Endodualisable and endoprimal finite double Stone algebras

The finite endodualisable double Stone algebras are characterised, and every finite endoprimal double Stone algebra is shown to be endodualisable. Received February 24, 1999; accepted in final form May 10, 1999.     

Projections in a Synaptic Algebra

A synaptic algebra is an abstract version of the partially ordered Jordan algebra of all bounded Hermitian operators on a Hilbert space. We review the basic features of a synaptic algebra and then focus on the interaction between a synaptic algebra and its orthomodular lattice of projections. Each element in a synaptic algebra determines and is determined by a one-parameter family of projections—its spectral resolution. We observe that a synaptic algebra is commutative if and only if its projection lattice is boolean, and we prove that any commutative synaptic algebra is isomorphic to a subalgebra of the Banach algebra of all continuous functions on the Stone space of its boolean algebra of projections. We study the so-called range-closed elements of a synaptic algebra, prove that (von Neumann) regular elements are range-closed, relate certain range-closed elements to modular pairs of projections, show that the projections in a synaptic algebra form an M-symmetric orthomodular lattice, and give several sufficient conditions for modularity of the projection lattice.     

On algebras of polynomially compact operators

In this paper, we find sufficient and necessary conditions for a triangularizable closed algebra of polynomially compact operators to be commutative modulo the radical. We also prove that an algebraic algebra of operators of a bounded degree on a Banach space is triangularizable under some mild additional conditions. As a special case we obtain a result stating that every algebraic algebra of operators of bounded degree is triangularizable whenever its commutators are nilpotent operators.     

Endodualisable and endoprimal finite double Stone algebras

The finite endodualisable double Stone algebras are characterised, and every finite endoprimal double Stone algebra is shown to be endodualisable. The authors wish to express their gratitude to B. A. Davey and T. Katriňák for their helpful remarks and to J. G. Pitkethly for her assistance with the pictures. A support by Slovak grants VEGA 1/4057/97, 1/3026/06 and APVV-51-009605 is acknowledged by the first author who also wishes to thank the Mathematical Institute of the University of Oxford and the School of Mathematical and Statistical Sciences of La Trobe University for their hospitality.     

Stone代数的Cayley表示定理

首先通过集代数得到了Stone代数的表示定理,然后证明了每一个Stone代数均嵌入到某个集合X上的一个Stone映射类S中.     

Injective and projective regular double Stone algebras

We characterize the injective and projective regular double Stone algebras, and describe those regular double Stone algebras which are also projective in the category of double Stone algebras.Presented by Bjarni Jónsson.     

Projectivity,prime ideals,and chain conditions of Stone algebras

Some properties of projective stone algebras are exhibited, which are connected with the ordered set of prime ideals. From this we derive a simple characterization of finite projective Stone algebras, and of those projective Stone algebras, whose centre is a projective Boolean algebra, and whose dense set is a projective Stone algebras, whose centre is a projective Boolean algebra, and whose dense set is a projective distributive lattice. Finally, we give some conditions under which a Stone algebra has no chains of type λ, where λ is an infinite regular cardinal. The results of this paper are part of the author's Ph.D. Thesis written under the direction of S. Koppelberg. The author wishes to express his gratitude to Prof. Koppelberg for her guidance and her patience. Presented by K. A. Baker.     

Banach algebra generated by a finite number of bergman polykernel operators,continuous coefficients,and a finite group of shifts

We study the Banach algebra generated by a finite number of Bergman polykernel operators with continuous coefficients that is extended by operators of weighted shift that form a finite group. By using an isometric transformation, we represent the operators of the algebra in the form of a matrix operator formed by a finite number of mutually complementary projectors whose coefficients are Toeplitz matrix functions of finite order. Using properties of Bergman polykernel operators, we obtain an efficient criterion for the operators of the algebra considered to be Fredholm operators.     

Spectra of weighted algebras of holomorphic functions

We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach–Stone type question is addressed.     

双重Stone代数的主同余关系

给出了双重Stone代数的主同余关系θ(a,b)(a≤b)的等式刻划以及其它的一些性质,由此得到了主同余关系θ(a,b)(a≤b)存在Boo le-补的若干充分条件.     

CCC forcing and splitting reals

The results of this paper were motivated by a problem of Prikry who asked if it is relatively consistent with the usual axioms of set theory that every nontrivial ccc forcing adds a Cohen or a random real. A natural dividing line is into weakly distributive posets and those which add an unbounded real. In this paper I show that it is relatively consistent that every nonatomic weakly distributive ccc complete Boolean algebra is a Maharam algebra, i.e. carries a continuous strictly positive submeasure. This is deduced from theP-ideal dichotomy, a statement which was first formulated by Abraham and Todorcevic [AT] and later extended by Todorcevic [T]. As an immediate consequence of this and the proof of the consistency of theP-ideal dichotomy we obtain a ZFC result which says that every absolutely ccc weakly distributive complete Boolean algebra is a Maharam algebra. Using a previous theorem of Shelah [Sh1] it also follows that a modified Prikry conjecture holds in the context of Souslin forcing notions, i.e. every nonatomic ccc Souslin forcing either adds a Cohen real or its regular open algebra is a Maharam algebra. Finally, I also show that every nonatomic Maharam algebra adds a splitting real, i.e. a set of integers which neither contains nor is disjoint from an infinite set of integers in the ground model. It follows from the result of [AT] that it is consistent relative to the consistency of ZFC alone that every nonatomic weakly distributive ccc forcing adds a splitting real.     

Vertex algebras associated to modified regular representations of the Virasoro algebra

We give an abstract construction, based on the Belavin–Polyakov–Zamolodchikov equations, of a family of vertex algebras with conformal elements of rank 26 associated to the modified regular representations of the Virasoro algebra. The vertex operators are obtained from the products of intertwining operators for a pair of Virasoro algebras. We explicitly determine the structure coefficients that yield the axioms of vertex algebras. In the process of our construction, we obtain new hypergeometric identities.     

Combinatorics of rooted trees and Hopf algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities.

Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.

     

Volterra type integral operators with homogeneous kernels in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We consider multidimensional integral Volterra type operators with kernels homogeneous of degree (?n); the operators act in L _p -spaces with a submultiplicative weight. For these operators we obtain necessary and sufficient conditions of their invertibility. Besides, we describe the Banach algebra generated by the operators. For this algebra we construct the symbolic calculus, in terms of which we obtain an invertibility criterion of the operators.     

Dynamic effect algebras

We introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.     

Double Basic Algebras

To any bounded lattice with section antitone bijections can be assigned an algebra with two binary, two unary and a nullary operations satisfying similar axioms as a basic algebra. Due to the doubled similarity type, this algebra is called a double basic algebra. Also conversely, every double basic algebra induces a bounded lattice equipped with antitone bijections in every section. We study properties of double basic algebras, their interval algebras and several conclusions of the so-called pseudo-commutativity. This work is supported by the Research Project MSM 6198959214 by Czech Government.     

Algebra of Bergman Operators with Automorphic Coefficients and Parabolic Group of Shifts

We study the algebra of operators with the Bergman kernel extended by isometric weighted shift operators. The coefficients of the algebra are assumed to be automorphic with respect to a cyclic parabolic group of fractional-linear transformations of a unit disk and continuous on the Riemann surface of the group. By using an isometric transformation, we obtain a quasiautomorphic matrix operator on the Riemann surface with properties similar to the properties of the Bergman operator. This enables us to construct the algebra of symbols, devise an efficient criterion for the Fredholm property, and calculate the index of the operators of the algebra considered.     

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.     

The boundedness and the fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients

In the space L _p (? ⁿ ), 1 < p < ??, we study a new wide class of integral operators with anisotropically homogeneous kernels. We obtain sufficient conditions for the boundedness of operators from this class. We consider the Banach algebra generated by operators with anisotropically homogeneous kernels of compact type and multiplicatively slowly oscillating coefficients. We establish a relationship between this algebra and multidimensional convolution operators, and construct a symbolic calculus for it. We also obtain necessary and sufficient conditions for the Fredholm property of operators from this algebra.     

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.