Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

On the duals of Lp spaces with 0<p<1

Given a measure space < Ω,m,μ >, a locally bounded, Hausdorff topological linear space < X, τ > and a real number 0<p<1, one can define the space L_p(Ω,m,μ,X), which is, under certain assumptions, a Fréchet space if endowed with a suitable topology. M.M. Day [1] has given a necessary and sufficient condition, in terms of the properties of the measure space < Ω,m,μ >, for the dual of L_p(Ω,m,μ,C) to be trivial. In this paper a different proof along with a slight generalization is given for this result, using standard and elementary measure theoretic arguments. This revised version was published online in June 2006 with corrections to the Cover Date.     

Representation ofL p-norms and isometric embedding inL p-spaces

For fixed 1≦p<∞ theL _p-semi-norms onR ⁿ are identified with positive linear functionals on the closed linear subspace ofC(R ⁿ ) spanned by the functions |<ξ, ·>| ^p , ξ∈R ⁿ . For every positive linear functional σ, on that space, the function Φ_σ:R ⁿ →R given by Φ_σ is anL _p-semi-norm and the mapping σ→Φ_σ is 1-1 and onto. The closed linear span of |<ξ, ·>| ^p , ξ∈R ⁿ is the space of all even continuous functions that are homogeneous of degreep, ifp is not an even integer and is the space of all homogeneous polynomials of degreep whenp is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes linear isometric embeddability, in anyL _p unlessp=2. Supported by the National Science Foundation MCS-79-06634 at U.C. Berkeley.     

A Spectral Gap Property for Random Walks Under Unitary Representations

Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation $\pi ,\mathcal{H}Let G be a locally compact group and μ a probability measure on G, which is not assumed to be absolutely continuous with respect to Haar measure. Given a unitary representation p,H\pi ,\mathcal{H} of G, we study spectral properties of the operator π(μ) acting on H\mathcal{H} Assume that μ is adapted and that the trivial representation 1 _G is not weakly contained in the tensor product p?[`(p)]\pi\otimes \overline\pi We show that π(μ) has a spectral gap, that is, for the spectral radius r_spec(p(m))r_{\rm spec}(\pi(\mu)) of π(μ), we have r_spec(p(m)) < 1.r_{\rm spec}(\pi(\mu))< 1. This provides a common generalization of several previously known results. Another consequence is that, if G has Kazhdan’s Property (T), then r_spec(p(m)) < 1r_{\rm spec}(\pi(\mu))< 1 for every unitary representation π of G without finite dimensional subrepresentations. Moreover, we give new examples of so-called identity excluding groups.     

Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G ^N of orderp is permutable inN _G (P) and whenp = 2 either every cyclic subgroup ofP ∩G ^N of order 4 is permutable inN _G (P) orP is quaternion-free. Some applications of this result are given. The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China. The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).     

Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

Special orthogonal splittings of<Emphasis Type="Italic">L</Emphasis><Stack><Subscript>1</Subscript><Superscript>2<Emphasis Type="Italic">k</Emphasis></Superscript></Stack>

We show that for each positive integerk there is ak×k matrixB with ±1 entries such that puttingE to be the span of the rows of thek×2k matrix [√kI _k,B], thenE,E ^⊥ is a Kashin splitting: TheL ₁ ^2k and theL ₂ ^2k are universally equivalent on bothE andE ^⊥. Moreover, the probability that a random ±1 matrix satisfies the above is exponentially close to 1. Supported by the Israel Science Foundation.     

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.     

Rearrangement Invariant Continuous Linear Functionals on Weak <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We show that in the dual of Weak L¹ the subspace of all rearrangement invariant continuous linear functionals is lattice isometric to a space L¹(μ) and is the linear hull of the maximal elements of the dual unit ball. We also show that the dual of Weak L¹ contains a norm closed weak* dense ideal which is lattice isometric to an ℓ¹-sum of spaces of type C(K). Helmut H. Schaefer in memoriam     

On the <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group. For 1 < p < ∞, it is well-known that f * g exists and belongs to L^p(G) for all f, g L^p (G) if and only if G is compact. Here, for 2 < p < ∞, we show that f * g exists for all f, g L^p(G) if and only if G is compact. We also show that this result does not remain true for 1 < p ≤ 2. Received: 23 April 2006     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-conjecture for locally compact groups

Let G be a locally compact group, ω be a weight function on G and 1 < p < ∞. Here, we give a sufficient condition for that the weighted L ^p -space L ^p (G, ω) is a Banach algebra. Also, we get some necessary conditions on G and the weight function ω for L ^p (G, ω) to be a Banach algebra. As a consequence, we show that if G is abelian and L ^p (G, ω) is a Banach algebra, then G is σ-compact.     

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the L^p functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the L^p norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p ₀ = 2+4π ² /β ² ω ₂ , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p ₀ and p > p ₀ . We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.     

On some extensions of <Emphasis Type="Italic">p</Emphasis>-restricted completely splittable GL(<Emphasis Type="Italic">n</Emphasis>)-modules

In this paper, we calculate the space Ext _GL(n ¹ )(L _n (λ), L _n (μ)), where GL(n) is the general linear group of degree n over an algebraically closed field of positive characteristic, L _n (λ) and L _n (μ) are rational irreducible GL(n)-modules with highest weights λ and μ, respectively, the restriction of L _n (λ) to any Levi subgroup of GL(n) is semisimple, λ is a p-restricted weight, and μ does not strictly dominate λ. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 219–226, 2005.     

Gruppi Policiclici Dotati di un Automorfismo Uniforme di Ordinepq

An automorphismϕ of a groupG is said to be uniform il for everyg ∈G there exists anh ∈G such thatG=h ⁻¹ h ^ρ . It is a well-known fact that ifG is finite, an automorphism ofG is uniform if and only if it is fixed-point-free. In [7] Zappa proved that if a polycyclic groupG admits an uniform automorphism of prime orderp thenG is a finite (nilpotent)p′-group. In this paper we continue Zappa’s work considering uniform automorphism of orderpg (p andq distinct prime numbers). In particular we prove that there exists a constantμ (depending only onp andq) such that every torsion-free polycyclic groupG admitting an uniform automorphism of orderpq is nilpotent of class at mostμ. As a consequence we prove that if a polycyclic groupG admits an uniform automorphism of orderpq thenZ _μ (G) has finite index inG.

Al professore Guido Zappa per il suo 90⁰ compleanno     

A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday     

On the completeness of algebraic polynomials in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(ℝ, <Emphasis Type="Italic">d</Emphasis>μ)

We prove that the theorem on the incompleteness of polynomials in the space C ₀ ^w established by de Branges in 1959 is not true for the space L _p (ℝ, dμ) if the support of the measure μ is sufficiently dense.     

Crossed products over prime rings

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.     

Countable structures with a fixed group of automorphisms

We prove that, given a countable groupG, the set of countable structures (for a suitable languageL)U _G whose automorphism group is isomorphic toG is a complete coanalytic set and ifG ≄H thenU _G is Borel inseparable fromU _H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures forL whose automorphism group is isomorphic to ℤ _p ^ℕ ,p a prime number, is Π ₁ ¹ &σ ₁ ¹ -complete.     

On order preserving contractions

Let (Ω,Σ,μ) be a measure space and letP be an operator onL ₂(Ω,Σ,μ) with ‖P‖≦1,Pf≧0 a.e. wheneverf≧0. If the subspaceK is defined byK={x| ||P ⁿ x||=||P ^*n x||=||x||,n=1,2,...} thenK=L ₂(Ω,Σ₁,μ), where Σ₁ ⊂ Σ and onK the operatorP is “essentially” a measure preserving transformation. Thus the eigenvalues ofP of modulus one, form a group under multiplication. This last result was proved by Rota for finiteμ here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices. The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force.     

On the ideal centre of the space of vector valued integrable functions

Let E be a Banach lattice and L¹(μ, E) be the space of E-valued Bochner integrable functions. Some order properties of L¹(μ, E) are given. It is shown that L_s^∞(μ, Z(E)) is the ideal centre of L¹(μ, E) and it is obtained a Radon-Nikodym type theorem for B -integrable functions.