A strong hot spot theorem

A real number is said to be -normal if every -long string of digits appears in the base- expansion of with limiting frequency . We prove that is -normal if and only if it possesses no base- ``hot spot'. In other words, is -normal if and only if there is no real number such that smaller and smaller neighborhoods of are visited by the successive shifts of the base- expansion of with larger and larger frequencies, relative to the lengths of these neighborhoods.

     

On stable equivalences induced by exact functors

Let and be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence between and such that is induced by exact functors. We present a nice correspondence between indecomposable modules over and . As a consequence, we have the following: (1) If is a self-injective algebra, then so is ; (2) If and are finite dimensional algebras over an algebraically closed field , and if is of finite representation type such that the Auslander-Reiten quiver of has no oriented cycles, then and are Morita equivalent.

     

Fixed point spaces, primitive character degrees and conjugacy class sizes

Let be a finite group that acts on a nonzero finite dimensional vector space over an arbitrary field. Assume that is completely reducible as a -module, and that fixes no nonzero vector of . We show that some element has a small fixed-point space in . Specifically, we prove that we can choose so that , where is the smallest prime divisor of .

     

Fiber products, Poincaré duality and -ring spectra

For a Poincaré duality space and a map , consider the homotopy fiber product . If is orientable with respect to a multiplicative cohomology theory , then, after suitably regrading, it is shown that the -homology of has the structure of a graded associative algebra. When is the diagonal map of a manifold , one recovers a result of Chas and Sullivan about the homology of the unbased loop space .

     

The real rank zero property of crossed product

Let be a unital -algebra, and let be a -dynamical system with abelian and discrete. In this paper, we introduce the continuous affine map from the trace state space of the crossed product to the -invariant trace state space of . If is of real rank zero and is connected, we have proved that is homeomorphic. Conversely, if is homeomorphic, we also get some properties and real rank zero characterization of . In particular, in that case, is of real rank zero if and only if each unitary element in with the form can be approximated by the unitary elements in with finite spectrum, where , , and if moreover is a unital inductive limit of the direct sums of non-elementary simple -algebras of real rank zero, then the above can be cancelled.

     

Simultaneous non-vanishing of twists

Let be a newform of even weight , level and character and let be a newform of even weight , level and character . We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet -functions, in the context of twisted modular -functions associated to and . Using this result, we find a lower bound in terms of for the number of primitive Dirichlet characters modulo prime whose twisted product -functions are non-vanishing at a fixed point with .

     

Simple real rank zero algebras with locally Hausdorff spectrum

Let be a unital, simple, separable -algebra with real rank zero, stable rank one, and weakly unperforated ordered group. Suppose, also, that can be locally approximated by type I algebras with Hausdorff spectrum and bounded irreducible representations (the bound being dependent on the local approximating algebra). Then is tracially approximately finite dimensional (i.e., has tracial rank zero).

Hence, is an -algebra with bounded dimension growth and is determined by -theoretic invariants.

The above result also gives the first proof for the locally case.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

On chaotic -semigroups and infinitely regular hypercyclic vectors

A -semigroup on a Banach space is called hypercyclic if there exists an element such that is dense in . is called chaotic if is hypercyclic and the set of its periodic vectors is dense in as well. We show that a spectral condition introduced by Desch, Schappacher and Webb requiring many eigenvectors of the generator which depend analytically on the eigenvalues not only implies the chaoticity of the semigroup but the chaoticity of every . Furthermore, we show that semigroups whose generators have compact resolvent are never chaotic. In a second part we prove the existence of hypercyclic vectors in for a hypercyclic semigroup , where is its generator.

     

Almost simple groups of Suzuki type acting on polytopes

Let , with an odd power of two. For each almost simple group such that , we prove that is not a C-group and therefore is not the automorphism group of an abstract regular polytope. For , we show that there is always at least one abstract regular polytope such that . Moreover, if is an abstract regular polytope such that , then is a polyhedron.

     

Extensions of orthosymmetric lattice bimorphisms

Let be an Archimedean vector lattice, let be its Dedekind completion and let be a Dedekind complete vector lattice. If is an orthosymmetric lattice bimorphism, then there exists a lattice bimorphism that not just extends but also has to be orthosymmetric. As an application, we prove the following: Let be an Archimedean -algebra. Then the multiplication in can be extended to a multiplication in , the Dedekind completion of , in such a fashion that is again a -algebra with respect to this extended multiplication. This gives a positive answer to the problem posed by C. B. Huijsmans in 1990.

     

Entropy for automorphisms of free groups

Let be the automorphism of the free group which is arising from a permutation of the free generators of The naturally induces the automorphism of the reduced -algebra and also the automorphism of the group factor We show that the Brown-Germain entropy is zero. This implies that the Brown-Voiculescu topological entropy the Connes-Narnhofer-Thirring dynamical entropy and the Connes-Størmer entropy are all zero.

     

Fixed points of univalent functions II

For a closed nowhere dense subset of a bounded univalent holomorphic function on is found such that equals the cluster set of its fixed points.

     

Universal localization of triangular matrix rings

If is a triangular matrix ring, the columns and are f.g. projective -modules. We describe the universal localization of which makes invertible an -module morphism , generalizing a theorem of A. Schofield. We also describe the universal localization of -modules.

     

Greedy approximation with respect to certain subsystems of the Walsh orthonormal system

In an article that appeared in 1967, J.J. Price has shown that there is a vast family of subsystems of the Walsh orthonormal system each of which is complete on sets of large measure. In the present work it is shown that the greedy algorithm, when applied to functions in , is surprisingly effective for these nearly-complete families. Indeed, if is such a subsystem of the Walsh system, then to each positive , however small, there corresponds a Lebesgue measurable set such that for every , Lebesgue integrable on , the greedy approximants to , associated with , converge, in the norm, to an integrable function that coincides with on .

     

On the invariant translation approximation property for discrete groups

Recently J.Roe considered the question of whether for a discrete group the reduced group -algebra is the fixed point algebra of Ad acting on the uniform Roe algebra . is said to have the invariant translation approximation property in this case. We consider a slight generalization of this property which, for exact , is equivalent to the operator space approximation property of . We also give a new characterization of exactness and a short proof of the equivalence of exactness of and exactness of for discrete groups.

     

On maximal operators on -spheres in

A. Magyar's result on -bounds for a family of operators on -spheres () in is improved to match the corresponding theorem for -spheres.

     

Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

The weak Dirichlet problem for Baire functions

Let be a simplex and a compact subset of the set of all extreme points of . We show that any bounded function of Baire class on can be extended to a function of affine class on . Moreover, can be chosen in such a way that .

     

Hodge structures on posets

Let be a poset with unique minimal and maximal elements and . For each , let be the vector space spanned by -chains from to in . We define the notion of a Hodge structure on which consists of a local action of on , for each , such that the boundary map intertwines the actions of and according to a certain condition.

We show that if has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of into Hodge pieces.

We consider the case where is , the poset of subsets of with cardinality divisible by is fixed, and is a multiple of . We prove a remarkable formula which relates the characters of acting on the Hodge pieces of the homologies of the to the characters of acting on the homologies of the posets of partitions with every block size divisible by .