Third power associative composition algebras

Classical composition algebras, with a unit element, are well-known and can be obtained by means of the Cayley-Dickson doubling process. If the condition on the existence of unit element is dropped, many new algebras arise. However, it is shown in this paper that if such a weak condition as the associativity of third powers of any element is imposed, only the known flexible composition algebras appear. Partially supported by the DGICYT (PS 90-0129) and by the DGA (PCB-6/91) Supported by a grant from the ‘Plan de Formación del Personal Investigador’ (DGICYT, Spain) This article was processed by the authors using the Springer-Verlag TEX P Jour1g macro package 1991.     

Linear Representation of Medial-Like Algebras

In this paper we characterize the paramedial, comedial, and coparamedial pairs of quasigroup operations and paramedial, comedial, and coparamedial algebras with the quasigroup operations. Then we characterize paramedial, comedial, and coparamedial algebras with a unit element and an idempotent regular element.     

Igusa–Todorov Endomorphism Algebras

We study polynomial identities of algebras with adjoined external unit. For a wide class of algebras we prove that adjoining external unit element leads to increasing of PI-exponent precisely to 1. We also show that any real number from the interval [2,3] can be realized as PI-exponent of some unital algebra.     

Composition Algebras with Large Derivation Algebras

The finite dimensional flexible composition algebras include the Hurwitz algebras (composition algebras with unit element), but also other interesting classes of algebras: the para-Hurwitz and the Okubo algebras. The above mentioned algebras present many symmetries, and this is reflected in their large derivation algebras. In the present paper we study the opposite question: What can be said about the composition algebras if we have some information about their derivation algebras? Our main result is the classification of all the composition algebras with such large derivation algebras.     

Harmonic functions of maximal growth: invertibility and cyclicity in Bergman spaces

In the theory of commutative Banach algebras with unit, an element generates a dense ideal if and only if it is invertible, in which case its Gelfand transform has no zeros, and the ideal it generates is the whole algebra. With varying degrees of success, efforts have been made to extend the validity of this result beyond the context of Banach algebras. For instance, for the Hardy space on the unit disk, it is known that all invertible elements are cyclic (an element is cyclic if its polynomial multiples are dense), but cyclic elements need not be invertible. In this paper, we supply examples of functions in the Bergman and uniform Bergman spaces on the unit disk which are invertible, but not cyclic. This answers in the negative questions raised by Shapiro, Nikolskii, Shields, Korenblum, Brown, and Frankfurt.

     

Novikov–Poisson Algebras

We classify Novikov–Poisson algebras whose Novikov algebras are simple with an idempotent element. Moreover, a class of simple Novikov algebras without idempotent elements is constructed through Novikov–Poisson algebras. Certain new simple Lie superalgebras induced by Novikov–Poisson algebras are introduced.     

Dimension theory for generalized effect algebras

In this paper, we define and study dimension generalized effect algebras (DGEAs), i.e., Dedekind orthocomplete and centrally orthocomplete generalized effect algebras equipped with a dimension equivalence relation. Our theory is a bona fide generalization of the theory of dimension effect algebras (DEAs), i.e., it is formulated so that if a DGEA happens to be an effect algebra (i.e., it has a unit element), then it is a DEA. We prove that a DGEA decomposes into type I, II, and III DGEAS in a manner analogous to the type I/II/III decomposition of a DEA.     

Idempotent et cohomologie de Hochschild

Any idempotent element e of an (associative) algebra T defines an algebra $A=eTe$ with unit e. We show that the morphism which compares their Hochschild cohomology algebras is a Gerstenhaber algebras morphism. Moreover, this morphism factorizes through the cohomological algebras of many triangular algebras. To cite this article: B. Bendiffalah, D. Guin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Algebras Defined from Ordered Sets and the Varieties they Generate

We investigate ways of representing ordered sets as algebras and how the order relation is reflected in the algebraic properties of the variety (equational class) generated by these algebras. In particular we consider two different but related methods for constructing an algebra with one binary operation from an arbitrary ordered set with a top element. The two varieties generated by all these algebras are shown to be well-behaved in that they are locally finite, finitely based, and have an equationally definable order relation. We exhibit a bijection between the subdirectly irreducible algebras in each variety and the class of all ordered sets with top element. We determine the structure and cardinality of the free algebra on n-free generators and provide sharp bounds on the number of n-generated algebras in each variety. These enumeration results involve the number of quasi-orders on an n-element set.     

Some New Class of Absolute Valued Algebras with Left Unit

Let A be an absolute valued algebra with left unit. We prove that if A contains a nonzero central element, then A is finite dimensional and is isomorphic to \mathbb R, \mathbb C{\mathbb {R}, \mathbb {C}} or new classes of four and eight–dimensional absolute valued algebras with left unit. This is more general than those results in [2] and [3].     

Ordered *-Algebras with an Order Unit

We develop a theory of ordered *-algebras with an order unit. These are complex algebras equipped with a conjugate-linear vector space involution whose hermitian elements form an ordered real algebra with an order unit but not necessarily with a multiplicative identity. Our main result is a representation theorem for ordered *-algebras with an order unit. This can be viewed as the complex and the non-unital version of the representation theorem of Stone and Kadison for ordered real algebras containing an order unit which is a multiplicative identity. A key role in our approach is played by certain weighted function algebras.     

Extreme points in triangular UHF algebras

We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points.

     

MV-algebras: a variety for magnitudes with archimedean units

Chang’s MV-algebras, on the one hand, are the algebras of the infinite-valued Łukasiewicz calculus and, on the other hand, are categorically equivalent to abelian lattice-ordered groups with a distinguished strong unit, for short, unital ℓ-groups. The latter are a modern mathematization of the time-honored euclidean magnitudes with an archimedean unit. While for magnitudes the unit is no less important than the zero element, its archimedean property is not even definable in first-order logic. This gives added interest to the equivalent representation of unital ℓ-groups via the equational class of MV-algebras. In this paper we survey several applications of this equivalence, and various properties of the variety of MV-algebras.Dedicated to the Memory of Wim BlokReceived August 26, 2003; accepted in final form October 3, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Adjoining units to residuated Boolean algebras

We consider a varietyV ofr-algebras, — residuated Boolean algebras, — and ask under what conditions a memberA ofV can be embedded in a memberA' having a unit element. The answer, although quite simple, is somewhat surprising for two reasons. First, to a large extent the answer is independent of the varietyV, as long asV is closed under canonical extensions. This is so because if any extension ofA has a unit, then the canonical extension has a unit. The second surprise is that, for varietiesV closed under canonical extensions, the members for which this extension has a unit form a subvariety with a very simple equational basis relatively toV. Applied to the variety of all relation algebras, this latter result solves a problem of long standing due toA. Tarski. This problem was solved independently by H. Andréka and I. Németi.Presented by J. Sichler.     

Primitive Elements and Automorphisms of Free Algebras of Schreier Varieties

In this article, we review results on primitive elements of free algebras of main types of Schreier varieties of algebras. A variety of linear algebras over a field is Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. A system of elements of a free algebra is primitive if it is a subset of some set of free generators of this algebra. We consider free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. We present matrix criteria for systems of elements of elements. Primitive elements distinguish automorphisms: endomorphisms sending primitive elements to primitive elements are automorphisms. We give a series of examples of almost primitive elements (an element of a free algebra is almost primitive if it is not a primitive element of the whole algebra, but it is a primitive element of any proper subalgebra which contains it). We also consider generic elements and Δ-primitive elements. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 74, Algebra-15, 2000.     

Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators

We present here a quite unexpected result: Apart from already known commutative C*-algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C*-algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols.     

标度广义效应代数和标度效应代数的结构

研究了标度广义效应代数与标度效应代数的代数结构,给出了比较完整的结果.通过引入全标度广义代数的概念,本文证明了区间[0,1)上的标度广义效应代数和单位区间[0,1]上的标度效应代数完全由单位区间上的阿基米德余模确定,标度广义效应代数恰同构于全标度广义代数的下集.若标度广义代数满足局部有限条件,则它同构于实数加法群的子群代数.满足(S)条件的标度效应代数同构于实数加法群的子群代数和全标度广义代数的字典序乘积的子代数.     

Representing measures and infinite-dimensional holomorphy

We consider some applications of the Bishop-De Leeuw Theorem about representing measures for some algebras of analytic functions on unit balls of Banach spaces. In particular, we investigate Hardy spaces H² associated with corresponding algebras. Some examples are considered.     

Bifunctional-elementary relation algebras

A relation algebra is bifunctional-elementary if it is atomic and for any atom a, the element a;1;a is the join of at most two atoms, and one of these atoms is bifunctional (an element x is bifunctional if ’). We show that bifunctional-elementary relation algebras are representable. Our proof combines the representation theorems for: pair-dense relation algebras given by R. Maddux; relation algebras generated by equivalence elements provided corresponding relativizations are representable by S. Givant; and strong-elementary relation algebras dealt with in our earlier work. It turns out that atomic pair-dense relation algebras are bifunctional elementary, showing that our theorem generalizes the representation theorem of atomic pair-dense relation algebras. The problem is still open whether the related classes of rather elementary, functional-elementary, and strong functional-elementary relation algebras are representable. Received July 15, 2007; accepted in final form March 17, 2008.