Crossed products by finite group actions with the Rokhlin property

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: (a) AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. (b) Simple unital AH algebras with slow dimension growth and real rank zero. (c) C*-algebras with real rank zero or stable rank one. (d) Simple C*-algebras for which the order on projections is determined by traces. (e) C*-algebras whose quotients all satisfy the Universal Coefficient Theorem. (f) C*-algebras with a unique tracial state. Along the way, we give a systematic treatment of the derivation of direct limit decompositions from local approximation conditions by homomorphic images which are not necessarily injective.     

The complex separation of real surfaces and extensions of Rokhlin congruence

Summary The subject of this paper is the problem of topological arrangement of real algebraic curves on real algebraic surfaces. In this paper I extend Rokhlin, Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences and give some applications of this extension. Among these applications there are new restrictions for topological arrangement of real algebraic curves of a given degree on hyperboloid and ellipsoid, new restrictions for complex orientations of curves on a hyperboloid and the topological classification of pairs made of a non-connected cubic surfaces in P ³ and its regular intersection with a quadric.Obalatum X-1992&14-10-1993     

Crossed products by finite group actions with certain tracial Rokhlin property

We introduce a special tracial Rokhlin property for unital C~*-algebras. Let A be a unital tracial rank zero C~*-algebra(or tracial rank no more than one C~*-algebra). Suppose that α : G → Aut(A) is an action of a finite group G on A, which has this special tracial Rokhlin property, and suppose that A is a α-simple C~*-algebra. Then, the crossed product C~*-algebra C~*(G, A, α) has tracia rank zero(or has tracial rank no more than one). In fact,we get a more general results.     

Relatively hyperbolic extensions of groups and Cannon-Thurston maps

Let 1 → (K, K ₁) → (G, N _G (K ₁)) → (Q, Q ₁) → 1 be a short exact sequence of pairs of finitely generated groups with K ₁ a proper non-trivial subgroup of K and K strongly hyperbolic relative to K ₁. Assuming that, for all g ∈ G, there exists k _g ∈ K such that gK ₁ g ⁻¹ = k _g K ₁ k _g⁻¹, we will prove that there exists a quasi-isometric section s: Q → G. Further, we will prove that if G is strongly hyperbolic relative to the normalizer subgroup N _G (K ₁) and weakly hyperbolic relative to K ₁, then there exists a Cannon-Thurston map for the inclusion i: Γ _K → Γ _G .     

Ergodicity and weak-mixing of homogeneous extensions of measure-preserving transformations with applications to Markov shifts

Given a homogeneous extensionS of a measure-preserving transformationT, we provide necessary and sufficient conditions for the ergodicity and weak-mixing ofS in terms of functional equations. We then apply our findings to the case whenT is a Markov shift and the associated skewing function ofS depends on a finite number of coordinates. In this case, we obtain a simplification to the appropriate functional equations.     

Group extensions and free actions by finite groups on solvmanifolds

The purpose of this paper is to generalize Yalçin's result [Trans. Amer. Math. Soc. 352 , 2689–2700 (2000)] from tori to solvmanifolds. As a result, we prove that if (?_p)^r acts freely on a solvmanifold M of dimension n, then r ≤ n (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Central extensions of Lie algebras graded by finite root systems

Lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by Berman and Moody, Benkart and Zelmanov, and Neher. In this paper we determine the central extensions of these Lie algebras and hence describe them completely up to isomorphism. Received: 22 May 1997 / in final form: 13 January 1999     

Universal finite group extensions and a non-splitting theorem

LetG andK be finite groups whose orders have a common prime divisor. Then there is a groupK ^* closely related toK for which there is a non-split extension ofK ^* byG. I wish to express thanks to the Mathematics Institute of the Hebrew University of Jerusalem for its hospitality from September to December 1972, and to Dr. Avinoam Mann for his helpful comments.     

Relatively prime polynomials and nonsingular Hankel matrices over finite fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field Fq to be relatively prime turns out to be identical with the probability for an n×n Hankel matrix over Fq to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over Fq of given degrees and for the number of n×n Hankel matrices over Fq of a given rank.     

Factor orbit equivalence of compact group extensions and classification of finite extensions of ergodic automorphisms

In §§1–5, we classifyn-point extensions of ergodic automorphisms up to factor orbit-equivalence (which is the natural analogue of factor isomorphism). This classification is in terms of conjugacy classes of subgroups of the symmetric group onn points, and parallels D. Rudolph’s classification ofn-point extensions of Bernoulli shifts up to factor isomorphism. In §6, we give another proof of A. Fieldsteel’s theorem on factor orbit-equivalence of compact group extensions.     

Hensel lifting and bivariate polynomial factorisation over finite fields

This paper presents an average time analysis of a Hensel lifting based factorisation algorithm for bivariate polynomials over finite fields. It is shown that the average running time is almost linear in the input size. This explains why the Hensel lifting technique is fast in practice for most polynomials.

     

The finiteness of the number of symmetrical extensions of a locally finite tree by a finite graph

The problem of incidence of an acoustic wave on the interface between media with impedance interface conditions is considered. An approximate method is proposed for calculating the result of diffraction under such conditions. The method is implemented as a computer program, and the result is compared with the analytical solution for the impedance conditions and with the calculations by a program for the contact boundary conditions. Good accuracy of the method and high computation speed are demonstrated, which allow one to apply the proposed approximate method to solving both direct and inverse problems of acoustics.     

Remainders in extensions and finite unions of locally compact sets

We discuss the relationship between properties of spaces and their remainders in extensions from the class P _fin of all finite unions of locally compact spaces. In particular, we show that a space X ∈ P _fin iff the remainder in each (some) compactification of X is in P _fin. Then we study the class P _fin and the relationship between the remainders of a space from this class in compact extensions and give a generalization of the theorem of Henriksen-Isbell.