Sur la Topologiem-Convexe d’une algèbre localementA-convexe

In a locallyA-convex algebra (E, τ) we consider the associatedm-convex topologym(τ). We show that the completion ofE with respect tom(τ) is always a locallyA-convex algebra contained in the complete locally convex space obtained from (E, τ). The topologym(τ) is also used to characterize locally boundedly multiplicatively convex algebras among locallyA-convex ones.     

Sur la Topologie m -Convexe d’une algèbre localement A -convexe

In a locallyA-convex algebra (E, τ) we consider the associatedm-convex topologym(τ). We show that the completion ofE with respect tom(τ) is always a locallyA-convex algebra contained in the complete locally convex space obtained from (E, τ). The topologym(τ) is also used to characterize locally boundedly multiplicatively convex algebras among locallyA-convex ones.

     

Boundedness of characters in locally a-convex algebras

We characterize advertibly complete topological algebras among those whose completions are algebras. We show that the set of non-zero characters in a commutative, advertibly complete,M-complete locallyA-convex algebra is equibounded whenever every element of the algebra is bounded.     

Topologiesm-Convexes dans les algebres a-convexes

We endow anyA-convex algebra (E, τ) with a multiplicatively-convex topologyM(τ) finer than τ. We then show that this is the weakestm-convex topology finer than τ. We are then led to consider two classes ofA-convex algebras examples and (or) characterizations of which are given.     

Operator Jensen＇s Inequality on C＊-algebras

A real continuous function which is defined on an interval is said to beA-convex if it is convex on the set of self-adjoint elements,with spectra in the interval,in all matrix algebras of the unital C-algebra A.We give a general formation of Jensen’s inequality for A-convex functions.     

Calcul fonctionnel harmonique dans les algèbres localementp-convexes

We define and study a harmonic functional calculus in locallyp-convex algebras in the case with a linear involution. Some applications are also provided.

     

Algèbres de Sobolev Généralisées

We examine the structure of a generalized Sobolev spaces. We also exhibit a class of locally uniformlyA-convex algebras, the unitization of which are not the same type.

     

Pseudo-Banach structure in locally convex algebras

We give a positive answer, in different settings, to a question of Oudadess (Q. A. Gen. Top., 13 (1995) 21–24) concerning the pseudo-Banach structure of a locallym-convex algebra whose elements are all bounded.     

On algebras of holomorphic functions of a given type

We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert–Schmidt bounded type, are locally m-convex Fréchet algebras. We prove that the spectrum of these algebras has a natural analytic structure, which we use to characterize the envelope of holomorphy. We also show a Cartan–Thullen type theorem.     

Complex convexity and monotonicity in quasi-Banach lattices

In this paper we study the monotonicity and convexity properties in quasi-Banach lattices. We establish relationship between uniform monotonicity, uniform ?-convexity, H-and PL-convexity. We show that if the quasi-Banach lattice E has α-convexity constant one for some 0 < α < ∞, then the following are equivalent: (i) E is uniformly PL-convex; (ii) E is uniformly monotone; and (iii) E is uniformly ?-convex. In particular, it is shown that if E has α-convexity constant one for some 0 < α < ∞ and if E is uniformly ?-convex of power type then it is uniformly H-convex of power type. The relations between concavity, convexity and monotonicity are also shown so that the Maurey-Pisier type theorem in a quasi-Banach lattice is proved.Finally we study the lifting property of uniform PL-convexity: if E is a quasi-Köthe function space with α-convexity constant one and X is a continuously quasi-normed space, then it is shown that the quasi-normed Köthe-Bochner function space E(X) is uniformly PL-convex if and only if both E and X are uniformly PL-convex.     

Character Connes amenability of dual Banach algebras

We study the notion of character Connes amenability of dual Banach algebras and show that if A is an Arens regular Banach algebra, then A^** is character Connes amenable if and only if A is character amenable, which will resolve positively Runde’s problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.     

Homotopy algebras and the inverse of the normalization functor

In this paper, we investigate multiplicative properties of the classical Dold-Kan correspondence. The inverse of the normalization functor maps commutative differential graded algebras to E_∞-algebras. We prove that it in fact sends algebras over arbitrary differential graded E_∞-operads to E_∞-algebras in simplicial modules and is part of a Quillen adjunction. More generally, this inverse maps homotopy algebras to weak homotopy algebras. We prove the corresponding dual results for algebras under the conormalization, and for coalgebra structures under the normalization resp. the inverse of the conormalization.     

Generalizations of the Sweedler Dual

As left adjoint to the dual algebra functor, Sweedler’s finite dual construction is an important tool in the theory of Hopf algebras over a field. We show in this note that the left adjoint to the dual algebra functor, which exists over arbitrary rings, shares a number of properties with the finite dual. Nonetheless the requirement that it should map Hopf algebras to Hopf algebras needs the extra assumption that this left adjoint should map an algebra into its linear dual. We identify a condition guaranteeing that Sweedler’s construction works when generalized to noetherian commutative rings. We establish the following two apparently previously unnoticed dual adjunctions: For every commutative ring R the left adjoint of the dual algebra functor on the category of R-bialgebras has a right adjoint. This dual adjunction can be restricted to a dual adjunction on the category of Hopf R-algebras, provided that R is noetherian and absolutely flat.     

Retractions and Gorenstein Homological Properties

We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for example, being Gorenstein algebras or CM-free algebras). We apply the results to Nakayama algebras. It turns out that for a connected Nakayama algebra A, there exists a connected self-injective Nakayama algebra A′ such that there is a sequence of left retractions linking A to A′; in particular, the singularity category of A is triangle equivalent to the stable category of A′. We classify connected Nakayama algebras with at most three simple modules according to Gorenstein homological properties.     

A note onm-convex and pseudo-Banach structures

We prove the functional boundedness of any commutative and completem-convex algebra every element of which is bounded. Moreover such algebras are pseudo-Banach if, and only if, the inverse mapx→x ^?1 is Mackey continuous. Other questions are considered.     

On functional representation of normed algebras

Let A be a commutative algebra over complex numbers with a space norm ‖⋅‖ making the multiplication on A separately continuous. We will study the Gelfand representation of this type of normed algebra. In particular, we look at the cases where the standard Gelfand representation (i.e., the use of supremum-norm on the Gelfand transform algebra ) gives different properties from the original algebra (A,‖⋅‖). We show that there are even Banach algebras for which this type of difficulty may happen. We will provide with some weighted supremum-norm and by using these weights we can avoid the difficulties mentioned above. For the definition of these weights we adopt the ideas of Cochran represented in [A.C. Cochran, Representation of A-convex algebras, Proc. Amer. Math. Soc. 30 (1973) 473-479].     

Generalized Lie derivations on skew elements of prime algebras with involution

We characterize generalized Lie derivations on skew elements of prime algebras A with involution, provided that A does not satisfy polynomial identities of low degree. The analogous result for matrix algebras is also described.     

Generalized Lie derivations on triangular algebras

Let A be a unital algebra and let M be a unitary A-bimodule. We consider generalized Lie derivations mapping from A to M. Our results are applied to triangular algebras, in particular to nest algebras and (block) upper triangular matrix algebras. We prove that under certain conditions each generalized Lie derivation of a triangular algebra A is the sum of a generalized derivation and a central map which vanishes on all commutators of A.     

The subalgebra lattice of a finite algebra

The aim of this paper is to characterize pairs (L, A), where L is a finite lattice and A a finite algebra, such that the subalgebra lattice of A is isomorphic to L. Next, necessary and sufficient conditions are found for pairs of finite algebras (of possibly distinct types) to have isomorphic subalgebra lattices. Both of these characterizations are particularly simple in the case of distributive subalgebra lattices. We do not restrict our attention to total algebras only, but we consider the more general case of partial algebras. Moreover, we use connections between algebras and hypergraphs to solve these problems.     

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.