Properly embedded least area planes in Gromov hyperbolic -spaces

Let be a Gromov hyperbolic -space with cocompact metric, and the sphere at infinity of . We show that for any simple closed curve in , there exists a properly embedded least area plane in spanning . This gives a positive answer to Gabai's conjecture from 1997. Soma has already proven this conjecture in 2004. Our technique here is simpler and more general, and it can be applied to many similar settings.

     

Amenability of Banach and C-algebras generated by unitary representations

We study the amenability of a locally compact group in relation to the amenability properties of a variety of -algebras and (quantized/dual) Banach algebras naturally associated to a unitary representation of .

     

Monotonicity of the principal eigenvalue of the -Laplacian in an annulus

In this note we prove a monotonicity result related to the principal eigenvalue of the -Laplacian in an annulus in .

     

Every

It is known that congruence lattices of algebras in -permutable varieties satisfy non-trivial identities; however, the identities discovered so far are rather artificial and seem to have little intrinsic interest.

We show here that every -permutable variety satisfies the well-known and well-studied congruence identity . We also get a new condition equivalent to -permutability.

     

Finite rank Toeplitz operators on the Bergman space

Given a complex Borel measure with compact support in the complex plane the sesquilinear form defined on analytic polynomials and by , determines an operator from the space of such polynomials to the space of linear functionals on . This operator is called the Toeplitz operator with symbol . We show that has finite rank if and only if is a finite linear combination of point masses. Application to Toeplitz operators on the Bergman space is immediate.

     

Norming algebras and automatic complete boundedness of isomorphisms of operator algebras

We combine the notion of norming algebra introduced by Pop, Sinclair and Smith with a result of Pisier to show that if and are operator algebras, then any bounded epimorphism of onto is completely bounded provided that contains a norming -subalgebra. We use this result to give some insights into Kadison's Similarity Problem: we show that every faithful bounded homomorphism of a -algebra on a Hilbert space has completely bounded inverse, and show that a bounded representation of a -algebra is similar to a -representation precisely when the image operator algebra -norms itself. We give two applications to isometric isomorphisms of certain operator algebras. The first is an extension of a result of Davidson and Power on isometric isomorphisms of CSL algebras. Secondly, we show that an isometric isomorphism between subalgebras of -diagonals () satisfying extends uniquely to a -isomorphism of the -algebras generated by and ; this generalizes results of Muhly-Qiu-Solel and Donsig-Pitts.

     

Local and global -regularity

A bounded domain is called -regular if the plurisubharmonic envelope of every continuous function on extends continuously to . We show using Gauthier's Fusion Lemma that a domain is locally -regular if and only if it is -regular.

     

Rational functions with linear relations

We find all polynomials over a field such that and are linear and . We also solve the same problem for rational functions , in case the field is algebraically closed.

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

Only `free' measures are admissable on is incomplete

Using elementary arguments and without having to recall the Gleason Theorem, we prove that the existence of a nonsingular measure on the lattice of orthogonally closed subspaces of an inner product space is a sufficient (and of course, a necessary) condition for to be a Hilbert space.

     

Groups which do not admit ghosts

A ghost in the stable module category of a group is a map between representations of that is invisible to Tate cohomology. We show that the only non-trivial finite -groups whose stable module categories have no non-trivial ghosts are the cyclic groups and . We compare this to the situation in the derived category of a commutative ring. We also determine for which groups the second power of the Jacobson radical of is stably isomorphic to a suspension of .

     

Hypersurfaces whose tangent geodesics do not cover the ambient space

Let be an immersion of an -dimensional connected manifold in an -dimensional connected complete Riemannian manifold without conjugate points. Assume that the union of geodesics tangent to does not cover . Under these hypotheses we have two results. The first one states that is simply connected provided that the universal covering of is compact. The second result says that if is a proper embedding and is simply connected, then is a normal graph over an open subset of a geodesic sphere. Furthermore, there exists an open star-shaped set such that is a manifold with the boundary .

     

On sums of powers of inverse complete quotients

For an irrational number , let denote its -th continued fraction inverse complete quotient, obtained by deleting the first partial quotients. For any positive real number , we establish the optimal linear bound on the sum of the -th powers of the first complete quotients. That is, we find the smallest constants such that for all and all irrationals .

     

The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups and exceptional Lie groups

For a fixed prime , we compute the Brown-Peterson cohomologies of classifying spaces of and exceptional Lie groups by using the Adams spectral sequence. In particular, we see that and are even dimensionally generated.

     

Codes over rings of size four, Hermitian lattices, and corresponding theta functions

Let be an imaginary quadratic field with ring of integers , where is a square free integer such that , and let is a linear code defined over . The level theta function of is defined on the lattice , where is the natural projection. In this paper, we prove that:

i) for any such that , and have the same coefficients up to ,

ii) for , determines the code uniquely,

iii) for , there is a positive dimensional family of symmetrized weight enumerator polynomials corresponding to .

     

Homogeneous Hilbert scheme

Let be a locally noetherian scheme and an -graded -algebra of finite type. We say that is a homogeneous variety over . In this paper we prove that the functor

is representable by an -scheme that is a disjoint union of locally projective schemes over . The proof is very simple, and it only makes use of the theory of graded modules and standard flatness criteria. From this, one obtains an elementary construction (which does not make use of cohomology) of the ordinary Hilbert scheme of a locally projective -scheme.

     

On the Pytkeev property in spaces of continuous functions

Answering a question of Sakai, we show that the minimal cardinality of a set of reals such that does not have the Pytkeev property is equal to the pseudo-intersection number . Our approach leads to a natural characterization of the Pytkeev property of by means of a covering property of , and to a similar result for the Reznichenko property of .

     

On Generic differential -extensions

Let be an algebraically closed field with trivial derivation and let denote the differential rational field , with , , , , differentially independent indeterminates over . We show that there is a Picard-Vessiot extension for a matrix equation , with differential Galois group , with the property that if is any differential field with field of constants , then there is a Picard-Vessiot extension with differential Galois group if and only if there are with well defined and the equation giving rise to the extension .

     

Projective normality of ruled surfaces

In this article we study normal generation of irrational ruled surfaces. When is a smooth curve of genus , Green and Lazarsfeld proved that a very ample line bundle    Pic with deg   Cliff is normally generated where Cliff denotes the Clifford index of the curve (Green and Lazarsfeld, 1986). We generalize this to line bundles on a ruled surface over .

     

A polarized partition relation for cardinals of countable cofinality

We prove that if and , then

for all . This polarized partition relation holds if for every partition either there are and with or there are and with .