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If M is a commutative W*-algebra of operators and if ReM is the Dedekind complete Riesz space of self-adjoint elements of M, then it is shown that the set of densely defined self-adjoint transformations affiliated with ReM is a Dedekind complete, laterally complete Riesz algebra containing ReM as an order dense ideal. The Riesz algebra of densely defined orthomorphisms on ReM is shown to coincide with , and via the vector lattice Randon-Nikodym theorem of Luxemburg and Schep, it is shown that the lateral completion of ReM may be identified with the extended order dual of ReM.
Some results on fixed points of certain involutions in Banach spaces have been obtained, and whence a few coincidence theorems are also derived. These are indeed generalization of previously known results due to Browder, Goebel-Zlotkiewicz and Iséki. Illustrative examples are also given.
The author presents a proof that a partially ordered strongly regular ring S which has the additional property that the square of each member of S is greater than or equal to zero cannot have nontrivial positive derivations.
Vertices u0, u1, …, uk−1 of a graph X are mutually pseudo-similar if X − u0 ≌ X − u1 ≌ … ≌ X − uk−1, but no two of the vertices are related by an automorphism of X. We describe a method for constructing graphs with a set of k≥2 mutually pseudo-similar vertices, using a group with a special subgroup. We show that in all graphs with pseudo-similar vertices, the vertices are pseudo-similar due to the action of a group on the cosets of some subgroup.
In this note semi inner product spaces over quaternions are studied. We investigate the properties of complex strict convexity over C, left and right Q-complex strict convexity over Q, and left and right quaternion strict convexity. Finally we discuss the permanence properties with respect to formation of products.
The special Denjoy-Bochner integral (the D*B-integral) which are generalisations of Lebesgue-Bochner integral are discussed in [7, 6, 5]. Just as the concept of numerical almost periodicity was extended by Burkill  to numerically valued D*- or D-integrable function, we extend the concept of almost periodicity for Banach valued function to Banach valued D*B-integrable function. For this purpose we introduce as in  a distance in the space of all D*B-integrable functions with respect to which the D*B-almost periodicity is defined. It is shown that the D*B-almost periodicity shares many of the known properties of the almost periodic Banach valued function [1, 4].
Let E be an Archimedean Riesz space and let Orth∞(E) be the f-algebra consisting of all extended orthomorphisms on E, that is, of all order bounded linear operators T:D→E, with D an order dense ideal in E, such that T(B∩D) ⊆ B for every band B in E. We give conditions on E and on a Riesz subspace F of E insuring that every T ∈ Orth∞(F) can be extended to some ∈ Orth∞(E), and we also consider the problem of inversing an extended orthomorphism on its support. The same problems are also studied in the case of σ-orthomorphisms, that is, extended orthomorphisms with a super order dense domain. Furthermore, some applications are given.
The problem is the reconstruction of the shape of an object, whose shell is a surface star-shaped with respect to a point 0, from the knowledge of the volume of every “half-object” obtained by taking any plane through 0. Conditions for the existence and uniqueness of the solution are given. The main result consists in showing that any uniform a-priori bound on the mean curvature of the shell reestablishes continuous dependence on the data for bodies satisfying a certain symmetry condition.
A new arrow notation is used to describe biordered sets. Biordered sets are characterized as biordered subsets of the partial algebras formed by the idempotents of semigroups. Thus it can be shown that in the free semigroup on a biordered set factored out by the equations of the biordered set there is no collapse of idempotents and no new arrows.
We find the atoms of certain subclasses of varieties of finite semigroups and the corresponding varieties of languages. For example we give a new description of languages whose syntactic monoids are R-trivial and idempotent. We also describe the least variety containing all commutative semigroups and at least one non-commutative semigroup. Finally we extend to varieties of finite semigroups some classical results about semilattice decomposition of semigroups.
Let f and g denote immersions of the n-manifolds M and N, respectively, in Rn+1. We say that f is athwart to g if f(M) and g(N)m have no tangent hyperplane in common. In this paper necessary conditions for athwartness are obtained.