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 .

     

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.

     

On Voiculescu's double commutant theorem

For a separable infinite-dimensional Hilbert space , we consider the full algebra of bounded linear transformations and the unique non-trivial norm-closed two-sided ideal of compact operators . We also consider the quotient -algebra with quotient map

For any -subalgebra of , the relative commutant is given by for all in . It was shown by D. Voiculescu that, for any separable unital -subalgebra of ,

In this note, we exhibit a non-separable unital -subalgebra of for which (VDCT) fails.

     

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

     

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.

     

Vanishing of the leading term in Harish-Chandra's local character expansion

Harish-Chandra's formula for the character of an irreducible smooth representation of a reductive -adic group expresses near as a linear combination of the Fourier transforms of nilpotent -orbits in the Lie algebra of . In this note, we prove that if is tempered but not in the discrete series, then the coefficient attached to the zero nilpotent orbit vanishes.

     

A topological characterization of linearity for quasi-traces

Let be a -algebra, and let be a (local) quasi-trace on . Then is linear if, and only if, the restriction of to the closed unit ball of is uniformly weakly continuous.

     

On a polynomial inequality of Kolmogoroff's type

We prove an inequality of the form

for polynomials of degree and any fixed . Here is the -norm on with a weight . The coefficients and are given explicitly and depend on and only. The equality is attained for the Hermite orthogonal polynomials .

     

A rigidity theorem for the Clifford tori in

Let be the unit hypersphere in the 4-dimensional Euclidean space defined by . For each with , we denote by the Clifford torus in given by the equations and . The Clifford torus is a flat Riemannian manifold equipped with the metric induced by the inclusion map . In this note we prove the following rigidity theorem: If is an isometric embedding, then there exists an isometry of such that . We also show no flat torus with the intrinsic diameter is embeddable in except for a Clifford torus.

     

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

     

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.

     

Amenability and weak amenability of second conjugate Banach algebras

For a Banach algebra , amenability of necessitates amenability of , and similarly for weak amenability provided is a left ideal in . For a locally compact group, indeed more generally, is amenable if and only if is finite. If is weakly amenable, then is weakly amenable.

     

On rigidity of affine surfaces

Rigidity of nondegenerate Blaschke surfaces in is studied. The rigidity criteria are given in terms of , where is the curvature of the Blaschke connection . If the rank of is 2, then the surface is rigid. If , it is nonrigid. In the case where the rank of is 1 there are both rigid and nonrigid surfaces. This case is discussed for various types of surfaces.

     

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 .

     

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).

     

Minimal upper bounds of commuting operators

Let be a finite collection of commuting self-adjoint elements of a von Neumann algebra . Then within the (abelian) C*-algebra they generate, these elements have a least upper bound . We show that within , is a minimal upper bound in the sense that if is any self-adjoint element such that for all , then . The corresponding assertion for infinite collections is shown to be false in general, although it does hold in any finite von Neumann algebra. We use this sort of result to show that if are von Neumann algebras, is a faithful conditional expectation, and is positive, then converges in the strong operator topology to the ``spectral order majorant' of in .

     

On the size of lemniscates of polynomials in one and several variables

In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set and , for a polynomial of degree . Usually, is taken to be monic, and either Cartan's Lemma or potential theory is used to estimate the size of , in terms of Hausdorff contents, planar Lebesgue measure , or logarithmic capacity cap. Here we normalize and show that cap and are the sharp estimates for the size of . Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on or product capacity and Favarov's capacity. Several of our estimates are sharp with respect to order in and .

     

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: .

     

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 .

     

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.