The Rokhlin property and the tracial topological rank

Let A be a unital separable simple C∗-algebra  with TR(A)?1 and α be an automorphism. We show that if α satisfies the tracially cyclic Rokhlin property then . We also show that whenever A has a unique tracial state and α^m is uniformly outer for each m(≠0) and α^r is approximately inner for some r>0, α satisfies the tracial cyclic Rokhlin property. By applying the classification theory of nuclear C∗-algebras, we use the above result to prove a conjecture of Kishimoto: if A is a unital simple -algebra of real rank zero and α∈Aut(A) which is approximately inner and if α satisfies some Rokhlin property, then the crossed product is again an -algebra of real rank zero. As a by-product, we find that one can construct a large class of simple C∗-algebras with tracial rank one (and zero) from crossed products.     

Semiprime ideals in rings with finite group actions

LetR be a ring, G a finite group of automorphisms acting on R, and R^G the-fixed subring of R. We prove that if R is semiprime with no additive ¦ G¦-torsion, then R is left Goldie if and only if R^G is left Goldie. By coupling this with an examination of the prime ideal structures of R^G and R, we are able to prove that if ¦G ¦ is invertible in R and R^G is left Noetherian, then R satisfies the-ascending chain condition on semiprime ideals, every semiprime factor ring of R is left Goldie, and nil subrings of R are nilpotent. For the pair R^G and R, we also consider various other properties of prime and maximal ideals such as lying over, going up, going down, and incomparability.     

Equivariant autoequivalences for finite group actions

The familiar Fourier-Mukai technique can be extended to an equivariant setting where a finite group G acts on a smooth projective variety X. In this paper we compare the group of invariant autoequivalences Aut(DbG(X)) with the group of autoequivalences of DG(X). We apply this method in three cases: Hilbert schemes on K3 surfaces, Kummer surfaces and canonical quotients.     

Crossed products by a coalgebra

We introduce the notion of a crossed product of an al¬gebra by a coalgebra C, which generalises the notion of a crossed product by a bialgebra well-studied in the theory of Hopf algebras. The result of such a crossed product is an algebra which is also a right C-comodule. We find the necessary and sufficient conditions for two coalgebra crossed products be equivalent. We show that the two-dimensional quantum Euclidean group is a coalgebra crossed product. The paper is completed with an appendix describing the dualisation of construction of coalgebra crossed products.     

Relatively finite measure-preserving extensions and lifting multipliers by Rokhlin cocycles

We show that under some natural ergodicity assumptions, extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L^∞-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank modules associated to an extension of measure-preserving systems in the setting of a non-singular base.     

Large orbit sizes in finite group actions

In this paper, we study the relations of the sizes of various sections of finite linear groups and the largest orbit size of the linear group actions. We also study various applications of those orbit theorems.     

Stable Rank One and Real Rank Zero for Crossed Products by Finite Group Actions with the Tracial Rokhlin Property

The authors prove that the crossed product of an infinite dimensional simple separable unital C＊-algebra with stable rank one by an action of a finite group with the tracial Rokhlin property has again stable rank one. It is also proved that the crossed product of an infinite dimensional simple separable unital C＊-algebra with real rank zero by an action of a finite group with the tracial Rokhlin property has again real rank zero.     

Finite group actions on products of spheres

In this work we make some contributions to the theory of actions of finite groups on products of spheres. Suppose that the groupZ _q ^r acts freely on the product of k copies of spheres. Question: Isr≤k? We solve this question for several values ofr andk.     

Diffeomorphism classification of finite group actions on disks

In this paper we discuss the diffeomorphism classification of finite group actions on disks. We answer the question when an action on a space M can be extended to an action on a disk such that the action is free away from M. Let the singular set consist of the points with nontrivial isotropy group. We show (under some dimension assumptions) that disks with diffeomorphic neighborhoods of the singular set can be imbedded into each other. As a consequence we find a classification of group actions on disks in terms of the neighborhood of the singular set and an element in the Whitehead group of G.     

Extending finite group actions from surfaces to handlebodies

We show that every action of a finite dihedral group on a closed orientable surface extends to a 3-dimensional handlebody , with . In the case of a finite abelian group , we give necessary and sufficient conditions for a -action on a surface to extend to a compact -manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order of on a surface of genus does not extend to any compact 3-manifold with , thus resolving the only case of Hurwitz actions of type of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137--151).

     

Fourier transforms on finite group actions and bent functions

Journal of Algebraic Combinatorics - Highly nonlinear functions (bent functions, perfect nonlinear functions, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in...     

Reconstructing finite group actions and characters from subgroup information

A finite group G is said to be action reconstructible if, for any action of G on a finite set, the numbers of orbits under restriction to each subgroup always give enough information to reconstruct the action up to equivalence. G is character reconstructible if, given any matrix representation of G, the mean value of the character on each subgroup always gives enough information to reconstruct the character. The conjugacy matrix of G is the matrix whose (ij) entry is the number of elements of the jth conjugacy class belonging to a typical subgroup of the ith subgroup conjugacy class. It is shown that G is action reconstructible if and only if the rows of this matrix are linearly independent (which is in turn true if and only if G is cyclic), and is character reconstructible if and only if the columns are linearly independent (which is true if and only if any two elements of G which generate conjugate cyclic subgroups are themselves conjugate).     

A property of the character table for a finite group

A character table X of a finite group is broken up into four squares: A, B, C, and D. We establish relations via which ranks of the matrices inX are connected. In particular, ifX is an l × l-matrix, A is an s × t-matrix, and, moreover, the squares A and C are opposite, thenr(A)=r(C) + s + t − l; here.r(M) is the rank of a matrix M. Associated with such each block ofX is some integral nonnegative parameter m, and we have m=0 iff A, B, C, and D are active fragments ofX. Supported by RFFR grant No. 96-01-00488. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 273–279, May–June, 2000.     

The free group does not have the finite cover property

We prove that the first order theory of nonabelian free groups eliminates the ?^∞-quantifier (in eq). Equivalently, since the theory of nonabelian free groups is stable, it does not have the finite cover property.