Showing papers in "Israel Journal of Mathematics in 1978"

Making the supercompactness of κ indestructible under κ-directed closed forcing

Abstract: A model is found in which there is a supercompact cardinal κ which remains supercompact in any κ-directed closed forcing extension.

Produits infinis de resolvantes

Abstract: This paper is concerned with the convergence of the sequence χ n =(I+λ n A)−1χ n−1 whereA is maximal monotone and λ n >0. Various assumptions onA and λ n are considered.

A weak version of ◊ which follows from 2ℵ 0<2ℵ 1

Abstract: We prove that if CH holds (or even if 2ℵ0 < 2ℵ1), then a weak version of ◊ holds. This weak version of ◊ is a ◊-like principle, and is strong enough to yield some of the known consequences of ◊.

Minimal degree for a permutation representation of a classical group

Abstract: The minimal degree for a permutation representation of the finite linear groups, and finite classical groups is determined.

On an inverse problem for nonnegative and eventually nonnegative matrices

Abstract: Let σ= (λ1,···λn)⊂C. We discuss conditions for which σ is the spectrum of a nonnegative or eventually nonnegative matrix. This brings us to study rational functions with nonnegative Maclaurin coefficients. A conjecture for special sets σ is stated and some evidence in support of this conjecture is given.

A weak generalization of MA to higher cardinals

Abstract: We generalized MA e.g., to ℵ1-complete forcing, by strengthening the ℵ2-C.C. condition which occurs in many proofs. We show some consequences of MA generalized, and show that we get a model of ZFC in which the modadic theory ofω 2 is decidable.

Van der corput’s difference theorem

Abstract: We obtain a sufficient condition for a subsetH of positive integers to satisfy that the equidistribution (mod 1) of the sequences (u n+h − u n; n=1, 2, ···) for allh ∈H implies the equidistribution of (u n). Our condition is satisfied, for example, for the following sets: (1)H={n − m; n ∈ I, m ∈ I, n>m}, whereI is any infinite subset of integers; (2)H={| ψ (n)|; ψ(n)≠0,n ∈ Z}, where ψ is a nonconstant polynomial with integral coefficients having at least one integral zero (modq) for allq=2, 3, ···; (3)H={p+1;p is a prime} andH={p − 1;p is a prime}.

Some conditions which almost characterize Frobenius groups

Abstract: The main result of this paper is the following: LetG be a group with a proper non-trivial normal subgroupH such that each coset ofH distinct fromH is contained in a conjugacy class ofG. IfG is not a Frobenius group with kernelH then one ofH orG/H is ap-group. The hypothesis of this theorem is shown to be equivalent to a condition on characters ofG. The only group the author knows which satisfies this hypothesis and is not either Frobenius or ap-group is one of order 72.

Subspaces without the approximation property

Abstract: It is proved that the Banach spacel p with 1≦p<2 contains a subspace without AP (the case 2

Guessing the next output of a stationary process

Abstract: Suppose we start watching a stationary process at time 0. Then the conditional probability of a particular output at time −1, given the outputs at times 0 throughk, will converge. In this paper we will show that we can make a guess, depending only on the outputs from 0 throughk (and not, of course, on the process) that will converge to the above limit with probability one.

Multiplicative commutators in division rings II

Abstract: In this paper we study division rings in which the multiplicative commutators are periodic or periodic relative to the center.

Nonreflexive spaces of type 2

Abstract: The nonreflexive and uniformly nonoctahedral spacesX pgr are known to be of typep if 1≦p<2 and ρ is sufficiently large. It is shown thatX ρ is of type 2 if ρ>2.

The duality between Asplund spaces and spaces with the Radon-Nikodym property

Abstract: A Banach spaceX is an Asplund space (a strong differentiability space) if and only ifX * has the Radon-Nikodym property.

Nonlinear parabolic boundary value problems with upper and lower solutions

Abstract: The existence of at least one periodic solution of a very general second order nonlinear parabolic boundary value problem is proved under the assumption that a lower solution ϕ and an upper solution ψ with ϕ≦ψ are known

Central simple algebras

Abstract: Wedderburn’s factorization of polynomials over division rings is refined and used to prove that every central division algebra of degree 8, with involution, has a maximal subfield which is a Galois extension of the center (with Galois group Z2⊕Z2⊕Z2). The same proof, for an arbitrary central division algebra of degree 4, gives an explicit construction of a maximal subfield which is a Galois extension of the center, with Galois group Z2⊕Z2. Use is made of the generic division algebras, with and without involution.

On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak ω-limit set

Abstract: LetT be a nonexpansive self-mapping of a closed convex subsetC of a real Hilbert space. In this paper we deal with the structure of the weak ω-limit set of iterates {T nx}, establish conditions under which it is invariant underT, and show that {T nx} converges weakly iffT has a fixed-point andT nx-Tn+1x→0 weakly.

Crossed products over prime rings

Abstract: In this paper we obtain necessary and sufficient conditions for the crossed productR *G to be prime or semiprime under the assumption thatR is prime. The main techniques used are the Δ-methods which reduce these questions to the finite normal subgroups ofG and a study of theX-inner automorphisms ofR which enables us to handle these finite groups. In particular we show thatR *G is semiprime ifR has characteristic 0. Furthermore, ifR has characteristicp>0, thenR *G is semiprime if and only ifR *P is semiprime for all elementary abelianp-subgroupsP of Δ+(G) ∩G inn.

Horocycle flows are loosely Bernoulli

Abstract: It is proved that horocycle flows associated with transitiveC 2-Anosov flows are loosely Bernoulli with respect to their unique ergodic measures.

On dualL 1-spaces and injective bidual Banach spaces

Abstract: In a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l 1(τ) embeds in a Banach spaceX if and only ifL 1([0, 1]τ) embeds inX *. An extension (to a rather wider class of cardinals) of the basic lemma of that paper is here applied so as to yield an affirmative answer to a question posed by Rosenthal concerning dual ℒ1-spaces. It is shown that ifZ * is a dual Banach space, isomorphic to a complemented subspace of anL 1-space, and κ is the density character ofZ *, thenl 1(κ) embeds inZ *. A corollary of this result is that every injective bidual Banach space is isomorphic tol ∞(κ) for some κ. The second part of this article is devoted to an example, constructed using the continuum hypothesis, of a compact spaceS which carries a homogeneous measure of type ω1, but which is such thatl 1(ω1) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω1. The dual space ℰ(S)* is isometric to $$(L{}^1[0,1]^{\omega _1 } ) \oplus \left( {(\sum\limits_{\omega _1 } {{}^ \oplus L{}^1[0,1] \oplus l^1 (\omega _1 )} } \right)_1 ,$$ , and is a member of a new isomorphism class of dualL 1-spaces.

Unconditionality in tensor products

Abstract: It is proved that in order to study unconditional structures in tensor products of finite dimensional Banach spaces it is enough to consider a certain basis. This result is applied to spaces ofp-absolutely summing operators showing their “bad” structure.

Galois groups of intersections of local fields

Abstract: LetK be a denumerable Hilbertian field with separable algebraic closure $$\bar K$$ and Galois group $$\mathfrak{G}_K = Gal\left( {\bar K\left| K \right.} \right)$$ , letw 1,...w n be absolute values on $$\bar K$$ . Then for almost allσ ∈ G (in the sense of Haar measure) there are no relations between the decomposition groups G K (ω 1 σ 1),...,G K (w n σ n ) of the absolute valuesw 1 σ 1,...,w n σ n i.e. the subgroup of G K generated by these groups is the free product of these groups.

On Shelah’s compactness of cardinals

Abstract: We deal with the compactness property of cardinals presented by Shelah, who proved a compactness theorem for singular cardinals. We improve that result in eliminating axiom I there and show a new application of that theorem together with a straightforward proof of it for the special case discussed. We discuss compactness for regular cardinals and show some independence results: one of them, a part of which is due to A. Litman, is the independence from ZFC+GCH of the gap-one two cardinal problem for singular cardinals.

An extension theorem for separable Banach spaces

Abstract: We construct a totally disconnected ω*, norming subsetF of the unit ballB * of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ωω), is either isomorphic toC(ωω) or toc u.

Embedding (p,p−1) graphs in their complements

Abstract: A graphG is embeddable in its complement $$\bar G$$ ifG is isomorphic with a subgraph of $$\bar G$$ A complete characterization is given of those (p,p−1) graphs which are embeddable in their complements In particular, letG be a (p,p−1) graph wherep≧6 ifp is even andp≧9 ifp is odd; thenG is embeddable in $$\bar G$$ if and only ifG is neither the starK 1,p−1 norK 1,n ∪C 3 withn≧4

A Banach space with few operators

Abstract: Assuming the axiom (of set theory)V=L (explained below), we construct a Banach space with density character ℵ1 such that every (linear bounded) operatorT fromB toB has the formaI+T1, whereI is the identity, andT1 has a separable range. The axiomV=L means that all the sets in the universe are in the classL of sets constructible from ordinals; in a sense this is the minimal universe. In fact, we make use of just one consequence of this axiom, ℵ1 proved by Jensen, which is widely used by mathematical logicians.

Minimal functions with unbounded integral

Abstract: We prove theorems which imply the following results. (1) “Most” almost periodic functionsb(t) with unbounded integral oscillate in a strong sense. (2) IfB is a continuous function on a minimal flow (Ω,R), then either the time averages \((1/t)\smallint _o^t B(\omega \cdot s)ds\) all converge, or they diverge on a residual set.

Conjugacy classes in finite groups

Abstract: In the first part of this note, we give new proofs of known results regarding the class number of finite groups, adding a few related results. In the second part, we improve a result of Ito concerning a special class ofp-groups.

Unique ergodicity of flows on homogeneous spaces

Abstract: LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T a, G/Γ,θ) is weak-mixing. (2) (T a, G/Γ) is topologically weak-mixing. (3) (G u, G/Γ) is uniquely ergodic. (4) (G u, G/Γ,θ) is ergodic. (5) (G u, G/Γ) is point transitive. (6) (G u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T a, G/Γ,θ) is ergodic” may be added to the above list.

On spectral analysis of non-monic matrix and operator polynomials, I. Reduction to monic polynomials

Abstract: Three recent papers [1, 2, 3] developed the basic concepts of a spectral theory for matrix and operator monic polynomials. In this paper we continue the study, replacing the requirement of monicness by a weaker condition.