jordan algebra with respect to the sym¬metrized product x • y = ½(xy + yx) and M α (- 1) is a Lie algebra with pespect to the brackets [x,y] = xy - yx. Robinson and Stormer [RS] have initiated the classification of pairs of Jordan and Lie algebras which can occur in this manner by examining the case M = B(H) the algebra of all bounded linear operators on a complex Hilbert space H, i.e. when M is a type I factor. In this chapter we shall continue the study of Lie algebras M α (-1) for arbitrary factors M.

A note on multiplier convergent series

All Hartman–Mycielski groups are minimally almost periodic

The effect of structure-preserving perturbations on the solution to a linear system, matrix inversion, and distance to singularity is investigated. Particular attention is paid tolinear and nonlinear structures that form Lie algebras, Jordan algebras and automorphism groups of ascalar product. These include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, unitary, complex orthogonal and symplectic matrices. Under reasonable assumptions on the scalar product, it is shown that there is little or no difference between structured and unstructured condition numbers and distance to singularity for matrices in Lie and Jordan algebras. Hence, for these classes of matrices, the usual unstructured perturbation analysis is sufficient. It is shown that this is not true in general for structures in automorphism groups. Bounds and computable expressions for the structured condition numbers for a linear system and matrix inversion are derived for these nonlinear structures.Structured backward errors for the approximate solution of linear systems are also considered. Conditions are given for the structured backward error to be finite. For Lie and Jordan algebras it is proved that, whenever the structured backward error is finite, it is within a small factor of or equal to the unstructured one. The same conclusion holds for orthogonal and unitary structures but cannot easily be extended to other matrix groups. This work extends and unifies earlier analyses.

Let G be a reflexive topological group, and Gits group of characters, endowed with the compact open topology. We prove that the evaluation mapping from Gx G into the torus T is continuous if and only if G is locally compact. This is an analogue of a well-known theorem of Arens on admissible topologies on C(X) . DEFINITIONS AND REMARKS Let X, Y be topological spaces, and let Z be a subset of yX . A topology on Z is said to be admissible if the evaluation mapping from the product Z x X into Y, defined by w(f, x) = f(x), is continuous. Let (S, V) = {f E Z; f(S) C V}. The family {(S, V)}, where S runs over the collection of all compact subsets of X and V runs over a basis of open sets in Y, is a subbase for the compact open topology on Z. An admissible topology on Z must be finer than the compact open topology [8]. A result of Arens states that the existence of a coarsest admissible topology for the class of real continuous functions on a completely regular space X is equivalent to X being locally compact [1]. In this paper we are interested in reflexive topological groups. Answering in the negative a question of Megrelishvili [7], we will see that the evaluation map for those groups need not be continuous. In fact, as specified in the theorem, the continuity of the evaluation characterizes locally compact Hausdorff abelian groups among reflexive groups. We prove this result by means of convergence spaces. For an account of theory of convergence spaces the reader is referred to [3, 4]. We only give here needed definitions. Let G be a Hausdorff topological abelian group, and let FG be the set of all continuous homomorphisms from G into the unidimensional torus T. If addition is defined pointwise in FG, it becomes an abelian group. The continuous Received by the editors October 5, 1993 and, in revised form, April 25, 1994; the contents of this paper were presented to the First Ibero-American Conference on Topology and its Applications in Benicassim, Spain, on March 28-30, 1995. 1991 Mathematics Subject Classification. Primary 22A05.

The purpose of this work is to join Lie field structures with certain infinite-dimensional Lie algebras with locally convex topology. These topological Lie algebras allow topological groups which are a generalization of the connected nilpotent Lie groups. We showed the existence of the continuous unitary representations of the gained groups and then we proved the analogue of Gårding theorem. Using this theorem we established the existence of representations of Lie field structures into Lie algebras of skew-symmetric operators on Hilbert spaces.

For a Euclidean Jordan algebra V with the corresponding symmetric cone $\Omega$, we consider the semigroup $\Gamma\sb{\Omega}$ of elements in the automorphism group $G(T\sb{\Omega})$ of the tube domain $V$ + $i\Omega$ which can be extended to $\Omega$ and maps $\Omega$ into itself. A study of this semigroup was first worked out by Koufany in connection to Jordan algebra theory and Lie theory of semigroups. In this work we give a new proof of Koufany's results and generalize up to infinite dimensional Jordan algebras, so called $JB$-algebras. One of the nice examples of the semigroup $\Gamma\sb{\Omega}$ is from the Jordan algebra $Sym(n,\IR$) of symmetric matrices. However, $V\sb{\sigma}$ the set of all self-adjoint operators on $\IR\sp{n}$ with respect to a non-degenerated symmetric bilinear form $\sigma$, is a non-Euclidean Jordan algebra with a cone $\Omega\sb{\sigma}$ which is isomorphic to $\Omega$ of the symmetric cone of $Sym(n,\IR)$. We get an isomorphism of the automorphism groups between two tube domains which also induces an isomorphism between two Lie semigroups. The Lorentzian cone, which is one of the irreducible symmetric cones, is an essential tool in the study of semigroups in Mobius and Lorentzian geometry. J. D. Lawson studied the Mobius and Lorentzian semigroups with an Ol'shanskii decomposition even in the infinite dimensional cases. We study these semigroups via a Jordan algebra theory.

Given a ternary relation C on a set U and an algebra A, we present a construction of a convolution algebra A(U, C) of U = (U, C) over A. This generalises bothmatrix algebras and algebras obtained from convolution of monoids. To any class of algebras corresponds a class of convolution structures. Our study cases are the classes of commutative, associative, Lie, and Jordan algebras. In each of these classes we give conditions on (U, C) under which A(U, C) is in the same class as A. It turns out that in some situations these conditions are even necessary.

Given a ternary relation C on a set U and an algebra A, we present a construction of a convolution algebra A(U, C) of U = (U, C) over A. This generalises bothmatrix algebras and algebras obtained from convolution of monoids. To any class of algebras corresponds a class of convolution structures. Our study cases are the classes of commutative, associative, Lie, and Jordan algebras. In each of these classes we give conditions on (U, C) under which A(U, C) is in the same class as A. It turns out that in some situations these conditions are even necessary.

Metrizability of topological groups having weak topologies with respect to good covers

Standard and Non-Standard Methods in Uniform Topology

Gradings on simple Jordan and Lie algebras

This paper investigates the effect of structure‐preserving perturbations on the eigenvalues of linearly and nonlinearly structured eigenvalue problems. Particular attention is paid to structures that form Jordan algebras, Lie algebras, and automorphism groups of a scalar product. Bounds and computable expressions for structured eigenvalue condition numbers are derived for these classes of matrices, which include complex symmetric, pseudo‐symmetric, persymmetric, skew‐symmetric, Hamiltonian, symplectic, and orthogonal matrices. In particular we show that under reasonable assumptions on the scalar product, the structured and unstructured eigenvalue condition numbers are equal for structures in Jordan algebras. For Lie algebras, the effect on the condition number of incorporating structure varies greatly with the structure. We identify Lie algebras for which structure does not affect the eigenvalue condition number.