On the relative Wall-Witt groups

LetR* be a simplicial involutive ring. According to certain involutions onK(R*) and _ε L R _∗, there are 1/2-local splittings and . It is known [2] that _ε L _\ga ^α R _∗, the Wall-Witt group. SupposeI→R S is a split extension of discrete involutive rings withI ²=0, andI is a freeS-bimodule. Then we have and . The trace map Tr: Prim _n ∧*M(I ⊗ ℚ)→ ₀ ρ _n ;I ⊗ ℚ) is an isomorphism. We prove in Lemma 1 that the trace map Tr is ℤ/2-equivariant. In Theorem 2 we show that under a certain assumption the rational relative Wall-Witt group vanishes. Theorem 2 can be extended to a more general case (Theorem 3) by employing Goodwillie’s reduction technique [3]. This work was partially supported by KOSEF under Grant 923-0100-010-1.     

On Lie derivations of Lie ideals of prime algebras

LetF be a commutative ring with 1, letA, be a primeF-algebra with Martindale extended centroidC and with central closureA _c and letR be a noncentral Lie ideal of the algebraA generatingA. Further, letZ(R) be the center ofR, let be the factor Lie algebra and let δ: be a Lie derivation. Suppose that char(A) ≠ 2 andA does not satisfySt ₁₄, the standard identity of degree 14. We show thatR ΩC =Z(R) and there exists a derivation of algebrasD:A →A _c such that for allx∈R. Our result solves an old problem of Herstein.     

Isometric approximation property in euclidean spaces

We give a necessary and sufficient quantitative geometric condition for a compact setA⊂R ⁿ to have the following property with a givenc≥1: For everyɛ>0 and for every mapf: A→R ⁿ such that there is an isometryS: A→R ⁿ such that |Sx−fx|≤cɛ for allx∈A.     

Ratio limit theorems for Markov processes

Convergence of andμP ⁿ(B)/μP ⁿ(a) is established for a certain class of Markov operators,P, whereμ is a measure andB is a subset ofA. The results are proved under certain conditions onP and the setA.     

Spectra of ergodic group actions

LetG be a locally compact second countable abelian group, (X, μ) aσ-finite Lebesgue space, and (g, x) →gx a non-singular, properly ergodic action ofG on (X, μ). Let furthermore Γ be the character group ofG and let Sp(G, X) ⊂ Γ denote theL ^∞-spectrum ofG on (X, μ). It has been shown in [5] that Sp(G, X) is a Borel subgroup of Γ and thatσ (Sp(G, X))<1 for every probability measureσ on Γ with lim sup_g→∞Re (g)<1, where is the Fourier transform ofσ. In this note we prove the following converse: ifσ is a probability measure on Γ with lim sup_g→∞Re (g)<1 (g)=1 then there exists a non-singular, properly ergodic action ofG on (X, μ) withσ(Sp(G, X))=1.     

Exact asymptotic behaviour of the codimensions of some P.I. algebras

Letc _n (A) denote the codimensions of a P.I. algebraA, and assumec _n (A) has a polynomial growth: . Then, necessarily,q∈ℚ [D3]. If 1∈A, we show that , wheree=2.71…. In the non-unitary case, for any 0<q∈ℚ, we constructA, with a suitablek, such that . In memory of S. A. Amitsur, our teacher and friend Partially supported by Grant MM404/94 of Ministry of Education and Science, Bulgaria and by a Bulgarian-American Grant of NSF. Partially supported by NSF grant DMS-9101488.     

Denseness of holomorphic functions attaining their numerical radii

For two complex Banach spaces X and Y, (B _X; Y) will denote the space of bounded and continuous functions from B _X to Y that are holomorphic on the open unit ball. The numerical radius of an element h in (B _X; X) is the supremum of the set

Reflection equation, twist, and equivariant quantization

We prove that the reflection equation (RE) algebraL _R associated with a finite dimensional representation of a quasitriangular Hopf algebraH is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show thatL _R is a module algebra over the twisted tensor square and the double D( ). We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces. This research is partially supported by the Israel Academy of Sciences grant no. 8007/99-01.     

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that , and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.     

Iterated forcing and changing cofinalities

We weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some interations) are preserved, and it includes some forcing which changes the confinality of a regular cardinal >ℵ₁ to ℵ₀. So, using the right iteractions, we can iterate such forcing without collapsing ℵ₁. As a result, we solve the following problems of Friedman, Magidor and Avraham, by proving (modulo large cardinals) the consistency of the following with G.C.H.: (1) for everyS ⊑ ℵ₂,S or ℵ₂-S contains a closed copy of ω₁ (2) there is a normal precipitous filterD on (3) for every is regular inL (δ ∩A)} is statonary. The results can be improved to equi-consistency; this will be discussed in a future paper. The author thanks the United States-Israel Binational Science Foundation for supporting the research by grant 1110.     

Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

Ruzsa’s problem on sets of recurrence

The main purpose of this paper is to prove the existence of Poincaré sequences of integers which are not van der Corput sets. This problem was considered in I. Ruzsa’s expository article [R₁] (1982–83) on correlative and intersective sets. Thus the existence is shown of a positive non-continuous measureμ on the circle which Fourier transform vanishes on a set of recurrence, i.e.S={n ∈Z; (n)=0} is a set of recurrence but not a van der Corput set. The method is constructive and involves some combinatorial considerations. In fact, we prove that the generic density condition for both properties are the same.     

Quantization of orbit bundles in \mathfrakg\mathfrakln* (\mathbbC) \mathfrak{g}\mathfrak{l}_n^* (\mathbb{C})

Let G be the complex general linear group and its Lie algebra equipped with a factorizable Lie bialgebra structure; let U_ħ() be the corresponding quantum group. We construct explicit U_ħ()-equivariant quantization of Poisson orbit bundles O _λ → O _μ in *.     

r-Maximal major subsets

The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂_rm B) arose in attempts to extend Lachlan’s decision procedure for the αε-theory of ℰ^*, the lattice of r.e. sets modulo finite sets, and Soare’s theorem thatA andB are automorphic if their lattice of supersets ℒ^*(A) and ℒ^*(B) are isomorphic finite Boolean algebras. We characterize the r.e. setsA with someB⊂_rm A as those with a Δ₃ function that for each recursiveR _i specifiesR _i or as infinite on and to be preferred in the construction ofB. There are r.e.A andB with ℒ^*(A) and ℒ^*(B) isomorphic to the atomless Boolean algebra such thatA has anrm subset andB does not. Thus 〈ℰ^*,A〉 and 〈ℰ^*,B〉 are not even elementarily equivalent. In every non-zero r.e. degree there are r.e. sets with and withoutrm subsets. However the classF of degrees of simple sets with norm subsets satisfies . The authors were partially supported by NSF Grants MCS 76-07258, MCS 77-04013 and MCS 77-01965 respectively.     

Degenerate principal series representations of Sp(<Emphasis Type="Italic">p,q</Emphasis>)

Letp>q and letG=Sp(p, q). LetP=LN be the maximal parabolic subgroup ofG with Levi subgroupL≅GL _q (ℍ)×Sp(p−q). Forsεℂ andμ a highest weight of Sp(p−q), let п_s,μ be the representation ofP such that its restriction toN is trivial and ⊠T _p-q ^μ , where det _q is the determinant character of GL _q (ℍ) andT _p-q ^μ is the irreducible representation of Sp(p−q) with highest weightμ. LetI _p,q(s, μ) be the Harish-Chandra module of the induced representation Ind _P ^G . In this paper, we shall determine the module structure and unitarity ofI _{p, q}(s, μ). Partially supported by NUS grant R-146-000-026-112.     

Entropy and uniform distribution of orbits in\mathbb{T}

We present a class of integer sequences {c _n } with the property that for everyp-invariant and ergodic positive-entropy measure μ on L ₂ , {c _n x (mod 1)} is uniformly distributed for μ-almost everyx. This extends a result of B. Host, who proved this for the sequence {q ⁿ }, forq relatively prime top. Our class of sequences includes, for instance, the sequencec _n =Мf _i (n)q _i ⁿ , where the numbersq _i are distinct and are relatively prime top andf _i are any polynomials. More generally, recursion sequences for which the free coefficient of the recursion polynomial is relatively prime top are in this class as well, provided they satisfy a simple irreducibility condition. In the multi-dimensional case we derive sufficient conditions for a pair of endomorphisms (withA diagonal) and anA-invariant and ergodic measure μ, such thatB-orbits of the form {B ⁿ ω} are uniformly distributed for μ-almost every .     

An Operator Corona Theorem for Some Subspaces of <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

Let E, E_* be separable Hilbert spaces. If S is an open subset of , then denotes the space of all functions that are holomorphic in , and bounded and continuous on . In this article we prove the following results:

Extensions of the Hausdorff-Young theorem

In this paper the clasical Hausdorff-Young theorem, which states that iff ∈L ^p, 1≦p≦2, on the line and is its Fourier transform, then whereq ⁻¹+p ⁻¹=1, is extended in two ways for certain Orlicz spacesL ^Φ. IfL ^Φ is based on (G, μ), (1) an arbitrary compact topological group with Haar measure, and (2) a locally compact abelian topological group andμ is again the Haar measure, then the above inequality is extended to these cases. Various other related results and remarks are also included. Dedicated to the memory of my nephew, K. Ramakrishna, who appeared to be so brilliant. This research was supported by the NSF Grants GP-5921 and GP-7678.     

Concentration and influences

Consider the discrete cube Ω={0,1} ^N , provided with the uniform probabilityP. We denote byd(x, A) the Hamming distance of a pointx of Ω and a subsetA of Ω. We define the influenceI(A) of theith coordinate onA as follows. Forx in Ω, consider the pointT _i (x) obtained by changing the value of theith coordinate. Then We prove that we always have Since it is easy to see that , this recovers the well known fact that ∫_Ω d(x, A)dP(x) is at most of order whenP(A)≥1/2. The new information is that ∫_Ω d(x, A)dP(x) can be of order only ifA reassembles the Hamming ball {x; ∑_1≤N x _i ≥N/2}.