Volume 2008, Issue 621

For a set Ω ⊂ ℝ 3 which is diffeomorphic to a ball, we consider the problem of minimizing the area in the class of parametrized disks enclosing a prescribed volume with ∂Ω. Here the volume is defined only up to integer multiples of 3 (Ω) for topological reasons. We prove that the infimum is always realized by a system of finitely many disks, each of which is a parametric H -surface meeting ∂Ω orthogonally along its boundary.

A family of algebras, which we call topological conjugacy algebras, is associated with each proper continuous map on a locally compact Hausdorff space. Assume that is a continuous proper map on a locally compact Hausdorff space , for i = 1,2. We show that the dynamical systems and are conjugate if and only if some topological conjugacy algebra of is isomorphic as an algebra to some topological conjugacy algebra of . This implies as a corollary the complete classification of the semicrossed products , which was previously considered by Arveson and Josephson [ W. Arveson , K. Josephson, Operator algebras and measure preserving automorphisms II, J. Funct. Anal. 4 (1969), 100–134.], Peters [ J. Peters , Semicrossed products of C*-algebras, J. Funct. Anal. 59 (1984), 498–534.], Hadwin and Hoover [ D. Hadwin , T. Hoover , Operator algebras and the conjugacy of transformations, J. Funct. Anal. 77 (1988), 112–122.] and Power [ S. Power , Classification of analytic crossed product algebras, Bull. London Math. Soc. 24 (1992), 368–372.]. We also obtain a complete classification of all semicrossed products of the form , where denotes the disc algebra and a continuous map which is analytic on the interior. In this case, a surprising dichotomy appears in the classification scheme, which depends on the fixed point set of η . We also classify more general semicrossed products of uniform algebras.

The aim of this paper is to derive bounds on the critical line ℜ s = 1/2 for L -functions attached to twists f ⊗ χ of a primitive cusp form f of level N and a primitive character modulo q that break convexity simultaneously in the s and q aspects. If f has trivial nebentypus, it is shown that , where the implied constant depends only on ε > 0 and the archimedean parameter of f . To this end, two independent methods are employed to show and for any primitive cusp form g of level D and arbitrary nebentypus (not necessarily a twist f ⊗ χ of level D | Nq 2 ).

We show that if ( E , ∥ · ∥ E ) is a symmetric Banach sequence space then the corresponding space of operators on a separable Hilbert space, defined by if and only if , is a Banach space under the norm . Although this was proved for finite-dimensional spaces by von Neumann in 1937, it has never been established in complete generality in infinite-dimensional spaces; previous proofs have used the stronger hypothesis of full symmetry on E . The proof that is a norm requires the apparently new concept of uniform Hardy-Littlewood majorization; completeness also requires a new proof. We also give the analogous results for operator spaces modelled on a semifinite von Neumann algebra with a normal faithful semi-finite trace.

We show that the Fourier-Laplace transform of a regular holonomic module over the Weyl algebra of one variable, which generically underlies a variation of polarized Hodge structure, underlies itself an integrable variation of polarized twistor structure.

In this paper, we prove that the Strong Novikov Conjecture for a residually finite group is essentially equivalent to the Coarse Geometric Novikov Conjecture for a certain metric space associated to the group. As an application, we obtain the Coarse Geometric Novikov Conjecture for a large class of sequences of expanders.

We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. In particular, our results apply to the largest class of simple C*-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among -stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it resolves, for the class of algebras above, two conjectures of Blackadar and Handelman concerning the basic structure of dimension functions on C*-algebras. We also prove in passing that the Cuntz-Pedersen semigroup is recovered functorially from the Elliott invariant for a large class of simple unital C*-algebras.

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p ≧ 0. We study J.-P. Serre's notion of G -complete reducibility for subgroups of G . Specifically, for a subgroup H and a normal subgroup N of H , we look at the relationship between G -complete reducibility of N and of H , and show that these properties are equivalent if H/N is linearly reductive, generalizing a result of Serre. We also study the case when H = MN with M a G -completely reducible subgroup of G which normalizes N . In our principal result we show that if G is connected, N and M are connected commuting G -completely reducible subgroups of G , and p is good for G , then H = MN is also G -completely reducible.