Linear isometries between subspaces of continuous functions

We say that a linear subspace of is strongly separating if given any pair of distinct points of the locally compact space , then there exists such that . In this paper we prove that a linear isometry of onto such a subspace of induces a homeomorphism between two certain singular subspaces of the Shilov boundaries of and , sending the Choquet boundary of onto the Choquet boundary of . We also provide an example which shows that the above result is no longer true if we do not assume to be strongly separating. Furthermore we obtain the following multiplicative representation of : for all and all , where is a unimodular scalar-valued continuous function on . These results contain and extend some others by Amir and Arbel, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

     

Only single twists on unknots can produce composite knots

Let be a knot in the -sphere , and a disc in meeting transversely more than once in the interior. For non-triviality we assume that over all isotopy of . Let () be a knot obtained from by cutting and -twisting along the disc (or equivalently, performing -Dehn surgery on ). Then we prove the following: (1) If is a trivial knot and is a composite knot, then ; (2) if is a composite knot without locally knotted arc in and is also a composite knot, then . We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.

     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .

     

Automorphism Groups and Invariant Subspace Lattices

Let be a - dynamical system and let be the analytic subalgebra of . We extend the work of Loebl and the first author that relates the invariant subspace structure of for a -representation on a Hilbert space , to the possibility of implementing on We show that if is irreducible and if lat is trivial, then is ultraweakly dense in We show, too, that if satisfies what we call the strong Dirichlet condition, then the ultraweak closure of is a nest algebra for each irreducible representation Our methods give a new proof of a ``density' theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid -algebras.

     

Kaehler structures on

Let be a compact connected semi-simple Lie group, let , and let be an Iwasawa decomposition. To a given -invariant Kaehler structure on , there corresponds a pre-quantum line bundle on . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections as a -representation space. We defined a -invariant -structure on , and let denote the space of square-integrable holomorphic sections. Then is a unitary -representation space, but not all unitary irreducible -representations occur as subrepresentations of . This paper serves as a continuation of that work, by generalizing the space considered. Let be a Borel subgroup containing , with commutator subgroup . Instead of working with , we consider , for all parabolic subgroups containing . We carry out a similar construction, and recover in the unitary irreducible -representations previously missing. As a result, we use these holomorphic sections to construct a model for : a unitary -representation in which every irreducible -representation occurs with multiplicity one.

     

Randomness and semigenericity

Let contain only the equality symbol and let be an arbitrary finite symmetric relational language containing . Suppose probabilities are defined on finite structures with `edge probability' . By , the almost sure theory of random -structures we mean the collection of -sentences which have limit probability 1. denotes the theory of the generic structures for (the collection of finite graphs with hereditarily nonnegative). . , the almost sure theory of random -structures, is the same as the theory of the -generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.

     

A Characterization of Finitely Decidable Congruence Modular Varieties

For every finitely generated, congruence modular variety of finite type we find a finite family of finite rings such that the variety is finitely decidable if and only if is congruence permutable and residually small, all solvable congruences in finite algebras from are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra from is comparable with all congruences of , each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring from , the variety of -modules is finitely decidable.

     

Boundary limits and non-integrability of )

In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball in that are subharmonic with respect to the Laplace-Beltrami operator on . Since the operator is invariant under the group of holomorphic automorphisms of , functions that are subharmonic with respect to are usually referred to as -subharmonic functions. Our main result is as follows: Let be a non-negative -subharmonic function on satisfying

for some and some , where is the -invariant measure on . Suppose . Then for a.e. ,

uniformly as in each , where for ( when )

We also prove that for the only non-negative -subharmonic function satisfying the above integrability criteria is the zero function.

     

Symmetric powers of complete modules over a two-dimensional regular local ring

Let be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free -module , write for the th symmetric power of , mod torsion. We study the modules , , when is complete (i.e., integrally closed). In particular, we show that , for any minimal reduction and that the ring is Cohen-Macaulay.

     

Asymptotic behaviour of reproducing kernels of weighted Bergman spaces

Let be a domain in , a nonnegative and a positive function on such that is locally bounded, the space of all holomorphic functions on square-integrable with respect to the measure , where is the -dimensional Lebesgue measure, and the reproducing kernel for . It has been known for a long time that in some special situations (such as on bounded symmetric domains with and the Bergman kernel function) the formula

holds true. [This fact even plays a crucial role in Berezin's theory of quantization on curved phase spaces.] In this paper we discuss the validity of this formula in the general case. The answer turns out to depend on, loosely speaking, how well the function can be approximated by certain pluriharmonic functions lying below it. For instance, () holds if is convex (and, hence, can be approximated from below by linear functions), for any function . Counterexamples are also given to show that in general () may fail drastically, or even be true for some and fail for the remaining ones. Finally, we also consider the question of convergence of for , which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of : for instance, if is not real-analytic at some point, then must have zeroes for all sufficiently large.

     

homology over traced *-algebras

Given a unital complex *-algebra , a tracial positive linear functional on that factors through a *-representation of on Hilbert space, and an -module possessing a resolution by finitely generated projective -modules, we construct homology spaces for . Each is a Hilbert space equipped with a *-representation of , independent (up to unitary equivalence) of the given resolution of . A short exact sequence of -modules gives rise to a long weakly exact sequence of homology spaces. There is a Künneth formula for tensor products. The von Neumann dimension which is defined for -invariant subspaces of gives well-behaved Betti numbers and an Euler characteristic for with respect to and .

     

The generalized Berg theorem and BDF-theorem

Let be a separable simple -algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element , i.e., is normal ( is the quotient map), having the form for some normal element and We also show that a normal element can be quasi-diagonalized if and only if the Fredholm index for all In the case that is a simple -algebra of real rank zero, with stable rank one and with continuous scale, and has countable rank, we show that a normal element with zero Fredholm index can be written as

where is an (increasing) approximate identity for consisting of projections, is a bounded sequence of numbers and with for any given

     

Contractions on a manifold polarized by an ample vector bundle

A complex manifold of dimension together with an ample vector bundle on it will be called a generalized polarized variety. The adjoint bundle of the pair is the line bundle . We study the positivity (the nefness or ampleness) of the adjoint bundle in the case . If this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski.

If is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map from onto a normal projective variety with connected fiber and such that , for some ample line bundle on . We describe those contractions for which . We extend this result to the case in which has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which for every fiber and is birational.

     

A bound on the geometric genus of projective varieties verifying certain flag conditions

Fix integers and let be the set of all integral, projective and nondegenerate varieties of degree and dimension in the projective space , such that, for all , does not lie on any variety of dimension and degree . We say that a variety satisfies a flag condition of type if belongs to . In this paper, under the hypotheses , we determine an upper bound , depending only on , for the number , where denotes the geometric genus of . In case and , the study of an upper bound for the geometric genus has a quite long history and, for , and , it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data . For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension in as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in . Next we discuss how far is from and show a sort of lifting theorem which states that, at least in certain cases, the varieties of maximal geometric genus must in fact lie on a flag such as , where denotes a subvariety of of degree and dimension . We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.

     

Absolute Borel sets and function spaces

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Cech-complete spaces. We also show that the absolute Borel class of is determined by the uniform structure of the space of continuous functions ; however the case of absolute metric spaces is still open. More precisely, we prove that, for metrizable spaces and , if is a uniformly continuous surjection and is an absolute Borel set of multiplicative (resp., additive) class , , then is also an absolute Borel set of the same class. This result is new even if is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the \v{C}ech-completeness of a metric space is determined by the linear structure of .

     

Herz-Schur multipliers and weakly almost periodic functions on locally compact groups

For a locally compact group and , let be the Herz-Figà-Talamanca algebra and the Herz-Schur multipliers of , and the multipliers of . Let be the algebra of continuous weakly almost periodic functions on . In this paper, we show that (1), if is a noncompact nilpotent group or a noncompact [IN]-group, then contains a linear isometric copy of ; (2), for a noncommutative free group is a proper subset of .

     

The Mizohata complex

This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ at of a -valued, formally integrable (), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for -forms , of the equation

where is a given -form satisfying is estimated in terms of the De Rham cohomology relative to