Ideal structure of Beurling algebras on [FC]− groups

It is proved that for an [FC]^? group G, the Beurling algebra L_ω¹(G) is 1-regular if and only if ω is non-quasianalytic. As an application the Wiener property is deduced. Further for a σ-compact [FD]^? group G, points in Prim₁L_ω¹(G) are shown to be spectral provided that ω satisfies Shilov's conditions.     

Derived space of L p (G, H)

Let G denote a locally compact abelian group and H a separable Hilbert space. Let L _p (G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual L _p space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space L _p (G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then L _p ⁰ (G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let S _p (G, H) be the L ₁(G) Banach module of functions in L _p (G, H) having unconditionally convergent Fourier series in L _p -norm. We show that S _p (G, H) coincides with the derived space L _p ⁰ (G, H), as in the scalar valued case. We also show that if G is compact and abelian, then L _p ⁰ (G, H) = L ₂(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ L _p (G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in L _p -norm, then F ∈ L ₂(G, H). Let Ω be the set of all functions on Γ taking only the values 1, ?1 and Ω* be the set of all complex-valued functions on Γ having absolute value 1. As an application of the derived space L _p ⁰ (G, H), we prove the following main result of this paper. Let G be a compact abelian group and F be an H-valued function on the dual group Γ such that $$ \sum \omega (\gamma )F(\gamma )\gamma $$ is a Fourier-Stieltjes series of some measure µ ∈ M(G, H) for every scalar function ω such that |ω(γ)| = 1. Then F ∈ l ²(Γ, H).     

Признак Дини-Липшица и сходимость рядов Фу рье по мультипликативны м системам

Fourier serieses with respect to Vilenkin multiplicative systems are studied. It is proved that a Dini-Lipschitz condition imposed on the group or group integral modulus of continuity is sufficient for the convergence of the series in the normC(·) orL(·), respectively. Necessity in the classesH _ω andH _ω ¹ of that condition is also proved. A boundedness criterion of the sequence of partial sums inH _ω andH _ω ¹ is also obtained.     

Identifications in modular group algebras

Let G be a group, S a subgroup of G, and F a field of characteristic p. We denote the augmentation ideal of the group algebra FG by ω(G). The Zassenhaus-Jennings-Lazard series of G is defined by D_n(G)=G∩(1+ωⁿ(G)). We give a constructive proof of a theorem of Quillen stating that the graded algebra associated with FG is isomorphic as an algebra to the enveloping algebra of the restricted Lie algebra associated with the D_n(G). We then extend a theorem of Jennings that provides a basis for the quotient ωⁿ(G)/ωⁿ⁺¹(G) in terms of a basis of the restricted Lie algebra associated with the D_n(G). We shall use these theorems to prove the main results of this paper. For G a finite p-group and n a positive integer, we prove that G∩(1+ω(G)ωⁿ(S))=D_n+1(S) and G∩(1+ω²(G)ωⁿ(S))=D_n+2(S)D_n+1(S∩D₂(G)). The analogous results for integral group rings of free groups have been previously obtained by Gruenberg, Hurley, and Sehgal.     

A definable failure of the singular cardinal hypothesis

We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ ⁺). Then with further forcing we show that it is consistent that GCH fails at ? _ω , ? _ω strong limit, while there is a lightface definable wellorder of H(? _ω+1) (“definable failure” of the singular cardinal hypothesis at ? _ω ). The large cardinal hypothesis used is the existence of a κ ⁺⁺-strong cardinal, where κ is κ ⁺⁺-strong if there is an embedding j: V → M with critical point κ such that H(κ ⁺⁺) ? M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ ⁺⁺-strong cardinal is used throughout.     

Real loci of based loop groups

Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K is the fixed point set of an involutive automorphism σ. This induces an involutive automorphism τ of the based loop space Ω(G). There exists a maximal torus T ⊂ G such that the canonical action of T × S ¹ on Ω(G) is compatible with τ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat’s convexity theorem. Namely, the images of Ω(G) and Ω(G) ^τ (fixed point set of τ) under the T × S ¹ moment map on Ω(G) are equal. The space Ω(G) ^τ is homotopy equivalent to the loop space Ω(G/K) of the Riemannian symmetric space G/K. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in \mathbbZ₂ {\mathbb{Z}_2} of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [BS] had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe [HHP].     

Integrable cocycles and global deformations of Lie algebra of type G2 in characteristic 2

All classes of integrable cocycles in H²(L,L) are obtained for Lie algebra of type G₂ over an algebraically closed field of characteristic 2. It is proved that there exist only two orbits of classes of integrable cocycles with respect to automorphism group. The global deformation is shown to exist for any nontrivial class of integrable cocycles. These deformations are isomorphic to one of the two algebras of Cartan type, one of which being S(3:1,ω) while the other H(4:1,ω).     

Translation-invariant linear forms on C0(G), C(G), Lp(G) for noncompact groups

Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L¹(G), C(G), and L^∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C₀(G) and L^p(G) for 1 < p < ∞ have such forms if and only if $\text{G}\text{H}$ is a torsion group for some open σ-compact subgroup H of $\text{G}$. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.     

X-inner automorphisms of crossed products and semi-invariants of Hopf algebras

LetR*G be a crossed product of the groupG over the prime ringR and assume thatR*G is also prime. In this paper we study unitsq in the Martindale ring of quotientsQ ₀(R*G) which normalize bothR and the group of trivial units ofR*G. We obtain quite detailed information on their structure. We then study the group ofX-inner automorphisms ofR*G induced by such elements. We show in fact that this group is fairly close to the group of automorphisms ofR*G induced by certain trivial units inQ ₀(R)*G. As an application we specialize to the case whereR=U(L) is the enveloping algebra of a Lie algebraL. Here we study the semi-invariants forL andG which are contained inQ ₀(R*G) and we obtain results which extend known properties ofU(L). Finally, every cocommutative Hopf algebraH over an algebraically closed field of characteristic 0 is of the formH=U(L)*G. Thus we also obtain information on the semi-invariants forH contained inQ ₀(H). Research supported in part by N.S.F. Grant Nos. MCS 83-01393 and MCS 82-19678.     

Bimeasure algebras on locally compact groups

For locally compact groups G and H, let BM(G, H) denote the Banach space of bounded bilinear forms on C₀(G) × C₀(H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM(G, H) which generalize the convolution structure of M(G × H) and which make BM(G, H) into a K_G²-Banach ${}^1$-algebra, where K_G is Grothendieck's universal constant. Various topics relating to the ideal structure of BM(G, H) and the lifting of unitary representations of G × H to ${}^1$-representations of BM(G, H) are investigated.     

Operator space structure on Feichtinger's Segal algebra

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger's remarkable Segal algebra S₀(G). In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow S₀(G) with an operator space structure. With this structure S₀(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G), and of the group algebra L¹(G). We show that this operator space structure is consistent with the major functorial properties: (i) completely isomorphically (operator projective tensor product), if H is another locally compact group; (ii) the restriction map is completely surjective, if H is a closed subgroup; and (iii) is completely surjective, where N is a normal subgroup and . We also show that S₀(G) is an invariant for G when it is treated simultaneously as a pointwise algebra and a convolutive algebra.     

Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom

We consider a periodic magnetic Schrödinger operator Hh, depending on the semiclassical parameter h>0, on a noncompact Riemannian manifold M such that H¹(M,R)=0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface S. First, we prove upper and lower estimates for the bottom λ₀(Hh) of the spectrum of the operator Hh in L²(M). Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrarily large number of spectral gaps for the operator Hh in the region close to λ₀(Hh), as h→0. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.     

Modules over integer group rings of locally soluble groups with minimax restriction

Let ? be the ring of integers, and A be a ?G-module, where A/C _A (G) is not a minimax ?-module, C _G (A) = 1, and G is a locally soluble group. Let L _nm(G) be the system of all subgroups H ?? G such that quotient modules A/C _A (H) are not minimax Z-modules. The author studies ?G-modules A such that L _nm(G) satisfies the minimal condition as an ordered set. It is proved that a locally soluble group G with these conditions is soluble. The structure of the group G is described.     

On approximation of a class by another class and extremal subspaces inL 1

Получена оценка (в опр еделенном смысле неу лучшаемая) наилучшего приближе ния в метрикеL ₁=L₁(0,2π) 2π-перио дических функций кла сса W^rH^ω = {f:f∈C^r,ω(f ^(r),δ) ≦ ω(δ)}, r = 0, 1, ..., (ω(δ) — выпуклый вверх мо дуль непрерывности) ф ункциями класса W ₁ ^r+v N = {?: ?^r+v?1)(t) —локально абсолютн о непрерывна, ∥?(^r+v∥L₁≦N}, v≧2. Доказано, что каждое п одпространство нече тной размерности, реализу ющее поперечник (по Колмогорову) класс а W ₁ ^r+v в L₁, обладает аналог ичным свойством относител ьно класса W^rH^ω при любом выпуклом вверх ω(δ).     

Transformation de Fourier des distributions de type positif sur un groupe de Lie unimodulaire

Call a locally compact group G, C¹-unique, if L¹(G) has exactly one (separating) C¹-norm. It is easy to see that a ¹-regular group G is C¹-unique and that a C¹-unique group is amenable. For connected groups G it is proved that G is C¹-unique, if the interior R(G)⁰ of a certain part R(G) of Prim(G), called the regular part of Prim(G), is dense in Prim(G), and that C¹-uniqueness of G implies the density of R(G) in Prim(G). From this it is derived that a connected group of type I is C¹-unique if and only if R(G)⁰ is dense in Prim(G). For exponential G, a quite explicit version of this result in terms of the Lie algebra of G is given. As an easy consequence, examples of amenable groups, which are not C¹-unique, and C¹-unique groups, which are not ¹-regular are obtained. Furthermore it is shown that a connected locally compact group G is amenable if and only if L¹(G) has exactly one C¹-norm, which is invariant under the isometric ¹-automorphisms of L¹(G).     

On ideals in the bidual of the Fourier algebra and related algebras

Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group, A(G×H) the Fourier algebra of G×H, and UC₂(G×H) the space of uniformly continuous functionals in VN(G×H)=A^∗(G×H). We use weak factorization of operators in the group von Neumann algebra VN(G×H) to prove that there exist at least ²b(G)² left ideals of dimensions at least ²b(G)² in A(G×H)^∗∗ and in UC₂^∗(G×H). We show that every nontrivial right ideal in A(G×H)^∗∗ and in UC₂^∗(G×H) has dimension at least ²b(G)².     

Equivariant gerbes over compact simple Lie groups

Using groupoid S¹-central extensions, we present, for a compact simple Lie group G, an infinite dimensional model of S¹-gerbe over the differential stack G/G whose Dixmier–Douady class corresponds to the canonical generator of the equivariant cohomology H_G³(G). To cite this article: K. Behrend et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Riesz transforms on a solvable Lie group of polynomial growth

We study Riesz transforms associated with a sublaplacian H on a solvable Lie group G, where G has polynomial volume growth. It is known that the standard second order Riesz transforms corresponding to H are generally unbounded in L^p(G). In this paper, we establish boundedness in L^p for modified second order Riesz transforms, which are defined using derivatives on a nilpotent group G_N associated with G. Our method utilizes a new algebraic approach which associates a distinguished choice of Cartan subalgebra with the sublaplacian H. We also obtain estimates for higher derivatives of the heat kernel of H, and give a new proof (without the use of homogenization theory) of the boundedness of first order Riesz transforms. Our results can be generalized to an arbitrary (possibly non-solvable) Lie group of polynomial growth.     

New Orlicz-Hardy spaces associated with divergence form elliptic operators

Let L be the divergence form elliptic operator with complex bounded measurable coefficients, ω the positive concave function on (0,∞) of strictly critical lower type pω∈(0,1] and ρ(t)=t−1/ω−1(t−1) for t∈(0,∞). In this paper, the authors study the Orlicz-Hardy space Hω,L(Rn) and its dual space BMOρ,L^*(Rn), where L^* denotes the adjoint operator of L in L²(Rn). Several characterizations of Hω,L(Rn), including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(Rn) are also given. As applications, the authors show that the Riesz transform ∇L−1/2 and the Littlewood-Paley g-function gL map Hω,L(Rn) continuously into L(ω). The authors further show that the Riesz transform ∇L−1/2 maps Hω,L(Rn) into the classical Orlicz-Hardy space Hω(Rn) for and the corresponding fractional integral L−γ for certain γ>0 maps Hω,L(Rn) continuously into , where is determined by ω and γ, and satisfies the same property as ω. All these results are new even when ω(t)=tp for all t∈(0,∞) and p∈(0,1).     

Homogeneous almost normal Riemannian manifolds

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive Lie group G of isometries for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν such that the canonical projection p: (G, ν) → (G/H, µ) is a Riemannian submersion and the norm | · | of the product ν is at least the bi-invariant Chebyshev normon G defined by the space (M,µ).We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous G-normal Riemannian manifold with simple Lie group G, the unit ball of the norm | · | is a Löwner-John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group G. Some unsolved problems are posed.