ACTA SCIENTIARUM MATHEMATICARUM (Szeged)

Abstract. As an extension of a result in [1] about the semisimple case, left orders in the sense of Fountain and Gould in rings with local units, minimum condition for principal left ideals, and arbitrary radical are characterized by means of covering every finite subset of them by left orders in the classical sense in rings with identity and with minimum condition for principal left ideals.

AMS Subject Classification (1991): 16P70, 16N40, 16U99

Received October 13, 1993. (Registered under 5566/2009.)

Abstract. A refinement of Á. Szendrei's characterization theorem for simple surjective algebras is given. Moreover, it is shown that every at least three element finite simple nonunary algebra with transitive automorphism group is either functionally complete or affine.

AMS Subject Classification (1991): 08A40

Received January 7, 1994. (Registered under 5567/2009.)

Abstract. It is shown that every finite, simple, strongly Abelian algebra generating a minimal variety is term equivalent to a full matrix power of a $2$-element unary algebra. The proof is based on a classification of reducts of matrix powers of unary algebras.

AMS Subject Classification (1991): 08B05, 08A40, 08A05

Received January 10, 1994 and in revised form March 14, 1994. (Registered under 5568/2009.)

Abstract. A groupoid $(G;\cdot )$ is left uniquely divisible if for each $a,b\in G$ there exists exactly one $x\in G$ with $a\cdot x=b$. A semiloop is a left uniquely divisible groupoid with a right unit. We characterize ideals of semiloops which are kernels of congruences and we prove that the variety of all semiloops has ideal determined congruences.

AMS Subject Classification (1991): 08A30, 08B05, 20N05

Keyword(s): uniquely divisible groupoid, semiloop, ideal, kernel of congruence, ideal determined congruences

Received June 25, 1991 and in revised form November 29, 1993. (Registered under 5569/2009.)

Abstract. The present paper deals with some conditions characterizing consistence in lower continuous strongly dually atomic lattices.

AMS Subject Classification (1991): 06B35, 06C05

Received May 11, 1992 and in revised form January 14, 1994. (Registered under 5570/2009.)

Abstract. A finite distributive sublattice $T$ of a lattice $L$ is said to have the property $P$ if $T$ is contained in a Boolean sublattice of $L$. We consider the class $\Gamma $ of all lattices $L$ each of whose finite distributive sublattices has the property $P$. We prove that every relatively complemented modular lattice $L$ is in $\Gamma $. If, in addition, $L$ is atomic then we characterize subdirect irreducibility of $L$. A weaker result is given for semimodular lattices.

AMS Subject Classification (1991): 06D05, 06C05, 06C10

Received September 25, 1992 and in revised form December 13, 1993. (Registered under 5571/2009.)

Abstract. Let $L$ be a reduced, compactly generated multiplicative lattice in which 1 is compact and every finite product of compact elements is compact. In this setting we study quasiregular lattices (for every compact element $x\in L$, there exists a compact element $y\in L$ with $(0:(0:x))=(0:y))$, regular lattices (every compact element is complemented), and Baer lattices (for every compact element $x$, $(0:x)\vee(0:(0:x))=1)$.

AMS Subject Classification (1991): 06F10, 06E99

Keyword(s): quasiregular lattices, Baer lattices, multiplicative lattices, compact elements

Received October 26, 1992 (Registered under 5572/2009.)

Abstract. For an inverse semigroup $S$, we prove that the translational hull of the closure of the idempotents $\Omega(E_S\omega )$ is isomorphic to the closure of the idempotents $E_{\Omega(S)}\omega $ in the translational hull $\Omega(S)$. This characterization gives a new proof for the fact that the translational hull of an $E\omega $-Clifford semigroup is $E\omega $-Clifford, which was first proved by Reilly [3].

AMS Subject Classification (1991): 20M15, 20M18, 20M12

Received October 1, 1993. (Registered under 5573/2009.)

Abstract. The norm closure of the intermediate orbit of an essentially normal operator acting on an infinite dimensional separable Hilbert space need not contain any normal operator. Two instances of essentially normal operators where the norm closure of the intermediate orbit contains a normal operator are discussed.

AMS Subject Classification (1991): 47A58

Received August 24, 1992 and in revised form October 20, 1993. (Registered under 5574/2009.)

Abstract. Given Hilbert space operators $A$ and $B$, we characterize solvability of the operator equation $A = BJ + G$, where $J$ is a Fredholm operator of prescribed index and $G$ is an operator of prescribed finite rank. In the case when $A$ or $B$ has closed range, we characterize solvability of $A = BJ + K$, with $J$ Fredholm and $K$ compact. More generally, for elements $a$, $b$ of a $C^*$-algebra ${\cal A}$, we study solvability of the equation $a = bj$, $j$ invertible. There are natural majorization and annihilator conditions necessary for solvability, and we show that if ${\cal A}$ is Rickart, then these conditions are also sufficient.

AMS Subject Classification (1991): 47A68, 47A62

Received February 10, 1993. (Registered under 5575/2009.)

Abstract. In this paper the relation between the elements of a von Neumann algebra for which a suitable $L^2$ type norm is preserved by an $\omega $-conditional expectation to a von Neumann subalgebra and the relevant canonical state extension is discussed. The main tool used is a polar decomposition for operators depending on both the von Neumann algebra and a faithful normal state on it.

AMS Subject Classification (1991): 46L50

Received June 7, 1993 and in revised form December 28, 1993. (Registered under 5576/2009.)

Abstract. The author showed in [10] that a certain self-dual subnormal operator is represented as the sum of mutually commutative normal operator and a quasinormal operator and also showed that the converse of this theorem is not true, i.e., a pure operator $S$ which is the sum of mutually commutative normal operator and a quasinormal operator is not necessarily a self-dual subnormal operator, although $S$ is subnormal. In this paper we examine a pure operator $S$ of this type. We observe that this decomposition is unique. Moreover we show a necessary and sufficient condition for a subnormal operator to be of this type. This is an improvement of Brown's theorem [2] concerning quasinormal operators. We also give a necessary and sufficient condition for an operator of this type to be a self-dual subnormal operator.

AMS Subject Classification (1991): 47B20

Received August 11, 1993. (Registered under 5577/2009.)

Abstract. In this paper the discussion of normal spectral approximation in a unital $C^*$-algebra is continued. For the approximation by positive or self-adjoint elements and by positive contractions in the $C^*$-norm as well as in a second norm the extreme points of the convex set of all approximants are studied. The main purpose of this paper is to develop sufficient conditions for an approximant to be such an extreme point. As an application many extreme points for the approximation of normal elements are constructed. Moreover for the approximation in the $C^*$-algebra of all bounded linear operators on a complex Hilbert space the number of extreme points is completely determined.

AMS Subject Classification (1991): 47A58

Received August 30, 1993. (Registered under 5578/2009.)

Abstract. The main purpose is to give necessary and sufficient conditions for a metrizable topological vector space with a generalized form of Hahn--Banach extension property to be locally convex and describe the Mackey envelope of such spaces.

AMS Subject Classification (1991): 46A16, 46A22

Received September 23, 1993. (Registered under 5579/2009.)

Abstract. Toeplitz operators $T_\phi ^M$ are defined on invariant subspaces $M$ of an arbitrary uniform algebra $A$. We give a necessary and sufficient condition for uniform invertibility of $M_\phi ^M$ with respect to some family $\cal F$ of invariant subspaces $M$. This condition is the same as a classical one in case $A$ is the disc algebra. In special uniform algebras we can choose a small family, in fact, if $A$ is a disc algebra then $\cal F$ can be a single set. Then this generalizes the Widom-Devinatz Theorem. As an application, we study a $\cal F$-union of spectra of $\{T_\phi ^M;M\in{\cal F}\} $.

AMS Subject Classification (1991): 47B35, 47A10

Keyword(s): Toeplitz operator, invertible, uniform algebra, spectrum

Received November 10, 1993 and in revised form March 21, 1994. (Registered under 5580/2009.)

Abstract. In this note we discuss the Taylor--Weyl spectrum of a commuting pair of operators acting on a Hilbert space and compare the Taylor--Weyl spectrum with a joint Weyl spectrum due to Chō and Takaguchi.

AMS Subject Classification (1991): 47A13, 47A10, 47A53

Received December 20, 1993 and in revised form March 25, 1994. (Registered under 5581/2009.)

Abstract. By giving the sufficient and necessary condition of URED for Orlicz sequence spaces endowed with Orlicz norm, we solve the open problem posed in [3].

AMS Subject Classification (1991): 46E30

Keyword(s): Orlicz sequence space, uniform rotundity in every direction

Received January 6, 1992 and in revised from May 21, 1993. (Registered under 5582/2009.)

Abstract. We prove two general theorems. Applying these results we also prove certain ``converses'' of four theorems proved by the first author earlier. It is also verified that the new results, in general, can not be substantially improved.

AMS Subject Classification (1991): 40A05, 40A10, 40A99

Received April 19, 1993. (Registered under 5583/2009.)

Abstract. If we consider the class of (quasi-)monotone functions, then Hardy's classical inequalities hold also in the reversed direction for {\it some} constants. In this paper we present several proofs of these reversed Hardy's inequalities, which, in particular, give the best possible constants in all cases. The results obtained may be regarded as a unification and generalization of some results recently obtained in [2], [4] and [11].

AMS Subject Classification (1991): 26D15, 26D20

Received April 19, 1993. (Registered under 5584/2009.)

Abstract. Applying a recently developed axiomatic theory of non-absolutely convergent integrals we define a relatively simple and well-behaved integral over $n$-dimensional compact intervals for which the divergence theorem holds in its presently most general form.

AMS Subject Classification (1991): 26A39, 26B20

Received April 23, 1993. (Registered under 5585/2009.)

Abstract. We show that if a function $f$ defined on the real line ${\bf R}$ belongs to the Besov space $B^{1/p}_{p,1},$ where $1\le p< \infty,$ then its Dirichlet integral $s_T(f,x)$ converges uniformly as $T\to\infty.$ In the case where $1\le p\le2$, we even prove that $s_T(f,x)$ converges absolutely. It follows easily that any function $f$ in $B^{1/p}_{p,1}, 1\le p\le2,$ is a Fourier transform on $L^1({\bf R}).$ The counterparts of our results were proved by A. M. Garsia (1976) for functions defined on the unit circle ${\bf T}.$ Our proofs are basically different from those given there.

AMS Subject Classification (1991): 42A38, 46E35, 41A17

Keyword(s): Besov space, modulus of continuity, Fourier transform, Dirichlet integral, Riesz mean, Hardy-Littlewood inequality, L^1., multiplier of Fourier transform on

Received September 15, 1993. (Registered under 5586/2009.)

Abstract. In this paper we define and study exposed points of subsets in a smooth Riemannian manifold without focal points. Sectional curvature at exposed points is studied. Proofs of well-known theorems are given as corollaries of the established results.

AMS Subject Classification (1991): 53C42

Keyword(s): Manifolds without focal points, convex subsets, geodesics, sectional curvature

Received June 10, 1993 and in revised form November 24, 1993. (Registered under 5588/2009.)