Showing papers in "Israel Journal of Mathematics in 1979"
TL;DR: In this article, it was shown that if E is a uniformly rotund Banach space with a Frechet differentiable norm, and C is a bounded nonempty closed convex subset of E, and T: C→C is a contraction, then the iterates {Tnx} are weakly almost-convergent to a fixed-point of T.
Abstract: The following theorem is proven:if E is a uniformly rotund Banach space with a Frechet differentiable norm, C is a bounded nonempty closed convex subset of E, and T: C→C is a contraction, then the iterates {Tnx} are weakly almost-convergent to a fixed-point of T.
239 citations
TL;DR: In this article, the authors used disjoint sequence methods from the theory of Riesz spaces to study compact operators on Banach lattices and showed that each positive map from a Banach Lattice E to a BL T with compact majorant is itself compact provided the norms on E and T are order continuous.
Abstract: Disjoint sequence methods from the theory of Riesz spaces are used to study compact operators on Banach lattices. A principal new result of the paper is that each positive map from a Banach latticeE to a Banach latticeF with compact majorant is itself compact provided the norms onE′ andF are order continuous.
224 citations
189 citations
TL;DR: In this paper, a class of piecewise monotonically increasing functions f on the unit intervalI was considered and the measures with maximal entropy for these transformations were determined, where the supports of two of them have at most finitely many points in common.
Abstract: We consider a class of piecewise monotonically increasing functionsf on the unit intervalI. We want to determine the measures with maximal entropy for these transformations. In part I we construct a shift-space Σ
+
isomorphic to (I, f) generalizing the \-shift and another shift Σ
M
over an infinite alphabet, which is of finite type given by an infinite transition matrixM. Σ
M
has the same set of maximal measures as (I, f) and we are able to compute the maximal measures of maximal measures of. In part II we try to bring these results back to (I, f). There are only finitely many ergodic maximal measures for (I, f). The supports of two of them have at most finitely many points in common. If (I, f) is topologically transitive it has unique maximal measure.
185 citations
TL;DR: In this article, the authors studied a general condition on anm-accretive operators in a Banach space that guarantees strong convergence of the semigroup generated by A and related implicit and explicit iterative schemes to a zero.
Abstract: LetA be anm-accretive operator in a Banach spaceE. Suppose thatA
−10 is not empty and that bothE andE
* are uniformly convex. We study a general condition onA that guarantees the strong convergence of the semigroup generated by—A and of related implicit and explicit iterative schemes to a zero ofA. Rates of convergence are also obtained. In Hilbert space this condition has been recently introduced by A. Pazy. We also establish strong convergence under the assumption that the interior ofA
−10 is not empty. In Hilbert space this result is due to H. Brezis.
155 citations
TL;DR: In this paper, the relative projection constants of subspaces of a Banach spaceY in terms of geometrical properties of Y were estimated and it was shown that K-convex spaces are locally π-Euclidean.
Abstract: In this paper we obtain new estimates for the relative projection constants of subspaces of a Banach spaceY in terms of geometrical properties ofY. Our method gives thatK-convex spaces are locally π-Euclidean. We also get a version of Maurey’s extension theorem for spaces of typep<2.
124 citations
TL;DR: In this article, necessary and sufficient conditions for central simple algebras to have involutions of the first kind and to be tensor products of quaternion subalgeses were developed.
Abstract: We develop necessary and sufficient conditions for central simple algebras to have involutions of the first kind, and to be tensor products of quaternion subalgebras. The theory is then applied to give an example of a division algebra of degree 8 with involution (of the first kind), without quaternion subalgebras, answering an old question of Albert; another example is a division algebra of degree 4 with involution (*) has no (*)-invariant quaternion subalgebras.
102 citations
TL;DR: In this paper, a CR-submanifold of a Hermitian manifold is considered and various integrability theorems on the submanifolds are proved.
Abstract: In this paper we consider a CR-submanifold of a Hermitian manifold and prove various integrability theorems on the submanifold. When the ambient space is Kaehlerian a number of differential geometric results are also obtained.
91 citations
TL;DR: In this article, the relationship between the prime ideals of a finite group of automorphisms and the prime ranks of the fixed ring of a fixed automorphism over the ring is discussed.
Abstract: LetR * G be a crossed product of the finite groupG over the ringR. In this paper we discuss the relationship between the prime ideals ofR*G and theG-prime ideals ofR. In particular, we show that Incomparability and Going Down hold in this situation. In the course of the proof, we actually completely describe all the prime idealsP ofR*G such thatP∩R is a fixedG-prime ideal ofR. As an application, we prove that ifG is a finite group of automorphisms ofR, then the prime (primitive) ranks ofR and of the fixed ringRG are equal provided •G•−∈R. In an appendix, we extend some of these 3 results to crossed products of the infinite cyclic group.
87 citations
TL;DR: In this paper, a Markov operator P is defined to be ergodic and has no unimodular eigenvalues ≠ 1/ϵ ≠ 0 for every q ergodics with finite invariant measureP ×Q is defined.
Abstract: LetT be a Markov operator onL
1(X, Σ,m) withT*=P. We connect properties ofP with properties of all productsP ×Q, forQ in a certain class: (a) (Weak mixing theorem)P is ergodic and has no unimodular eigenvalues ≠ 1 ⇔ for everyQ ergodic with finite invariant measureP ×Q is ergodic ⇔ for everyu ∈L
1 with∝ udm=0 and everyf ∈L
∞ we haveN
−1Σ
≠1/
||→0. (b) For everyu ∈L
1 with∝ udm=0 we have ‖T
nu‖1 → 0 ⇔ for every ergodicQ, P ×Q is ergodic. (c)P has a finite invariant measure equivalent tom ⇔ for every conservativeQ, P ×Q is conservative. The recent notion of mild mixing is also treated.
82 citations
TL;DR: In this article, it was shown that under fairly general conditions on a compact metric spaceY there are minimal homeomorphisms on Z×Y of the form(z,y)→(σz, hz(y)) where (Z, σ) is a arbitrary metric minimal flow andz→hz is a continuous map from Z to the space of homeomorphism of Y.
Abstract: It is shown that under fairly general conditions on a compact metric spaceY there are minimal homeomorphisms onZ×Y of the form(z,y)→(σz, hz(y)) where (Z, σ) is a arbitrary metric minimal flow andz→hz is a continuous map fromZ to the space of homeomorphisms ofY. Similar results are obtained for strict ergodicity, topolotical weak mixing and some relativized concepts.
TL;DR: In this article, the authors give a description of all the continuous maps of the circle without periodic point, and show that all of these maps can be found in the Euclidean plane.
Abstract: This paper gives a description of all the continuous maps of the circle without periodic point.
TL;DR: In this paper, it was shown that irreducible finite state, Markov shifts of the same entropy and period are finitarily isomorphic, i.e., they are Markovian.
Abstract: It is shown that irreducible finite state, Markov shifts of the same entropy and period arefinitarily isomorphic.
TL;DR: In this article, it was shown that those minimal sets which have positive topological entropy and fail to be uniquely ergodic form a esidual set, i.e., a set of minimal sets whose points are almost automorphic symbolic minimal sets.
Abstract: Given a metrizable monothetic groupG with generatorg and a suitable closed nowhere dense subsetC of positive Haar measure, we associate a natural compact metric space whose points are almost automorphic symbolic minimal sets. It is then shown that those minimal sets which have positive topological entropy and fail to be uniquely ergodic form a esidual set. The example due to P. Julius [2] of a Toeplitz sequence of positive entropy which, is uniquely ergodic shows that the “residual” conclusion is sharp.
TL;DR: In this paper, the authors presented a classification of maximal amenable subgroups of a semi-simple group and gave necessary and sufficient conditions for a subgroup to have a fixed point in this compactification.
Abstract: We present a classification of maximal amenable subgroups of a semi-simple groupG. The result is that modulo a technical connectivity condition, there are precisely 2′ conjugacy classes of such subgroups ofG and we shall describe them explicitly. Herel is the split rank of the groupG. These groups are the isotropy groups of the action ofG on the Satake-Furstenberg compactification of the associated symmetric space and our results give necessary and sufficient conditions for a subgroup to have a fixed point in this compactification. We also study the action ofG on the set of all measures on its maximal boundary. One consequence of this is a proof that the algebraic hull of an amenable subgroup of a linear group is amenable.
TL;DR: In this article, it was shown that an algebra satisfies a Capelli identity if, and only if, all the Young diagrams associated with its cocharacters are of a bounded height.
Abstract: The sequence of cocharacters (c.c.s.) of a P.I. algebra is studied. We prove that an algebra satisfies a Capelli identity if, and only if, all the Young diagrams associated with its cocharacters are of a bounded height. This result is then applied to study the identities of certain P.I. algebras, includingF
k
.
TL;DR: In this paper, the authors established a theorem on l1-sequences obtained by averaging of semi-norms, applied in the study of weakly conditionally compact subsets of lx1, where X is a Banach space.
Abstract: We establish a theorem onl1-sequences obtained by averaging of semi-norms. The result is applied in the study of weakly conditionally compact subsets ofLx1, whereX is a Banach space.
[...]
TL;DR: In this paper, the authors give two constructions of thin αβ-sets, where α and β are elements of infinite order in T ifE⊂E−α). ∪ (E−β).
Abstract: A closed setE⊂T is an αβ-set, where α and β are elements of infinite order inT ifE⊂E−α). ∪ (E−β). We give two constructions of “thin” αβ-sets.
TL;DR: For operators A and B on a Hilbert space ℋ, let τ denote the operator on ℒ(ℋ) defined by τ(X)=AX−XB and several equivalent conditions are given for τ to be surjective or bounded below.
Abstract: For operatorsA andB on a Hilbert space ℋ, let τ denote the operator on ℒ(ℋ) defined by τ(X)=AX−XB. Several equivalent conditions are given for τ to be surjective or bounded below. Analogues of these results are given for the restrictions of τ to norm ideals, and the norms of these restrictions are estimated.
TL;DR: In this paper, it was shown that the Cartesian square of the classical horocycle flow is not loosely Bernoulli, and that it is algebraic non-loosely non-Bernoulli.
Abstract: We give an example of an algebraic non-loosely Bernoulli flow. Namely, we prove that the cartesian square of the classical horocycle flow is not loosely Bernoulli.
TL;DR: It has been long conjectured that ifn polynomialsf 1, …,f n inn variables have a (non-zero) constant Jacobian determinant, then every polynomial can be expressed as a polytope inf 1,..., f n as mentioned in this paper.
Abstract: It has been long conjectured that ifn polynomialsf 1, …,f n inn variables have a (non-zero) constant Jacobian determinant then every polynomial can be expressed as a polynomial inf 1, …,f n. In this paper, various extra assumptions (particularly whenn=2) are shown to imply the conclusion. These conditions are discussed algebraically and geometrically.
TL;DR: In this paper, the pointwise convergence of martingales in a von Neumann algebra has been proved, and it is shown that martingale convergence can also be shown in terms of convergence in a non-convex setting.
Abstract: We prove the pointwise convergence of martingales in a von Neumann algebra.
TL;DR: In a generalization of Radon's theorem, Tverberg as mentioned in this paper showed that each setS of at least (d+1) (r − 1)+1 points in Rd has anr-partition into (pair wise disjoint) subsetsS =S1 ∪ … ∪Srso that eachn members of the family {convSi∼i-1r have non-empty intersection, where 1<=n<=r.
Abstract: In a generalization of Radon’s theorem, Tverberg showed that each setS of at least (d+1) (r − 1)+1 points inRdhas anr-partition into (pair wise disjoint) subsetsS =S1 ∪ … ∪Srso that\(\bigcap
olimits_i^r {\underline{\underline {}} } _1 \) convSi# O. This note considers the following more general problems: (1) How large mustS σRdbe to assure thatS has anr-partitionS=S1∪ … ∪Srso that eachn members of the family {convSi∼i-1r have non-empty intersection, where 1<=n<=r. (2) How large mustS ∪Rd be to assure thatS has anr-partition for which\(\bigcap
olimits_i^r {\underline{\underline {}} } _1 \) convSr is at least 1-dimensional.
TL;DR: In this article, the concept of rational ergodicity is used to construct continuous measures on the circle, which show that the exceptional set in the weak mixing theorem may be rather large.
Abstract: Two ratio limit concepts for transformations preserving infinite measures, rational ergodicity and bounded rational ergodicity, are discussed and compared. The concept of rational ergodicity is used to construct some continuous measures on the circle, which show that the exceptional set in the weak mixing theorem may be rather large.
TL;DR: In this article, it was shown that in the division ring of quotients of the Weyl algebra, every maximal subfield has a transcendence degree at mostn over the center.
Abstract: In this paper we prove the following theorem: Let D be a division ring with center the field k, and let k (x
1, …, xn) denote the rational function field in n variables over k. If D contains a maximal subfield which has transcendence degree at least n over k, then D ⊗k k (x1, …, xn) is a simple Noetherian domain of Krull and global dimensions n. Rather surprisingly, the preceding result can be used to determine the maximum transcendence degrees of the commutative subalgebras of several classically studied division rings. Using the theorem we prove, for example, that in the division ring of quotients of the Weyl algebra,A
n, every maximal subfield has transcendence degree at mostn over the center.
TL;DR: In this article, the authors continue the investigation from [10], [11], and [12] on uncountable abelian groups and they tend more to group theory and was motivated by Nunke's statement (in [9]) that Whitehead problem, rephrased properly, is not solved yet.
Abstract: We continue the investigation from [10], [11], [12] on uncountable abelian groups. This paper tends more to group theory and was motivated by Nunke’s statement (in [9]) that Whitehead problem, rephrased properly, is not solved yet.
TL;DR: In this paper, the authors considered the problem of finding a solution to the following initial value problem: (du(t))/(dt) ∈Au(t) +F(u)(t),u(0)=u 0, has at least one integral solution on [0,T].
Abstract: LetX be a real Banach space,U ⊂X a given open set,A ⊂X×X am-dissipative set andF:C(0,a;U) →L
∞(0,a;X) a continuous mapping. Assume thatA generates a nonlinear semigroup of contractionsS(t): {ie221-2}) → {ie221-3}), strongly continuous at the origin, withS(t) compact for allt>0. Then, for eachu
0 ∈ {ie221-4}) ∩U there existsT ∈ ]0,a] such that the following initial value problem: (du(t))/(dt) ∈Au(t) +F(u)(t),u(0)=u
0, has at least one integral solution on [0,T]. Some extensions and applications are also included.
TL;DR: In this article, the authors established some number theoretical results about a continuous functionh from the circleT into itself, which generalize Kronecker's theorem in several ways, and used these results to characterize the almost periodic sets of the flow on the torusT 2 generated by (θ, φ), where α is irrational.
Abstract: We establish some “number theoretical” results about a continuous functionh from the circleT into itself, which generalize Kronecker’s theorem in several ways. These results are used to characterize the almost periodic sets of the flow on the torusT
2 generated by (θ, φ) → (θ+α, φ+h(θ)), where α is irrational. The almost periodic measures are characterized in the caseh(θ)=θ.
TL;DR: In this article, the authors introduce the notion of Whittaker models for representations of a metaplectic covering group of GLs and establish the uniqueness and existence of such models.
Abstract: We introduce the notion of Whittaker models for representations of a metaplectic covering group of GL (2) and establish the uniqueness and existence of such models. Our results generalize corresponding results of Jacquet-Langlands, but the methods are new.
TL;DR: For the notion of finitary isomorphism, which arises in many examples in ergodic theory, this article proved basic theorems about invariants, representations and the central limit theorem in shift spaces.
Abstract: For the notion of finitary isomorphism, which arises in many examples in ergodic theory, we prove some basic theorems about invariants, representations and the central limit theorem in shift spaces.