Serre duality for non-commutative -bundles

Let be a smooth scheme of finite type over a field , let be a locally free -bimodule of rank , and let be the non-commutative symmetric algebra generated by . We construct an internal functor, , on the category of graded right -modules. When has rank 2, we prove that is Gorenstein by computing the right derived functors of . When is a smooth projective variety, we prove a version of Serre Duality for using the right derived functors of .

     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

Global properties of the lattice of classes

Let be the lattice of classes of reals. We show there are exactly two possible isomorphism types of end intervals, . Moreover, finiteness is first order definable in .

     

Integral homology -spheres and the Johnson filtration

The mapping class group of an oriented surface of genus with one boundary component has a natural decreasing filtration , where is the kernel of the action of on the nilpotent quotient of . Using a tree Lie algebra approximating the graded Lie algebra we prove that any integral homology sphere of dimension has for some a Heegaard decomposition of the form , where and is such that . This proves a conjecture due to S. Morita and shows that the ``core' of the Casson invariant is indeed the Casson invariant.

     

Algebraic isomorphisms and -subspace lattices

The class of -lattices was originally defined in the second author's thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice on a Banach space which is also a -lattice is called a -subspace lattice, abbreviated JSL. It is demonstrated that every single element of has rank at most one. It is also shown that has the strong finite rank decomposability property. Let and be subspace lattices that are also JSL's on the Banach spaces and , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between and preserves rank. Finally we prove that every algebraic isomorphism between and is quasi-spatial.

     

On residualities in the set of Markov operators on

We show that the set of those Markov operators on the Schatten class such that , where is one-dimensional projection, is norm open and dense. If we require that the limit projections must be on strictly positive states, then such operators form a norm dense . Surprisingly, for the strong operator topology operators the situation is quite the opposite.

     

Invariance in and

We define , a substructure of (the lattice of classes), and show that a quotient structure of , , is isomorphic to . The result builds on the isomorphism machinery, and allows us to transfer invariant classes from to , though not, in general, orbits. Further properties of and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.

     

properties for Gaussian random series

Let be an arbitrary sequence of and let be a random series of the type

where is a sequence of independent Gaussian random variables and an orthonormal basis of (the finite measure space being arbitrary). By using the equivalence of Gaussian moments and an integrability theorem due to Fernique, we show that a necessary and sufficient condition for to belong to , for any almost surely is that . One of the main motivations behind this result is the construction of a nontrivial Gibbs measure invariant under the flow of the cubic defocusing nonlinear Schrödinger equation posed on the open unit disc of .

     

A new variational characterization of -dimensional space forms

A Riemannian manifold is associated with a Schouten -tensor which is a naturally defined Codazzi tensor in case is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional defined on , where is the space of smooth Riemannian metrics on a compact smooth manifold and is the elementary symmetric functions of the eigenvalues of with respect to . We prove that if and a conformally flat metric is a critical point of with , then must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of with characterized the three-dimensional space forms.

     

Boundary -algebras for acylindrical groups

Let be an infinite, locally finite tree with more than two ends. Let be an acylindrical uniform lattice. Then the boundary algebra is a simple Cuntz-Krieger algebra whose K-theory is determined explicitly.

     

New irrationality measures for -logarithms

The three main methods used in diophantine analysis of -series are combined to obtain new upper bounds for irrationality measures of the values of the -logarithm function

when and .

     

Average size of -Selmer groups of elliptic curves, I

In this paper, we study a class of elliptic curves over with -torsion group , and prove that the average order of the -Selmer groups is bounded.

     

An uncountable family of nonorbit equivalent actions of

For each , we construct an uncountable family of free ergodic measure preserving actions of the free group on the standard probability space such that any two are nonorbit equivalent (in fact, not even stably orbit equivalent). These actions are all ``rigid' (in the sense of Popa), with the IIfactors mutually nonisomorphic (even nonstably isomorphic) and in the class

     

The -theory of is undecidable

The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

Theorem. The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

     

The Dirichlet problem and nondivergence harmonic measure

We consider the Dirichlet problem

for two second-order elliptic operators , , in a bounded Lipschitz domain . The coefficients belong to the space of bounded mean oscillation with a suitable small modulus. We assume that is regular in for some , , that is, for all continuous boundary data . Here is the surface measure on and is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients that will assure the perturbed operator to be regular in for some , .

     

Full signature invariants for

Let be a number field closed under complex conjugation. Denote by the Witt group of hermitian forms over . We find full invariants for detecting non-zero elements in . This group plays an important role in topology in the work done by Casson and Gordon.

     

Tensor products of -weakly closed nest algebra submodules

In this paper we prove that for any unital -weakly closed algebra which is -weakly generated by finite-rank operators in , every -weakly closed -submodule has . In the case of nest algebras, if are nests, we obtain the following -fold tensor product formula:

where each is the -weakly closed Alg -submodule determined by an order homomorphism from into itself.

     

Spectra of operators with Bishop's property

Let be a Banach space and let be the class that consists of all operators such that for every , the range of has a finite-codimension when it is closed. For an integer , we define the class as an extension of . We then study spectral properties of such operators, and we extend some known results of multi-cyclic operators with .

     

(APD)-property of -algebras by extensions of -algebras with (APD)

A unital -algebra is said to have the (APD)-property if every nonzero element in has the approximate polar decomposition. Let be a closed ideal of . Suppose that and have (APD). In this paper, we give a necessary and sufficient condition that makes have (APD). Furthermore, we show that if and or is a simple purely infinite -algebra, then has (APD).