Weakly tight functions, their Jordan type decomposition and total variation in effect algebras

In the present paper, we have studied envelopes of a function m defined on a subfamily E (containing 0 and 1) of an effect algebra L. The notion of a weakly tight function is introduced and its relation to tight functions is investigated; examples and counterexamples are constructed for illustration. A Jordan type decomposition theorem for a locally bounded real-valued weakly tight function m defined on E is established. The notions of total variation |m| on the subfamily E and m-atoms on a sub-effect algebra E (along with a few examples of m-atoms for null-additive as well as non null-additive functions) are introduced and studied. Finally, it is proved for a real-valued additive function m on a sub-effect algebra E that, m is non-atomic if and only if its total variation |m| is non-atomic.     

Decomposition of ℓ-group-valued measures

We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete ?-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete ?-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for ?-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.     

Explicit Hilbert spaces for certain unipotent representations III

In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L²-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L²-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of these representations and describe their decomposition.     

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.     

Symplectic groups and the Klein-Gordon field

In this paper a Cohen factorization theorem x = a^t · x_t (t > 0) is proved for a Banach algebra A with a bounded approximate identity, where t ? a^t is a continuous one-parameter semigroup in A. This theorem is used to show that a separable Banach algebra B has a bounded approximate identity bounded by 1 if and only if there is a homomorphism θ from L¹($\text{R}$⁺) into B such that ∥ θ ∥ = 1 and θ(L¹($\text{R}$⁺)). B = B = B · θ(L¹($\text{R}$⁺)). Another corollary is that a separable Banach algebra with bounded approximate identity has a commutative bounded approximate identity, which is bounded by 1 in an equivalent algebra norm.     

Polar factorization and monotone rearrangement of vector-valued functions

Given a probability space (X, μ) and a bounded domain Ω in ?^d equipped with the Lebesgue measure |·| (normalized so that |Ω| = 1), it is shown (under additional technical assumptions on X and Ω) that for every vector-valued function u ∈ L^p (X, μ; ?^d) there is a unique “polar factorization” u = ?Ψs, where Ψ is a convex function defined on Ω and s is a measure-preserving mapping from (X, μ) into (Ω, |·|), provided that u is nondegenerate, in the sense that μ(u^?1(E)) = 0 for each Lebesgue negligible subset E of ?^d. Through this result, the concepts of polar factorization of real matrices, Helmholtz decomposition of vector fields, and nondecreasing rearrangements of real-valued functions are unified. The Monge-Ampère equation is involved in the polar factorization and the proof relies on the study of an appropriate “Monge-Kantorovich” problem.     

Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras

Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L²(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel. Engel’s work is extended here to L²(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued L²(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner.     

Vector measure Maurey–Rosenthal-type factorizations and -sums of -spaces

Consider a vector measure of bounded variation m with values in a Banach space and an operator T:XL¹(m), where L¹(m) is the space of integrable functions with respect to m. We characterize when T can be factorized through the space L²(m) by means of a multiplication operator given by a function of L²(|m|), where |m| is the variation of m, extending in this way the Maurey–Rosenthal Theorem. We use this result to obtain information about the structure of the space L¹(m) when m is a sequential vector measure. In this case the space L¹(m) is an ℓ-sum of L¹-spaces.     

Quotient-bounded elements in locally convex algebras. II

Consideration of quotient-bounded elements in a locally convexGB ^*-algebra leads to the study of properGB ^*-algebras viz those that admit nontrivial quotient-bounded elements. The construction and structure of such algebras are discussed. A representation theorem for a properGB ^*-algebra representing it as an algebra of unbounded Hilbert space operators is obtained in a form that unifies the well-known Gelfand-Naimark representation theorem forC ^*-algebra and two other representation theorems forb ^*-algebras (also calledlmc ^*-algebras), one representinga b ^*-algebra as an algebra of quotient bounded operators and the other as a weakly unbounded operator algebra. A number of examples are discussed to illustrate quotient-bounded operators. An algebra of unbounded operators constructed out of noncommutativeL ^p-spaces on a regular probability gauge space and the convolution algebra of periodic distributions are analyzed in detail; whereas unbounded Hilbert algebras andL ^w-integral of a measurable field ofC ^*-algebras are discussed briefly.     

Orderamarts: A class of asymptotic martingales

We extend the notion of real-valued asymptotic martingales to the Banach lattice valued case. Unlike the other extensions, the notion of “orderamart” preserves the lattice property of real amarts. We show also, a Riesz decomposition, a weak and strong convergence theorem, a probabilistic characterization of A-L spaces from which we can prove that a Banach lattice with the shur property and a quasi-interior point in the dual is an l¹(Γ).     

On a singular integral on product spaces

We show that if K(x,y)=Ω(x,y)/|x|ⁿ|y|^m is a Calder n-Zygmund kerned on Rⁿ×R^m, where Ω∈L²(Sⁿ⁻¹×S^m−1) and b(x,y) is any bounded function which is radial with x∈Rⁿ and y∈R^m respectively, then b(x,y)K(x,y) is the kernel of a convolution operator which is bounded on L^p(Rⁿ×R^m) for 1<p<∞ and n≧2, m≧2. Project supported by NSFC     

Lie algebras admitting a metacyclic frobenius group of automorphisms

Suppose that a Lie algebra L admits a finite Frobenius group of automorphisms FH with cyclic kernel F and complement H such that the characteristic of the ground field does not divide |H|. It is proved that if the subalgebra C _L (F) of fixed points of the kernel has finite dimension m and the subalgebra C _L (H) of fixed points of the complement is nilpotent of class c, then L has a nilpotent subalgebra of finite codimension bounded in terms of m, c, |H|, and |F| whose nilpotency class is bounded in terms of only |H| and c. Examples show that the condition of F being cyclic is essential.     

DECOMPOSITION OF BV FUNCTIONS IN CARNOT-CARATHÉODORY SPACES

The aim of this paper is to get the decomposition of distributional derivatives of functions with bounded variation in the framework of Carnot-Carathéodory spaces (C-C spaces in brievity) in which the vector fields are of Carnot type. For this purpose the approximate continuity of BV functions is discussed first, then approximate differentials of L¹ functions are defined in the case that vector fields are of Carnot type and finally the decomposition Xu = ∇u · Lⁿ + X^sи is proved, where и ∈ BVx(Ω) and ∇u denotes the approximate differential of u.     

On normed Jordan algebras which are Banach dual spaces

Alfsen, Shultz, and Størmer have defined a class of normed Jordan algebras called JB-algebras, which are closely related to Jordan algebras of self-adjoint operators. We show that the enveloping algebra of a JB-algebra can be identified with its bidual. This is used to show that a JB-algebra is a dual space iff it is monotone complete and admits a separating set of normal states; in this case the predual is unique and consists of all normal linear functionals. Such JB-algebras (“JBW-algebras”) admit a unique decomposition into special and purely exceptional summands. The special part is isomorphic to a weakly closed Jordan algebra of self-adjoint operators. The purely exceptional part is isomorphic to C(X, M₃⁸) (the continuous functions from X into M₃⁸).     

A decomposition into atoms of distributions on spaces of homogeneous type

A maximal function is introduced for distributions acting on certain spaces of Lipschitz functions defined on spaces of homogeneous type. A decomposition into atoms for distributions whose maximal functions belong to L^p, p ? 1, is obtained, as well as, an approximation theorem of these distributions by Lipschitz functions.     

Applications of the complex interpolation method to a von Neumann algebra: Non-commutative Lp-spaces

Non-commutative L^p-spaces, 1 < p < ∞, associated with a von Neumann algebra are considered. The paper consists of two parts. In part I, by making use of the complex interpolation method, non-commutative L^p-spaces are defined as interpolation spaces between the von Neumann algebra in question and its predual. Also, all expected properties (such as duality and uniform convexity) are proved in the frame of interpolaton theory and relative modular theory. In part II, these L^p-spaces are compared with Haagerup's L^p-spaces. Based on this comparison, a non-commutative analogue of the classical Stein-Weiss interpolation theorem is obtained.     

The eigenvalue distribution of products of Toeplitz matrices - Clustering and attraction

We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h, is described by the function h in the “Szegö way”. Then, using Mergelyan’s theorem, we extend the result to the more general case where h belongs to the Tilli class. The same technique gives us the analogous result for sequences belonging to the algebra generated by Toeplitz sequences, if the symbols associated with the sequences are bounded and the global symbol h belongs to the Tilli class. A generalization to the case of multilevel matrix-valued symbols and a study of the case of Laurent polynomials not necessarily belonging to the Tilli class are also given.     

Table Bases as Unions of Proper Closed Subsets

If the distinguished basis of a table algebra is an irredundant union of n proper closed subsets, and if the positive structure constants of the quotient table algebra (rescaled to be standard) modulo the intersection of these closed subsets are all at least 1, then it is proved that the order of this quotient algebra is bounded above by a function of n. This generalizes a result of B. H. Neumann for finite groups, applies directly to association schemes, and also yields the following result: if G is a finite group, ?? is the set of minimal members (with respect to containment) of the set of kernels of irreducible characters of G, and N = ∏ _K∈?? K, then |N| is bounded above by a function of |??|. Table algebras where the table basis is a union of three or four proper closed subsets are characterized as well.     

The sharp quantitative Sobolev inequality for functions of bounded variation

The classical Sobolev embedding theorem of the space of functions of bounded variation BV(Rn) into Ln^′(Rn) is proved in a sharp quantitative form.     

Produkte positiver elemente in formal-reellen Jordan-Algebren

A generalization of the Rayleigh quotient defined for real symmetric matrices to the elements of a formally real Jordan algebra is used here to give a generalization to formally real Jordan algebras of the theorem that for any real symmetric matrix C with tr C > 0 there are positive definite real symmetric matrices A and B with C = AB + BA.