Transformations conjugate to their inverses have even essential values

Let be an ergodic automorphism defined on a standard Borel probability space for which and are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of . It was shown in Ergodic transformations conjugate to their inverses by involutions by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97--124) that if is ergodic having simple spectrum and isomorphic to its inverse, and if is a conjugation between and (i.e. satisfies ), then , the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation , the unitary operator induced by on must have a multiplicity function whose essential values on the ortho-complement of the subspace are always even. In particular, we see that can be weakly mixing, so the corresponding must have even maximal spectral multiplicity (regarding as an even number).

     

On fully invariant ideals of the free group algebra

Let be the group algebra of a free noncyclic group over an integral domain . It is proved that if is not a field, then there exists a fully invariant ideal of such is torsion-free but not projective as an -module. In other words, there exists a pure nonprojective variety of group representations over .

     

The statistics of continued fractions for polynomials over a finite field

Given a finite field of order and polynomials of degrees respectively, there is the continued fraction representation . Let denote the number of such pairs for which and for . We give both an exact recurrence relation, and an asymptotic analysis, for . The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of . Averaged over all and as above, this presents no difficulties. The average value of is , and there is full information about the distribution. When is fixed and only is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of that differs from this average by more than

     

Derivations with Engel conditions on multilinear polynomials

Let be a prime algebra over a commutative ring with unity and let be a multilinear polynomial over . Suppose that is a nonzero derivation on such that for all in some nonzero ideal of , with fixed. Then is central--valued on except when char and satisfies the standard identity in 4 variables.

     

The local cohomology modules of Matlis reflexive modules are almost cofinite

We show that if and are Matlis reflexive modules over a complete Gorenstein local domain and is an ideal of such that the dimension of is one, then the modules are Matlis reflexive for all and if . It follows that the Bass numbers of are finite. If is not a domain, then the same results hold for .

     

Oscillatory singular integrals on and Hardy spaces

We consider boundedness properties of oscillatory singular integrals on and Hardy spaces. By constructing a phase function, we prove that boundedness may fail while boundedness holds for all . This shows that the theory and theory for such operators are fundamentally different.

     

The local zeta function for the non-trivial characters associated with the singular Jordan algebras

This paper investigates the local integrals

where represents the integers of a composition algebra over a non-archimedean local field and is a non-trivial character on the units in the ring of integers of extended to by setting . The local zeta function for the trivial character is known for all composition algebras . In this paper, we show in the quaternion case that for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for equal to the quadratic character. In this latter case, for any character of order greater than .

     

On the higher delta invariants of a Gorenstein local ring

Let be a Gorenstein complete local ring. Auslander's higher delta invariants are denoted by for each module and for each integer . We propose a conjecture asking if for any positive integers and . We prove that this is true provided the associated graded ring of has depth not less than . Furthermore we show that there are only finitely many possibilities for a pair of positive integers for which .

     

On nonlinear -widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

     

Multipliers of weak type on locally compact Vilenkin groups

Let be a locally compact Vilenkin group with dual group . We give a sufficient condition for to be a multiplier of weak type on . Some applications of our result are given. We also prove that our result is sharp.

     

A Wolff-Denjoy theorem for infinitely connected Riemann surfaces

We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let be a non-parabolic Riemann surface with Martin boundary . Suppose each Martin function , , extends continuously to and vanishes there. We show that if is an endomorphism of and the iterates of converge to the point at infinity, then the iterates converge locally uniformly to a point in . As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on but none on itself.

     

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).

     

Embedding of a Banach algebra

Given a Banach space , we have shown in 1994 that a product can be defined on it in such a way that the resulting Banach algebra is isomorphic to a compact subalgebra of the algebra of all bounded linear operators on the topological dual of . Our purpose here is to prove that, more generally, any Banach algebra admitting a left approximate identity, is isomorphic to a subalgebra of , the isomorphism being isometric, provided the approximate identity is bounded by 1. As a consequence, we get a factorization through , of the elements in .

     

On a pattern of reflexive operator spaces

A linear subspace is a separating subspace for an operator space if the only member of annihilating is 0. It is proved in this paper that if has a strictly separating vector and a separating subspace satisfying , then is reflexive. Applying this to finite dimensional leads to more results on reflexivity. For example, if dim , and every nonzero operator in has rank , then is reflexive.

     

Cohomological detection and regular elements in group cohomology

Suppose that is a compact Lie group or a discrete group of finite virtual cohomological dimension and that is a field of characteristic . Suppose that is a set of elementary abelian -subgroups such that the cohomology is detected on the centralizers of the elements of . Assume also that is closed under conjugation and that is in whenever some subgroup of is in . Then there exists a regular element in the cohomology ring such that the restriction of to an elementary abelian -subgroup is not nilpotent if and only if is in . The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.

     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

Operator versions of the Kantorovich inequality

The Operator Kantorovich Inequality

holds for a wide class of operators on a Hilbert space and all operators for which is a partial isometry, being the range projection of

     

On purely inseparable extensions and their generators

Let be a field of characteristic and a polynomial algebra in two variables. By a -generator of we mean an element of for which there exist and such that . We also define a -line of to mean any element of whose coordinate ring is that of a -generator. Then we prove that if is such that is a -line of (where is an indeterminate over ), then is a -generator of . This is analogous to the well-known fact that if is such that is a line of , then is a variable of . We also prove that if is a -line of for which there exist and such that , then is in fact a -generator of .

     

sets of unbounded Loeb measure

For every and non-Borel subset of an internal set in a saturated nonstandard universe there exists an internal, unbounded, non-atomic measure so that is not finite for any Borel set in

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .