On a maximal torus in subgroups of a general linear group

Let k be a field, K/k a finite extension of it of degree n. We denote G=Aut(kK), G_o=Aut(k K) and fix in K a basis ω₁,...,ω_n over k. In this basis, to any automorphism group of kK there corresponds a matrix group, which is denoted by the same symbol. Let G′≤G., In this paper, the conditions under which G′⊎G_o is a maximal torus in G′ are studied. The calculation of N_G′(G′⊎G_o) is carried out, provided that thee conditions are fulfilled. The case G′=SL (kK) is of particular interset. It is known that for Galois extensions and for extensions of algebraic number fields, G′⊎G_o is a maximal torus in G′. Bibligraphy: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995. pp. 15–22.     

An extension theorem for separable Banach spaces

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(ω^ω), is either isomorphic toC(ω^ω) or toc _u.     

Contractions of infrainvariant systems of subgroups

We create a method which allows an arbitrary group G with an infrainvariant system ℒ(G) of subgroups to be embedded in a group G* with an infrainvariant system ℒ(G*) of subgroups, so that G _α^* ∩G ∈ ℒ(G) for every subgroup G _α^* ∩G ∈ ℒ(G*) and each factor B/A of a jump of subgroups in ℒ(G*) is isomorphic to a factor of a jump in ℒ(G), or to any specified group H. Using this method, we state new results on right-ordered groups. In particular, it is proved that every Conrad right-ordered group is embedded with preservation of order in a Conrad right-ordered group of Hahn type (i.e., a right-ordered group whose factors of jumps of convex subgroups are order isomorphic to the additive group ℝ); every right-ordered Smirnov group is embedded in a right-ordered Smirnov group of Hahn type; a new proof is given for the Holland–McCleary theorem on embedding every linearly ordered group in a linearly ordered group of Hahn type.     

Homology of the double and the triple loop spaces¶of E6, E7, and E8

We study the mod p homology of the double and the triple loop spaces of exceptional Lie groups E ₆, E ₇, and E ₈ through the Eilenberg–Moore spectral sequence and the Serre spectral sequence using homology operations. The Bockstein actions on them are also determined. As a result, the Eilenberg–Moore spectral sequences of the path loop fibrations converging to H _*(Ω² G;? _p ) and H _*(Ω³ G;? _p ) collapse at the E ²-term for any compact simple Lie group G. Received: 11 November 1999     

On the Oka principle in a Banach space,II

 Let X be one of the Banach spaces c ₀ , ℓ _p , 1≤p<∞; Ω⊂X pseudoconvex open, a holomorphic Banach vector bundle with a Banach Lie group G ^* for structure group. We show that a suitable Runge-type approximation hypothesis on X, G ^* (which we also prove for G ^* a solvable Lie group) implies the vanishing of the sheaf cohomology groups H ^q (Ω, 𝒪 ^E ), q≥1, with coefficients in the sheaf of germs of holomorphic sections of E. Further, letting 𝒪^Γ (𝒞^Γ) be the sheaf of germs of holomorphic (continuous) sections of a Banach Lie group bundle Γ→Ω with Banach Lie groups G, G ^* for fiber group and structure group, we show that a suitable Runge-type approximation hypothesis on X, G, G ^* (which we prove again for G, G ^* solvable Lie groups) implies the injectivity (and for X=ℓ₁ also the surjectivity) of the Grauert–Oka map H ¹ (Ω, 𝒪^Γ)→H ¹ (Ω, 𝒞^Γ) of multiplicative cohomology sets. Received: 1 March 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 32L20, 32L05, 46G20 RID="*" ID="*" Kedves Laci Móhan kisfiamnak. RID="*" ID="*" To my dear little Son     

Quantum cohomology of <Emphasis Type="Italic">G/P</Emphasis> and homology of affine Grassmannian

Let G be a simple and simply-connected complex algebraic group, P ⊂ G a parabolic subgroup. We prove an unpublished result of D. Peterson which states that the quantum cohomology QH ^*(G/P) of a flag variety is, up to localization, a quotient of the homology H _*(Gr _G ) of the affine Grassmannian Gr _G of G. As a consequence, all three-point genus-zero Gromov–Witten invariants of G/P are identified with homology Schubert structure constants of H _*(Gr _G ), establishing the equivalence of the quantum and homology affine Schubert calculi.     

Weight functions on groups and an amenability criterion for Beurling algebras

The paper is devoted to the study of weights on groups. A connection between weight functions and harmonic functions is established. A relationship between the weight theory on groups with the “Tychonoff property” and the theory of bounded cohomology is presented. It is proved that the Beurling algebraℓ1 (G, ω) constructed for the weightω is amenable if and only if the groupG is amenable and the weightω is equivalent to a multiplicative characterχ:G→ℝ₊. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 448–460, September, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-00974 and by the INTAS Foundation under grant No. 94-3420.     

On amenable groups of automorphisms on Von Neumann algebras

LetG ⊂ Aut ℳ be a countable group, ℳ a Von Neumann algebra. LetE be a set of pure states on ℳ such thatG*E ⊂E, S ^G be the set ofG invariant states on ℳ andS _E ^G =S ^G ∩w* cl coE. We investigate in this paper some geometric properties for the setS _E ^G which turn out to be equivalent to amenability for the groupG. For example, we show thatS _E ^G ⊂ ℳ_* (S _E ^G has the WRNP) implies that ℳ contains minimal projections (ê containsfinite G invariant orbits) hold true, for all ℳ iffG is amenable. Furthermore we show that ifG is amenable thenS ^G ∩M _* ^⊥ contains a big set, thus improving results obtained by Ching Chou in [2]. These results imply that no action of an amenable countable groupG on an arbitraryW* algebra ℳ iss — strongly ergodic. Moreover cardS ^G ∩M _* ^⊥ ≧2 ^c (see M. Choda [4], K. Schmidt [21] and compare with A. Connes and B. Weiss [5]). The author gratefully acknowledges the support of an Izaak Walton Killam Memorial Senior Fellowship.     

Hamiltonism and Partially Square Graphs

 Given a graph G, we define its partially square graph G ^* as the graph obtained by adding edges uv whenever the vertices u and v have a common neighbor x satisfying the condition N _G[x]⊆N _G[u]∪N _G [v], where N _G[x]=N _G(x)∪{x}. In particular, this condition is satisfied if x does not center a claw (an induced K _1,3). Obviously G⊆G ^*⊆G ², where G ² is the square of G. We prove that a k-connected graph (k≥2) G is hamiltonian if the independence number α(G ^*) of G ^* does not exceed k. If we replace G ^* by G we get a well known result of Chvátal and Erdo?s. If G is claw-free and G ^* is replaced by G ² then we obtain a result of Ainouche, Broersma and Veldman. Relationships between connectivity of G and independence number of G ^* for other hamiltonian properties are also given in this paper. Received: June 17, 1996 Revised: October 30, 1998     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

Spaces with “small” annihilators

In this note one investigates the properties of subspaces G of C(S), such that G¹ is “not a very large part” of the space C(S)^*. The fundamental result is: if G¹ is reflexive, then every operator from G^* into ℓ₂ is absolutely summable. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 65, pp. 192–195, 1976.     

On a width of verbal subgroups of certain free constructions

We study into widths of verbal subgroups of HNN-extensions, and of groups with one defining relation. It is proved that if a group G^* is an HNN-extension and the connected subgroups in G^* are distinct from a base of the extension, then every verbal subgroup V(G^*) has infinite width relative to a finite proper set V of words. A similar statement is proven to hold for groups presented by one defining relation and ≥3 generators. to Yurii I. Merzlyakov dedicated Supported by RFFR grant No. 93-01-01513. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 494–517, September–October, 1997.     

Bounds for exponential sums modulo <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

In this paper we consider exponential sums over subgroups G ⊂ ℤ _q ^* . Using Stepanov’s method, we obtain nontrivial bounds for exponential sums in the case where q is a square of a prime number. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 81–94, 2005.     

Maxwell equations and the direction of the electromagnetic field

Let M₀ be the Minkowski space, let Λ₂(M₀) be the space of bivectors in M₀, and let G¹ ⊂ Λ₂(M₀) be the manifold of directions of the physical space, consisting of simple bivectors with square −1. A mapping F: U → Λ₂(M₀), U ⊂ ℝ⁴, satisfying the Maxwell equations is regarded as the tensor of an electromagnetic field in vacuum. The field is described on the basis of a special decomposition F = eω + h(*ω), where the mapping ω: U → G¹ is called the direction of the field, and e: U → (0, +∞) and h: U → ℝ are the electric and magnetic coefficients of the field. The Maxwell equations are reformulated in terms of ω, e, and h. Electromagnetic fields whose set of directions is a point or a one-dimensional subset of G¹ are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 118–146.     

Strong Arens irregularity of Beurling algebras with a locally convex topology

Let G be a locally compact group with a weight function ω. Recently, we have shown that the Banach space L₀^∞ (G,1/ω) can be identified with the strong dual of L¹(G, ω)equipped with some locally convex topologies τ. Here we use this duality to introduce an Arens multiplication on (L¹(G, ω), τ)**, and prove that the topological center of (L¹(G, ω), τ)** is (L¹(G, ω); this enables us to conclude that (L¹(G, ω), τ) is Arens regular if and only if G is discrete. We also give a characterization for Arens regularity of L₀^∞ (G, 1/ω)¹. Received: 8 March 2005     

Characterizations of finite groups by sets of orders of their elements

For a finite group G, ω(G) denotes the set of orders of its elements. If ω is a subset of the set of natural numbers, h(ω) stands for the number of pairwise nonisomorphic finite groups G for which ω(G)=ɛ. We prove that h(ω(G))=1, if G is isomorphic to S₉, S₁₁, S₁₂, S₁₃, or A₁₂, and h(ω(G))=2 if G is isomorphic to S₂(6) or to O ₈ ⁺ (2). 01 Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 37–53, January–February, 1997.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function X ? ^*\Bbb CX \rightarrow {}{^{\ast}{\Bbb C}} . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient M_w(X)/M₀(X){\cal M}_{\omega}(X)/{\cal M}_0(X) , for certain external subspaces M₀(X), M_w(X){\cal M}_0(X), {\cal M}_{\omega}(X) of the hyperfinite dimensional Banach space ^*\Bbb C^X{}{^{\ast}{\Bbb C}}^X , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and M_w(G)/M₀(G){\cal M}_{\omega}(G)/{\cal M}_0(G) are isometrically isomorphic as Banach algebras.     

Decomposable graphs and definitions with no quantifier alternation

Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D₀(G) be the version of D(G) where we do not allow quantifier alternations in Φ. Define q₀(n) to be the minimum of D₀(G) over all graphs G of order n.We prove that for all n we have

log^*n−log^*log^*n−2≤q₀(n)≤log^*n+22,