The purpose of this paper is to consider, in rings with involution, the structure of those subrings which are invariant under Lie commutation with [K, K]. Our goal is to find conditions which force such subrings to contain a noncentral ideal of the ring. Of course, the subring itself may lie in the center. Orders in 4 × 4 matrix rings over fields are known to provide examples of invariant subrings which are not central and contain no ideal (see [1, 40] or [5]). Except for subdirect products of these two kinds of “counter-examples”, we show that in semi-prime rings, invariant subrings do contain noncentral ideals. This generalizes work of Herstein [4] in two directions by considering semi-prime rings rather than simple rings, and by using [K, K] instead of K.

In this paper we investigate methods for estimating the dimension of the null space of operators in a finite W* algebra, the dimension being measured by the trace τ. For the most part we are concerned with operators A which are a finite linear combination of orthogonal unitaries. We give various results which show how certain information about the unitaries and the coefficients can be utilized to derive an upper bound for τ(NA) where NA is the null space of A.

Let X be an infinite topological space, let 𝔫 be an infinite cardinal number with 𝔫 ≦ |X|. The basic problem in this paper is to find the number of closed sets in X of cardinality 𝔫. A complete answer to this question for the class of metrizable spaces has been given by A. H. Stone in [31], where he proves the following result. Let X be an infinite metrizable space of weight 𝔪, let 𝔫 ≦ |X|.

It is known that the Pn-tegral as originally defined is not additive on abutting intervals. This paper offers a slight modification in the definition of the integral and develops necessary and sufficient conditions for the integral to be additive.

Let Tn denote a tree with n nodes that is rooted at node r. (For definitions not given here see [4] or [10].) The altitude of a node u in Tn is the distance α = α (u, Tn) between r and u in Tn. The width of Tn at altitude is the number Wk = Wk(Tn) of nodes at altitude in Tn, where = 0, 1, …

If UR is a real subspace of a finite dimensional vector space VC over the field C of complex numbers, then there exists a basis ﹛e1, … , en﹜ of VG such that

In [10], Witt laid the foundation for the study of quadratic forms over fields. Suppose Q is a quadratic form defined on a finite dimensional vector space V over a field of characteristic not equal to 2. Witt showed that non-zero vectors x and y in V satisfying Q(x) = Q(y) can be mapped into each other via an isometry of the vector space V. More generally, if τ : W ⟶ W’ is an isometry between subspaces of V, then τ extends to an isometry ϕ of V.

All rings in this paper are assumed to be commutative with identity, and the undefined terminology is the same as that in [3].

In 1956, in an important paper [2], M. Nagata constructed an example which showed (among other things): (i) a maximal chain of prime ideals in an integral extension domain R' of a local domain (R, M) need not contract in R to a maximal chain of prime ideals; and, (ii) a prime ideal P in R' may be such that height P < height P ∩ R. In his example, Rf was the integral closure of R and had two maximal ideals. In this paper, by using Nagata's example, we show that there exists a finite local integral extension domain of D = R[X](M,X) for which (i) and (ii) hold (see (2.8.1) and (2.10)).

In an effort to unify the arithmetic of cardinal and ordinal numbers, Garrett Birkhoff [2; 3; 4; 5] (cf. [6]) defined several operations on partially ordered sets of which at least one, (cardinal) exponentiation, is of considerable independent interest: for partially ordered sets P and Q let PQ denote the set of all order-preserving maps of Q to P partially ordered by f ≦ g if and only if f(x) ≦ g(x) for each x ∈ Q.

Generalized weighing (GW) matrices are orthogonal matrices whose nonzero entries are roots of unity. Several families are constructed with the aid of finite geometries which include as special cases interesting examples of conference matrices and weighing matrices.

Throughout this paper, it is assumed that A is the complete, equicharacteristic, local ring of an algebraic curve at a one-branch singularity whose residue field is algebraically closed and contained in A. Hence, the domain A is dominated by only one valuation ring in its quotient held F, and if t is a uniformizing parameter, then the integral closure of A in F, denoted by Ā, is [[t]].

The main result of this paper is the following:

THEOREM. Every finitely generated orthomodular lattice L with finitely manymaximal Boolean subalgebras (blocks) is finite.

If L has one block only, our theorem reduces to the well-known fact that every finitely generated Boolean algebra is finite. On the other hand, it is known that a finitely generated orthomodular lattice without any further restrictions can be infinite. In fact, in [2] we constructed an orthomodular lattice which is generated by a three-element set with two comparable elements, has infinitely many blocks and contains an infinite chain.

Let G1 and G2 be compact groups and T : Lp(G1) → LP(G2) (1 ≦ P ≦ ∞ ) be an algebra homomorphism. If || T || ≦ 1 and T is either a monomorphism of an epimorphism then T can in many cases be explicitly characterized (see [4 ; 8 ; 9 ; 11 ; 13 ; 14]). Excluding p = 2, the outstanding cases are 1 < p < ∞ for monomorphisms and 2 < p < ∞ for epimorphisms (cf. [14]). One aim of the present note is to complete this work. We also consider the problem of extending these results in some form to homomorphisms on ideals of group algebras; the only known result in this area is for abelian groups [3].

Notation. For S a subset of the extended plane C* = C ∪ (∞ ) let S0 be its interior, its closure in C* and its boundary. H(S) and M(S) will denote the sets of all restrictions to S of functions which are, on a neighborhood of S, holomorphic or meromorphic, respectively. A (S) will stand for the set of functions from S to C* which are continuous on S and whose restriction to S° are holomorphic.

This paper studies the construction of Frobenius algebras. We begin with a description of when a graded -algebra has a Frobenius algebra as a homomorphic image. We then turn to the question of actual constructions of Frobenius algebras. We give a, method for constructing Frobenius algebras as factor rings of special tensor algebras. Since the representation theory of special tensor algebras has been studied intensively ([6], see also [2; 3; 4]), our results permit the construction of Frobenius algebras which have representations with prescribed properties. Such constructions were successfully used in [9].

There are three lattice-theoretic properties that are generally used to open a discussion on sublattices of a free lattice:

(W) for all a, b, c, d, a Λ b ≦ c V d implies a A b ≦ c, a Λ b g rf, a ≦cVd,orb ≦ c V d;

(SDv) for all a, b, c, a V b = a V c implies a V b = a V (b Λ c ) ;

(SDΛ) for all a, b, c, a Λ b = a Λ c implies a Λ b = a Λ (b V c).

Many results concerning the automatic continuity of linear functions intertwining continuous linear operators on Banach spaces have been obtained, chiefly by B. E. Johnson and A. M. Sinclair [1; 2; 3; 5]. The purpose of this paper is essentially to extend this automatic continuity theory to the situation of Fréchet spaces. Our motive is partly to be able to handle the more general situation, since for example, questions about Fréchet spaces and LF spaces arise in connection with the functional calculus.

In 1964, J. J. Charatonik in [1] introduced a new class of mappings, the so-called confluent mappings, which comprises the classes of open, monotone and quasi-interior mappings (see [20]). In 1966, A. Lelek started working on confluent mappings with applications to continua theory (see [7]). He introduced two other classes of mappings, the so-called weakly confluent and pseudo confluent mappings, he proved the invariance of rational continua under open, monotone and quasi-interior mappings and he asked about their invariance under confluent mappings. In 1976, E. D. Tymchatyn gave an example of a confluent mapping, which does not preserve the rationality of a curve (see [18]).

During the early nineteenth century much effort was spent on attempts to find a “universal comparison test“: i.e., a sequence in l1 that dominates every other member of l1. The nonexistence of such a series converging at a minimal rate was demonstrated by Abel, et al. [1; 4; 7; 9, pp. 298-304].

A conference matrix of order n is a square matrix C with zeros on the diagonal and ±1 elsewhere, which satisfies the orthogonality condition CCT = (n — 1)I. If in addition C is symmetric, C =CT, then its order n is congruent to 2 modulo 4 (see [5]). Symmetric conference matrices (C) are related to several important combinatorial configurations such as regular two-graphs, equiangular lines, Hadamard matrices and balanced incomplete block designs [1; 5; and 7, pp. 293-400]. We shall require several definitions.