On universal Korovkin sets w.r.t. positive spectral contractions

We prove that there exists a finite universal Korovkin set with respect to positive spectral contraction operators for a Segal algebra on a locally compact abelian groupG if and only ifĜ is finite dimensional separable metrizable space.     

On existence of finite universal Korovkin sets in the centre of group algebras

In this paper, we characterize compact groupsG as well as connected central topological groupsG for which the centreZ(L ¹(G)) admits a finite universal Korovkin set. Also we prove that ifG is a non-connected central topological group which has a compact open normal subgroupK such thatG=KZ, thenZ(L ¹(G)) admits a finite universal Korovkin set if is a finite-dimensional separable metric space or equivalentlyG is separable metrizable andG/K has finite torsion-free rank.     

A characterization of a class of [Z] groups via Korovkin theory

We characterize all the central topological groupsG for which the centreZ(L ¹(G)) of the group algebra admits a finite universal Korovkin set. It is proved thatZ(L ¹(G)) has a finite universal Korovkin set iffĜ is a finite dimensional, separable metric space. This is equivalent to the fact thatG is separable, metrizable andG/K has finite torsion free rank, whereK is a compact open normal subgroup of certain direct summand ofG.     

Compact Hypergroup Characters and Finite Universal Korovkin Sets in L 1-Hypergroup Algebra

Let K be a compact hypergroup.We investigate Trig(K), the linear span of coordinate functions of the irreducible representations of K. Contrary to the group case, Trig(K) endowed with the usual multiplication does not bear an algebra structure, but it has a natural normed algebra structure when it inherits the convolution from , the algebra of all bounded Radon measures. We characterize the center of the algebras , L _p (K) and Trig(K) respectively, and consequently we obtain, for a certain class of hypergroups, the correspondence between the structure space of the center of L ₁(K) and the center of Trig(K). As an application we study the existence of a finite universal Korovkin set w.r.t. positive operators in the center of L ₁(K), in particular in L ₁(K), whenever K is commutative. The author was partially supported by the Romanian Academy under the grant No. 22/2007.     

Stability results for approximation by positive definite functions on SO(3)

We consider interpolation methods defined by positive definite functions on a locally compact group G. Estimates for the smallest and largest eigenvalue of the interpolation matrix in terms of the localization of the positive definite function on G are presented, and we provide a method to get positive definite functions explicitly on compact semisimple Lie groups. Finally, we apply our results to construct well-localized positive definite basis functions having nice stability properties on the rotation group SO(3).     

Universal free G-spaces

For a compact Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free G-spaces. We show that such a universal free G-space cannot be compact.     

Approximate Identities for Ideals of Segal Algebras on A Compact Group

We show that every closed ideal of a Segal algebra on a compact group admits a central approximate identity which has the property, called condition (U), that the induced multiplication operators converge to the identity operator uniformly on compact sets of the ideal. This result extends a known one due to H. Reiter who has considered the problem under the condition that the Segal algebra is symmetric. We prove further that a closed right ideal of a Segal algebra on a compact group admits a left approximate identity satisfying condition (U) if and only if it is approximately complemented as a subspace of the Segal algebra; if in addition the Segal algebra is symmetric, then a closed left ideal admits a right approximate identity satisfying condition (U) if and only if it is approximately complemented.     

Matrix Summability and Positive Linear Operators

In the present paper we prove a Korovkin type approximation theorem for a sequence of positive linear operators acting from a weighted space C_ρ1 into a weighted space B_ρ2 with the use of a matrix summability method which includes both convergence and almost convergence. We also study the rates of convergence of these operators.     

Continuity of epimorphisms and derivations on vector-valued group algebras

This paper deals with the automatic continuity theory for the convolution algebra of all Bochner integrable functions from a locally compact abelian group G into an arbitrary unital complex Banach algebra A. For non-compact G, it is shown that all epimorphisms and all derivations on this vector-valued group algebra are necessarily continuous while for compact G, such results depend heavily on the automatic continuity properties of the range algebra a. Dedicated to Heinz Konig on the occasion of his 65th birthday Research supported by Grant SNF 11-1015 from the Danish Science Research Council.     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Biflatness and biprojectivity of the Fourier algebra

We show that the biflatness—in the sense of A. Ya. Helemskiĭ—of the Fourier algebra A(G) of a locally compact group G forces G to either have an abelian subgroup of finite index or to be non-amenable without containing as a closed subgroup. An analogous dichotomy is obtained for biprojectivity. Received: 4 August 2008     

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.Research supported by NSERC under grant no. 90749-00.Research supported by NSERC under grant no. 227043-00.     

The interior of the Šilov boundary of M(G) is trivial

Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the ?ilov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.     

Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group. S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.     

On the index of nonlocal elliptic operators for compact Lie groups

We consider a class of nonlocal operators associated with a compact Lie group G acting on a smooth manifold. A notion of symbol of such operators is introduced and an index formula for elliptic elements is obtained. The symbol in this situation is an element of a noncommutative algebra (crossed product by G) and to obtain an index formula, we define the Chern character for this algebra in the framework of noncommutative geometry.     

Stancu‐type generalization of Dunkl analogue of Szász–Kantorovich operators

The purpose of the paper is to introduce Stancu‐type linear positive operators generated by Dunkl generalization of exponential function. We present approximation properties with the help of well‐known Korovkin‐type theorem and weighted Korovkin‐type theorem and also acquire the rate of convergence in terms of classical modulus of continuity, the class of Lipschitz functions, Peetre's K‐functional, and second‐order modulus of continuity by Dunkl analogue of Szász operators. Copyright © 2015 John Wiley & Sons, Ltd.     

On the semidistributivity of elements in weak congruence lattices of algebras and groups

Weak congruence lattices and semidistributive congruence lattices are both recent topics in universal algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included. This research of the second and third authors was partially supported by Serbian Ministry of Science and Environment, Grant No. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, grant ”Lattice methods and applications”.     

The Baire set fixing property and uniform continuity in topological groups

Every topological group that satisfies theM-Baire set fixing property and is left separable isdense in a retract of a product of metrizable groups. Since compact groups satisfy MBSFP, it is a corollary that compact groups are dyadic spaces (Kuz'minov). IfG satisfies MBSFP, then the left uniformity is separable if and only ifG satisfies the countable chain condition. Dense subgroups of left separable MBSFP groups satisfy MBSFP, and any product of left separable MBSFP groups satisfies MBSFP. A left separable group satisfies MBSFP if and only if the left uniformity is weak generated by natural maps to metrizable quotient groups. If for each ,A is a denseC-embedded set in the left separable MBSFP groupG , then the real-compactification A = A . It is a corollary that the product of pseudocompact groups is pseudocompact (Comfort and Ross). A locally compact group satisfies MBSFP if and only if it is metrizable or sigma compact (Ross and Stromberg, forG Abelian).     

A class of Banach algebras whose duals have the Radon-Nikodym property

Let A be a complex, commutative Banach algebra and let M_A be the structure space of A. Assume that there exists a continuous homomorphism h : L¹(G) → A with dense range, where L¹(G) is the group algebra of a locally compact abelian group G. The main results of this paper can be summarized as follows: (a) If the dual space A* has the Radon-Nikodym property, then M_A is scattered (i.e., it has no nonempty perfect subset) and . (b) If the algebra A has an identity, then the space A* has the Radon-Nikodym property if and only if . Furthermore, any of these conditions implies that M_A is scattered. Several applications are given. Received: 29 September 2005