Search Results (6)

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and $m^{\ast }\in M^{\ast }$ is a continuous linear functional on M which is open as a map between topological spaces, then ${\left(m^{\ast }\right)}^{-1}\left(\mathrm{int}\left(B\right)\right)$ is regular open and ${\left(m^{\ast }\right)}^{-1}\left(B\right)$ is regular closed, for B any closed-unit zero-neighborhood in R. Full article

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space $\mathcal{H}$ and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space, ${\mathsf{\Phi }}^{\times }$ , of a rigged Hilbert space, $\mathsf{\Phi }\subset \mathcal{H}\subset {\mathsf{\Phi }}^{\times }$ . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors $\mathsf{\Phi }$ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on $\mathsf{\Phi }$ with its own topology, so that they admit continuous extensions to the dual ${\mathsf{\Phi }}^{\times }$ and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider $SO\left(2\right)$ and functions on the unit circle, $SU\left(2\right)$ and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions, $SO(3,2)$ and spherical harmonics, $su(1,1)$ and Laguerre functions, $su(2,2)$ and algebraic Jacobi functions and, finally, $su(1,1)\oplus su(1,1)$ and Zernike functions on a circle. Full article

A dynamical system is a triple $(A,G,\alpha )$ consisting of a unital locally convex algebra A, a topological group G, and a group homomorphism $\alpha :G\to Aut\left(A\right)$ that induces a continuous action of G on A. Furthermore, a unital locally convex algebra A is called a continuous inverse algebra, or CIA for short, if its group of units $A^{\times }$ is open in A and the inversion map $\iota :A^{\times }\to A^{\times }$ , $a\mapsto a^{-1}$ is continuous at $1_A$ . Given a dynamical system $(A,G,\alpha )$ with a complete commutative CIA A and a compact group G, we show that each character of the corresponding fixed point algebra can be extended to a character of A. Full article

A well-known result of Ferri and Galindo asserts that the topological group $c_0$ is not reflexively representable and the algebra WAP $\left(c_0\right)$ of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame $\left(c_0\right)$ of tame functions. Respectively, it is an open question if $c_0$ is representable on a Rosenthal Banach space. In the present work we show that Tame $\left(c_0\right)$ is small in a sense that the unit sphere S and $2S$ cannot be separated by a tame function f ∈ Tame $\left(c_0\right)$ . As an application we show that the Gromov’s compactification of $c_0$ is not a semigroup compactification. We discuss some questions. Full article

Three-dimensional (3D) cadastral data models that are based on Euclidean geometry (EG) are incapable of providing a unified representation of geometry and topological relations for 3D spatial units in a cadastral database. This lack of unification causes problems such as complex expression structure and inefficiency in the updating of 3D cadastral objects. The inability of current cadastral data models to express cadastral objects in a unified manner can be attributed to the different expressions of dimensional objects. Because the hierarchical Grassmann structure corresponds to the hierarchical structure of dimensions in conformal geometric algebra (CGA), geometric objects in different dimensions can be constructed by outer products in a unified expression form, which enables the direct extension of two-dimensional (2D) spatial representations to 3D spatial representations. The multivector structure in CGA can be employed to organize and store different dimensional objects in a multidimensional and unified manner. With the advantages of CGA in multidimensional expressions, a new 3D cadastral data model that is based on CGA is proposed in this paper. The geometries and topological relations of 3D spatial units can be represented in a unified form within the multivector structure. Detailed methods for 3D cadastral data model design based on CGA and data organization in CGA are introduced. The new cadastral data model is tested and analyzed with experimental data. The results indicate that the geometry and topological relations of 3D cadastral objects can be represented in a multidimensional manner with an intuitive topological structure and a unified dimensional expression. Full article

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups. Full article