Lattice measures and topologies

Summary We abstract Frink's notion of a normal base of a topological space to an arbitrary lattice, and replace the notion of filters on a base by zero-one measures on a lattice. This offers analytical simplification and clarijication, and extends to arbitrary measures as well. By putting a topology on the set of measures, we generalize the notion of Wallman-type compactifications, and we look at relations between the compactifications by examining the underlying lattices. Entrata in Redazione il 20 gennaio 1975.     

On strongly replete lattices,support of A measure,and the wallman remainder

LetX be an abstract set andL a lattice of subsets ofX. To eachL-regular measure on the algebra generated byL, there are associated two measures on appropriate algebras of the Wallman space. In terms of these measures, we can obtain characterization for-smoothness,-smoothness, and tightness of the original measure. In particular, tight regular measures and their properties are investigated.     

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.     

Measures on minimally generated Boolean algebras

We investigate properties of minimally generated Boolean algebras. It is shown that all measures defined on such algebras are separable but not necessarily weakly uniformly regular. On the other hand, there exist Boolean algebras small in terms of measures which are not minimally generated. We prove that under CH a measure on a retractive Boolean algebra can be nonseparable. Some relevant examples are indicated. Also, we give two examples of spaces satisfying some kind of Efimov property.     

Symmetric Spaces of Matter and Real Fermion Manifolds

The geometric algebra Cl_3,1 generated by the Minkowski spacetime with signature {+++− } possesses a natural ternary partition which provides the Lie algebra of the standard model symmetry in an improved form. The symmetric spaces of matter embed a differentiable manifold of primitive idempotents which represents a real valued fermion space as an 8-dimensional real unitsphere in a 10-dimensional subspace with positive definite signature. The algebraic properties of the present theory of spacetime-matter are developed, beginning with the definiteness of the stabilizer algebra of neutrinos, investigating the orthogonality between fermions and neutrinos and ending with the curvature of the symmetric spaces of the strong force. The model brings together the quantum theory and relativity, as we conceive it at present, such that the standard model turns out to be a definite property of the spacetime algebra.     

Generalized Exponents and Forms

We consider generalized exponents of a finite reflection group acting on a real or complex vector space V. These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra. A basis for each isotypic component arises in a natural way from a basis of invariant generalized forms. We investigate twisted reflection representations (V tensor a linear character) using the theory of semi-invariant differential forms. Springer’s theory of regular numbers gives a formula when the group is generated by dim V reflections. Although our arguments are case-free, we also include explicit data and give a method (using differential operators) for computing semi-invariants and basic derivations. The data give bases for certain isotypic components of the coinvariant algebra.     

Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.     

Convergence of a sequence of weakly regular set functions

The present paper is devoted to generalizations of the Dieudonné theorem claiming that the convergence of sequences of regular Borelian measures is preserved under the passage from a system of open subsets of a compact metric space to the class of all Borelian subsets of this space. The Dieudonné theorem is proved in the case for which the set functions are weakly regular, nonadditive, defined on an algebra of sets that contains the class of open subsets of an arbitrary σ-topological space, and take values in a uniform space. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 103–110, July, 1997. Translated by O. V. Sipacheva     

A short zero-one law proof of a result of Abian

Summary In this note a new and very short zero-one law proof of the following theorem of Abian is presented. The subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use a given basis element of a prescribed Hamel basis, has outer Lebesgue measure one and inner measure zero.Let {a, b, c, ...} be a Hamel basis for the real numbers. LetA be the subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use the basis elementa. Sierpinski [4, p. 108] has shown thatA is nonmeasurable in the sense of Lebesgue. Abian [1] has improved Sierpinski's result by showing thatm* (A), the outer measure ofA, is one and thatm _* (A), the inner measure ofA, is zero. In this note a very short proof, using a zero-one law, of Abian's result will be presented.The following zero-one law is an immediate consequence of the Lebesgue Density Theorem [2, p. 290].     

Algebraic aspects of generalized approximation spaces

The concept of approximation spaces is a key notion of rough set theory, which is an important tool for approximate reasoning about data. This paper concerns algebraic aspects of generalized approximation spaces. Concepts of R-open sets, R-closed sets and regular sets of a generalized approximation space (U,R) are introduced. Algebraic structures of various families of subsets of (U,R) under the set-inclusion order are investigated. Main results are: (1) The family of all R-open sets (respectively, R-closed sets, R-clopen sets) is both a completely distributive lattice and an algebraic lattice, and in addition a complete Boolean algebra if relation R is symmetric. (2) The family of definable sets is both an algebraic completely distributive lattice and a complete Boolean algebra if relation R is serial. (3) The collection of upper (respectively, lower) approximation sets is a completely distributive lattice if and only if the involved relation is regular. (4) The family of regular sets is a complete Boolean algebra if the involved relation is serial and transitive.     

Intertwining operators for L 2(E)

Let (E,Q) be a finite dimensional quadratic vector space over a finite field. For the natural representation -π of the isometry group G of (E,Q) in the space L ²(E) of all complex valued functions on E, we analyse when the intertwining algebra of π is generated by just one averaging operator.     

Spaces of operator-valued measures on topological spaces

Summary Let X be a completely regular space and E a locally convex space. We study a space M(B, E′) of E′-valued measures defined on the algebra generated by the zero subsets of X. We also study certain subspaces of M(B, E′). Entrata in Redazione il 20 ottobre 1976.     

On the extension of subadditive measures in lattice ordered groups

A lattice ordered group valued subadditive measure is extended from an algebra of subsets of a set to a σ-algebra.     

Quotient spaces determined by algebras of continuous functions

We prove that if X is a locally compact σ-compact space, then on its quotient, γ(X) say, determined by the algebra of all real valued bounded continuous functions on X, the quotient topology and the completely regular topology defined by this algebra are equal. It follows from this that if X is second countable locally compact, then γ(X) is second countable locally compact Hausdorff if and only if it is first countable. The interest in these results originated in [1] and [7] where the primitive ideal space of a C*-algebra was considered.     

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.     

The banach algebra A and its properties

Beurling’s algebra is considered. A* arises quite naturally in problems of summability of the Fourier series at Lebesgue points, whereas Wiener’s algebra A of functions with absolutely convergent Fourier series arises when studying the norm convergence of linear means. Certainly, both algebras are used in some other areas. A* has many properties similar to those of A, but there are certain essential distinctions. A* is a regular Banach algebra, its space of maximal ideals coincides with[−π, π], and its dual space is indicated. Analogs of Herz’s and Wiener-Ditkin’s theorems hold. Quantitative parameters in an analog of the Beurling-Pollard theorem differ from those for A. Several inclusion results comparing the algebra A* with certain Banach spaces of smooth functions are given. Some special properties of the analogous space for Fourier transforms on the real axis are presented. The paper ends with a summary of some open problems.     

Operator-valued positive definite kernels on tubes

In this paper we generalize the concept of an infinite positive measure on a -algebra to a vector valued setting, where we consider measures with values in the compactification of a convex coneC which can be described as the set of monoid homomorphisms of the dual coneC ^* into [0, ]. Applying these concepts to measures on the dual of a vector space leads to generalizations of Bochner's Theorem to operator valued positive definite functions on locally compact abelian groups and likewise to generalizations of Nussbaum's Theorem on positive definite functions on cones. In the latter case we use the Laplace transform to realize the corresponding Hilbert spaces by holomorphic functions on tube domains.     

Extensions of group-valued set functions

The problem of common extension ofcharges (finitely additive measures) is generalised to include group-valued functions defined on a system of sets (u-systems). To eachu-systemU an Abelian groupH(U) is attached. Every Abelian group is isomorphic to one of the formH(U). The groupH(U) is an indicator for extendability of charges fromU to the Boolean algebra generated byU. AllG-valued measures extend if and only if Ext(H(U),G)=0, for instance. Supported as van Vleck visiting professor at Wesleyan University, Connecticut in 1993. Partially supported by the Graduierten KollogTheoretische und experimentelle Methoden der reinen Mathematik of Essen University, a project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development and by the German Academic Exchange, DAAD 1994.     

Automatic boundedness of quantum measures

LetG be a commutative Hausdorff topological group. Letm be aG-valued, completely additive measure on a complete orthomodular posetL. It is shown, among other results, that when the centre ofL is non-atomic thenm must be strictly bounded. WhenL is specialised to being the lattice of projections in a von Neumann algebra this extends some results known for real valued measures. The first author was partially supported by GNAFA and by the project Analisi Real of MURST.     

T-orthogonally scattered measures and a Radon-Nikodym theorem

Banach space valued T-orthogonally scattered measures admitting a R-N derivative are characterized. One of the corollaries of this theorem improves a theorem of Masani on Hilbert space valued orthogonally scattered measures.