New Class Of Examples Research Articles

Merging of positive maps: A construction of various classes of positive maps on matrix algebras

For two positive maps ϕi:B(Ki)→B(Hi), i=1,2, we construct a new linear map ϕ:B(H)→B(K), where K=K1⊕K2⊕C, H=H1⊕H2⊕C, by means of some additional ingredients such as operators and functionals. We call it a merging of maps ϕ1 and ϕ2. The properties of this construction are discussed. In particular, conditions for positivity of ϕ, as well as for 2-positivity, complete positivity, optimality and indecomposability, are provided. In particular, we show that for a pair composed of 2-positive and 2-copositive maps, there is an indecomposable merging of them. One of our main results asserts, that for a canonical merging of a pair composed of completely positive and completely copositive extremal maps, their canonical merging is an exposed positive map. This result provides a wide class of new examples of exposed positive maps. As an application, new examples of entangled PPT states are described.

A complete classification of Blaschke parallel submanifolds with vanishing Möbius form

The Blaschke tensor and the Mobius form are two of the fundamental invariants in the Mobius geometry of submanifolds; an umbilic-free immersed submanifold in real space forms is called Blaschke parallel if its Blaschke tensor is parallel. We prove a theorem which, together with the known classification result for Mobius isotropic submanifolds, successfully establishes a complete classification of the Blaschke parallel submanifolds in Sn with vanishing Mobius form. Before doing so, a broad class of new examples of general codimensions is explicitly constructed.

Small product sets in compact groups

This paper addresses the structure of pairs (A B) of Borel sets in a compact and second countable group G with Haar probability measure mG which have positive mG-measures and satisfy the identity mG(AB) = mG(A) + mG(B) < When G is a compact, abelian and connected group, M. Kneser provided a very satisfactory description of such pairs, which roughly says that both A and B must be co-null subsets of pre-images of two closed intervals under the same surjective homomorphism onto the circle group. Much more recently, Griesmer was able to divide the set of all possible such pairs in any compact and abelian group into four (not completely disjoint) subclasses. Motivated by some applications to small product sets in countable amenable groups with respect to the left upper Banach density, we continue this study, and offer a description of pairs of Borel sets as above in any compact and second countable group with an abelian identity component. The relevance of these compact groups to the study of product sets in countable amenable groups will be explained. The main result of this paper asserts that in this non-abelian situation, essentially only one new class of examples of pairs as above can occur. Unfortunately, due to a wide variety of technicalities we will have to spend a non-trivial part of the introduction to carefully set up the tools and concepts necessary for the formulation of the main results and to prepare for their proofs.

Suppressing structure formation at dwarf galaxy scales and below: Late kinetic decoupling as a compelling alternative to warm dark matter

Warm dark matter cosmologies have been widely studied as an alternative to the cold dark matter paradigm, the characteristic feature being a suppression of structure formation on small cosmological scales. A very similar situation occurs if standard cold dark matter particles are kept in local thermal equilibrium with a, possibly dark, relativistic species until the universe has cooled down to keV temperatures. We perform a systematic phenomenological study of this possibility, and classify all minimal models containing dark matter and an arbitrary radiation component that allow such a late kinetic decoupling. We recover explicit cases recently discussed in the literature and identify new classes of examples that are very interesting from a model-building point of view. In some of these models dark matter is inevitably self-interacting, which is remarkable in view of recent observational support for this possibility. Hence, dark matter models featuring late kinetic decoupling have the potential not only to alleviate the missing satellites problem but also to address other problems of the cosmological concordance model on small scales, in particular the cusp-core and too-big-too-fail problems, in some cases without invoking any additional input.

Ladders and simplicity of derived module categories

Recollements of derived module categories are investigated, using a new technique, ladders of recollements, which are maximal mutation sequences. The position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category. Ladders also turn out to control derived simplicity on all levels. An algebra is derived simple if its derived category cannot be deconstructed, that is, if it is not the middle term of a non-trivial recollement whose outer terms are again derived categories of algebras. Derived simplicity on each level is characterised in terms of heights of ladders.These results are complemented by providing new classes of examples of derived simple rings, in particular indecomposable commutative rings, as well as by a finite dimensional counterexample to the Jordan–Hölder problem for derived module categories. Moreover, recollements are used to compute homological and K-theoretic invariants.

Tautological sheaves: Stability, moduli spaces and restrictions to generalised Kummer varieties

Results on stability of tautological sheaves on Hilbert schemes of points are extended to higher dimensions and to the restriction of tautological sheaves to generalised Kummer varieties. This provides a big class of new examples of stable sheaves on higher dimensional irreducible symplectic manifolds. Some aspects of deformations of tautological sheaves are studied.

On the correlation of increasing families

The classical correlation inequality of Harris asserts that any two monotone increasing families on the discrete cube are nonnegatively correlated. In 1996, Talagrand [19] established a lower bound on the correlation in terms of how much the two families depend simultaneously on the same coordinates. Talagrand's method and results inspired a number of important works in combinatorics and probability theory.In this paper we present stronger correlation lower bounds that hold when the increasing families satisfy natural regularity or symmetry conditions. In addition, we present several new classes of examples for which Talagrand's bound is tight.A central tool in the paper is a simple lemma asserting that for monotone events noise decreases correlation. This lemma gives also a very simple derivation of the classical FKG inequality for product measures, and leads to a simplification of part of Talagrand's proof.

On model-theoretic connected components in some group extensions

We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these 2-cocycles, when the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some situations, we also describe the quotient of these two connected components. Using our general results about extensions of groups together with Matsumoto–Moore theory or various quasi-characters considered in bounded cohomology, we obtain new classes of examples of groups whose smallest type-definable subgroup of bounded index differs from the smallest invariant subgroup of bounded index. This includes the first known example of a group with this property found by Conversano and Pillay, namely the universal cover of [Formula: see text] (interpreted in a monster model), as well as various examples of different nature, e.g. some central extensions of free groups or of fundamental groups of closed orientable surfaces. As a corollary, we get that both non-abelian free groups and fundamental groups of closed orientable surfaces of genus [Formula: see text], expanded by predicates for all subsets, have this property, too. We also obtain a variant of the example of Conversano and Pillay for [Formula: see text] instead of [Formula: see text], which (as most of our examples) was not accessible by the previously known methods.

Anti-self-dual orbifolds with cyclic quotient singularities

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with cyclic quotient singularities is proved. We present two applications of this theorem. The first is to compute the dimension of the deformation space of the Calderbank–Singer scalar-flat Kähler toric ALE spaces. A corollary of this is that, except for the Eguchi–Hanson metric, all of these spaces admit non-toric anti-self-dual deformations, thus yielding many new examples of anti-self-dual ALE spaces. For our second application, we compute the dimension of the deformation space of the canonical Bochner-Kähler metric on any weighted projective space \mathbb {CP}^2_{(r,q,p)} for relatively prime integers 1 &lt; r &lt; q &lt; p . A corollary of this is that, while these metrics are rigid as Bochner–Kähler metrics, infinitely many of these admit non-trivial self-dual deformations, yielding a large class of new examples of self-dual orbifold metrics on certain weighted projective spaces.

Strict deformation quantization of locally convex algebras and modules

In this work various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffel’s strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of R n . Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactly supported R n -actions on R n is constructed.

Para-CR structures on almost paracontact metric manifolds

Abstract Almost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and sufficient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that normal almost paracontact metric manifolds are para-CR. We establish necessary and sufficient conditions for paracontact metric manifolds as well as for almost para-cosymplectic manifolds to be para-CR. We find also basic curvature identities for para-CR paracontact metric manifolds and study their consequences. Among others, we prove that any para-CR paracontact metric manifold of constant sectional curvature and of dimension greater than 3 must be para-Sasakian and its curvature equal to -1. The last assertion does not hold in dimension 3. We show that a conformally flat para-Sasakian manifold is of constant sectional curvature equal to -1. New classes of examples of para-CR manifolds are established.

On Kazhdan's Property (T) for the special linear group of holomorphic functions

We investigate when the group $SL_n(\mathcal{O}(X))$ of holomorphic maps from a Stein space $X$ to $SL_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. This provides a new class of examples of non-locally compact groups having Kazhdan's property (T). In particular we prove that the holomorphic loop group of $SL_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. Our result relies on the method of Shalom to prove Kazhdan's property (T) and the solution to Gromov's Vaserstein problem by the authors.

On Kazhdan's Property (T) for the special linear group of holomorphic functions

We investigate when the group SLn (O(X)) of holomorphic maps from a Stein space X to SLn (C) has Kazhdan's property (T) for n >= 3. This provides a new class of examples of non-locally compact groups having Kazhdan's property (T). In particular we prove that the holomorphic loop group of SLn (C) has Kazhdan's property (T) for n >= 3. Our result relies on the method of Shalom to prove Kazhdan's property (T) and the solution to Gromov's Vaserstein problem by the authors.

Pseudo-Anosov flows in toroidal manifolds

We first prove rigidity results for pseudo-Anosov flows in prototypes of toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically equivalent to a geodesic flow and we show that a pseudo-Anosov flow in a solv manifold is topologically equivalent to a suspension Anosov flow. Then we study the interaction of a general pseudo-Anosov flow with possible Seifert fibered pieces in the torus decomposition: if the fiber is associated with a periodic orbit of the flow, we show that there is a standard and very simple form for the flow in the piece using Birkhoff annuli. This form is strongly connected with the topology of the Seifert piece. We also construct a large new class of examples in many graph manifolds, which is extremely general and flexible. We construct other new classes of examples, some of which are generalized pseudo-Anosov flows which have one prong singularities and which show that the above results in Seifert fibered and solvable manifolds do not apply to one prong pseudo-Anosov flows. Finally we also analyse immersed and embedded incompressible tori in optimal position with respect to a pseudo-Anosov flow.

On the Hofer geometry for weakly exact Lagrangian submanifolds

We use spectral invariants in Lagrangian Floer theory in order to show that there exist isometric embeddings of normed linear spaces (finite or infinite-dimensional, depending on the case) into the space of Hamiltonian deformations of certain weakly exact Lagrangian submanifolds in tame symplectic manifolds. In addition to providing a new class of examples in which the Lagrangian Hofer metric can be computed explicitly, we refine and generalize some known results about it.

Braidings on the Category of Bimodules, Azumaya Algebras and Epimorphisms of Rings

Let $A$ be an algebra over a commutative ring $k$. We prove that braidings on the category of $A$-bimodules are in bijective correspondence to canonical R-matrices, these are elements in $A\ot A\ot A$ satisfying certain axioms. We show that all braidings are symmetries. If $A$ is commutative, then there exists a braiding on ${}_A\Mm_A$ if and only if $k\to A$ is an epimorphism in the category of rings, and then the corresponding $R$-matrix is trivial. If the invariants functor $G = (-)^A:\{}_A\Mm_A\to \Mm_k$ is separable, then $A$ admits a canonical R-matrix; in particular, any Azumaya algebra admits a canonical R-matrix. Working over a field, we find a remarkable new characterization of central simple algebras: these are precisely the finite dimensional algebras that admit a canonical R-matrix. Canonical R-matrices give rise to a new class of examples of simultaneous solutions for the quantum Yang-Baxter equation and the braid equation.

Amenability and vanishing of [formula omitted]-Betti numbers: An operator algebraic approach

We introduce a Følner condition for dense subalgebras in finite von Neumann algebras and prove that it implies dimension flatness of the inclusion in question. It is furthermore proved that the Følner condition naturally generalizes the existing notions of amenability and that the ambient von Neumann algebra of a Følner algebra is automatically injective. As an application, we show how our techniques unify previously known results concerning vanishing of L2-Betti numbers for amenable groups, quantum groups and groupoids and moreover provide a large class of new examples of algebras with vanishing L2-Betti numbers.

Six-Dimensional Nearly Kähler Manifolds of Cohomogeneity One (II)

Let M be a six dimensional manifold, endowed with a cohomogeneity one action of G = SU2 × SU2, and \({M_{\rm reg} \subset M}\) its subset of regular points. We show that Mreg admits a smooth, 2-parameter family of G-invariant, non-isometric strict nearly Kahler structures and that a 1-parameter subfamily of such structures smoothly extends over a singular orbit of type S3. This determines a new class of examples of nearly Kahler structures on T S3.

Box spaces, group extensions and coarse embeddings into Hilbert space

We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.

Homogeneous orbit closures and applications

AbstractWe give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in ℝd for d≥3 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit closures. We give Diophantine applications to the former; for instance, we show that, for all γ,δ∈ℝ, where 〈c〉 denotes the distance of a real number c to the integers.