Complete Vector Space Research Articles

An Araki–Elliott Theorem for Alternative C *-Algebras

Abstract: Araki and Elliott proved that if A is an associative complex *-algebra endowed with a complete vector space norm ∥ • ∥ such that ∥ a*a ∥ = ∥ a ∥2 for every a ∈ A , then the norm ∥ • ∥ is submultiplicative. In this paper we prove that the Araki–Elliott theorem remains true if 'associative' is relaxed to 'alternative'. We also prove that the Araki–Elliott theorem does not remain true if 'associative' is altogether removed.

Smooth embeddings of the Long Line and other non-paracompact manifolds into locally convex spaces

We show that every real finite dimensional Hausdorff (not necessarily paracompact, not necessarily second countable) Cr-manifold can be Cr-embedded into a weakly complete vector space, i.e. a locally convex space of the form RI for an uncountable index set I and determine the minimal cardinality of I for which such an embedding is possible.

Ordered Vector Valued Double Sequence Spaces

AbstractIn this paper we have introduced an order relation on convergent double sequences and have constructed an ordered vector space, Riesz space, order complete vector space in case of double sequences. We have verified the Archimedean property.

Ordered vector valued statistically convergent sequence space

In this paper we have introduced an order relation on statistically convergent sequences and constructed statistically ordered vector space, statistically Riesz space, statistically order complete vector space and statistical Archimedean property.

Some basic results on the sets of sequences with geometric calculus

As an alternative to the classical calculus, Grossman and Katz [Non-Newtonian Calculus, Lee Press, Pigeon Cove, Massachusetts, 1972] introduced the non-Newtonian calculus consisting of the branches of geometric, anageometric and bigeometric calculus. Following Grossman and Katz, we construct the field C(G) of geometric complex numbers and the concept of geometric metric. Also we give the triangle and Minkowski's inequalities in the sense of geometric calculus. Later we respectively define the sets w(G), l∞(G), c(G), c0(G) and lp(G) of all, bounded, convergent, null and p-absolutely summable sequences, in the sense of geometric calculus and show that each of the set forms a complete vector space on the field C(G).

On the stability of functional equations

We give some theorems on the stability of the equation of homomorphism, of Lobacevski’s equation, of almost Jensen’s equation, of Jensen’s equation, of Pexider’s equation, of linear equations, of Schroder’s equation, of Sincov’s equation, of modified equations of homomorphism from a group (not necessarily commutative) into a $${\mathbb{Q}}$$ -topological sequentially complete vector space or into a Banach space, of the quadratic equation, of the equation of a generalized involution, of the equation of idempotency and of the translation equation. We prove that the different definitions of stability are equivalent for the majority of these equations. The boundedness stability and the stability of differential equations and the anomalies of stability are considered and open problems are formulated too.

Geometric interpretation of density displacements and charge sensitivities

The “geometric” interpretation of the electronic density displacements in the Hilbert space is given and the associated projection-operator partitioning of the hardness and softness operators (kernels) is developed. The eigenvectors |Æ〉= |α〉 of the hardness operator define the complete (identity) projector P =Σα| α〉〈α | = 1 for general density displacements, including thecharge-transfer (CT) component, while the eigenvectors |i〉= |i〉 of the linear response operator determine thepolarizational P-projector, Pp = Σi |i〉〈i| . Their difference thus defines the complementary CT-projector: PCT = 1-PP. The complete vector space for density displacements can be also spanned by supplementing the P-modes with the homogeneous CT-mode. These subspaces separate the integral (normalization) and local aspects of density shifts in molecular systems.

Transitive Actions of Compact Groups and Topological Dimension

There are many dimension functions defined on arbitrary topological spaces taking either a finite value or the value infinity. This paper defines a cardinal valued dimension function, dim. The Lie algebra L(G) of a compact group G is a weakly complete topological vector space. Quotient spaces of weakly complete spaces are weakly complete; the dimension of a weakly complete vector space is the linear dimension of its dual. Assume that a compact group G acts transitively on a given space X and that H is the isotropy group of the action at an arbitrary point; let L(G) and L(H) denote the Lie algebras of G, respectively, H. It is shown that dimX=dimL(G)/L(H). Moreover, such an X contains a space homeomorphic to [0,1]dimX; conversely, if X contains a homeomorphic copy of a cube [0,1]ℵ, then ℵ≤dimX. En route one establishes a good deal of information on the quotient spaces G/H; such information is of independent interest. Finally, these results are generalized to quotient spaces of locally compact groups. A generalization of a theorem of Iwasawa is instrumental; it is of independent interest as well.

Universal state space embeddability of Jordan-Banach algebras

We study extensions of states between projection structures of JB algebras and generalized orthomodular posets. It is shown that projection orthoposet of a JB algebra A admits the universal extension property if and only if the Gleason theorem is valid for A. As a consequence we get that any positive Stone algebra-valued measure on projection lattice of a quotient of a JBW algebra without type I2 direct summand extends to a positive measure on an arbitrary larger generalized orthomodular lattice. The classical extension theorem of Horn and Tarski [5] says that any probability measure on a Boolean algebra B extends to a probability measure on any larger Boolean algebra B' containing B. The first measure theoretic generalization of this result was given by P. Ptak [10] who showed that B' can be replaced by any quantum logic L with reasonably rich state space. We say in this case that Boolean algebras have the universal state extension property. In further development it has been proved in [6] that the universal state extension property for projection structures of unital C*-algebras is equivalent with the existence of non-commutative integral (so called Gleason property). As a consequence we get that projection lattices of von Neumann algebras without type I2 direct summand enjoy the universal state extension property. The aim of this note is to extend the above stated results for (not necessarily unital) Jordan Banach algebras and corresponding orthomodular structures not containing a largest element. This generalization is based on extension technique using determinacy of pure states on Jordan algebras. Moreover, general extension theorem for complete vector space valued measures on projections will be proved. First we recall a few notions and fix the notation. (Our standard references for Jordan algebras, ordered vector spaces, and quantum logics are [7, 1, 11], correspondingly.) * By a Jordan algebra we shall mean a real vector space A equipped with the product (i.e. bilinear form) written as (a, b) -? a o b and satisfying the following properties for all a, b E A: a o b = b o a, a o (bo a 2) = (aob) oa 2. A is said to be urnital if it admits a unit with respect to the Jordan product. In addition, if A is a Banach algebra with respect to the product o and if the norm I I satisfies the following conditions for all a,b E A: IIa211 < Ila2 + b211, Received by the editors May 1, 1997. 1991 Mathematics Subject Classification. Primary 46L70, 46L50, 28B15, 81P10.

Some nicely placed Hardy spaces

Let L 0 = L 0 (Ω, [sum ], μ) denote the vector space of (equivalence classes of) measurable functions on a measure space (Ω, [sum ], μ), taking values in a finite-dimensional Hilbert space H . We give L 0 the topology τ 0 of local convergence in measure[ratio ]τ 0 is a complete vector space topology, with base of neighbourhoods formula here where formula here

Saddlepoints for convex programming in order vector spaces

In this paper we characterize the existence of saddlepoints for optimization problems in order complete vector spaces by the existence of order automorphisms of vector space, by the existence of subgradients of the associated perturbing function, and by the existence of directional derivatives of the perturbing function, respectively. The statement of this paper is purely algebraic.

A Lagrange multiplier theorem and a sandwich theorem for convex relations.

We formulate and prove various separation principles for convex relations taking values in an order complete vector space. These principles subsume the standard ones.

Variational buildup of nuclear shell model bases

A method is proposed for accurately calculating a few of the low-lying states belonging to each nuclear spin within the shell model. A properly chosen initial vector is operated upon iteratively, generating a basis in which the Hamiltonian matrix is tridiagonal. The process may be halted when low-lying eigenvalues, up to a preset number, have been determined to a required accuracy, implying a truncation of the complete vector space. Simple formulae are given for the errors induced by this truncation, both for eigenvalues and eigenfunctions. Illustrative calculations are shown for the 0 + ( T = 0), 2 + ( T = 0), 4 + ( T = 0), 2 + ( T = 1) and + ( T = 1) states of 20Ne. By taking the leading SU 3 states as initial vectors, quite accurate low-lying eigenvalues and eigenfunctions are obtained with a small-sized truncation space. The T = 0 states mentioned allow a three dimensional truncation to yield lowest eigenvalues with an error of less than 0.05 MeV as well as corresponding eigenfunctions having an overlap of greater than 99.5% with the exact ones. Special eigenvalue-problem techniques are discussed in the appendix.

On Sequences of Operations in Complete Vector Spaces

On Sequences of Operations in Complete Vector Spaces

On Sequences of Operations in Complete Vector Spaces

On Sequences of Operations in Complete Vector Spaces