arXiv subjects

Ralf Meyer

Publications and source records attributed to Ralf Meyer.

At least 55 records · Page 3Linked to original sources

arXiv2018

Braided multiplicative unitaries as regular objects

We use the theory of regular objects in tensor categories to clarify the passage between braided multiplicative unitaries and multiplicative unitaries with projection. The braided multiplicative unitary and its semidirect product multiplicative unitary have the same Hilbert space representations. We also show that the multiplicative unitaries associated to two regular objects for the same tensor category are equivalent and hence generate isomorphic C*-quantum groups. In particular, a C*-quantum group is determined uniquely by its tensor category of representations on Hilbert space, and any functor between representation categories that does not change the underlying Hilbert spaces comes from a morphism of C*-quantum groups.

math.OA↗

arXiv2017

Weak completions, bornologies and rigid cohomology

Let $V$ be a complete discrete valuation ring with residue field $k$ of positive characteristic and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using completions of bornological $V$-algebras. This leads us to a functorial chain complex for a finitely generated commutative algebra over the residue field $k$ that computes its rigid cohomology in the sense of Berthelot.

math.AG↗

arXiv2017

Vibrational band structure of nanoscale phononic crystals

The vibrational properties of two-dimensional phononic crystals are studied with large-scale molecular dynamics simulations and finite element method calculation. The vibrational band structure derived from the molecular dynamics simulations shows the existence of partial acoustic band gaps along the $Γ$-M direction. The band structure is in excellent agreement with the results from the finite element model, proving that molecular dynamics simulations can be used to study the vibrational properties of such complex systems. An analysis of the structure of the vibrational modes reveals how the acoustic modes deviate from the homogeneous bulk behaviour for shorter wavelengths and hints towards a decoupling of vibrations in the phononic crystal.

cond-mat.mtrl-sci↗

arXiv2017

A bicategorical interpretation for relative Cuntz--Pimsner algebras

We interpret the construction of relative Cuntz-Pimsner algebras of correspondences in terms of the correspondence bicategory, as a reflector into a certain sub-bicategory. This generalises a previous characterisation of absolute Cuntz-Pimsner algebras of proper correspondences as colimits in the correspondence bicategory.

math.OA↗

arXiv2017

Nonarchimedean bornologies, cyclic homology and rigid cohomology

Let $V$ be a complete discrete valuation ring with residue field $k$ and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using spectral radius estimates for bounded subsets in complete bornological $V$-algebras. This leads us to a functorial chain complex for commutative $k$-algebras that computes Berthelot's rigid cohomology. This chain complex is related to the periodic cyclic homology of certain complete bornological $V$-algebras.

math.AG↗

arXiv2017

Representations of -algebras by unbounded operators: C-hulls, local-global principle, and induction

We define a C*-hull for a *-algebra, given a notion of integrability for its representations on Hilbert modules. We establish a local-global principle which, in many cases, characterises integrable representations on Hilbert modules through the integrable representations on Hilbert spaces. The induction theorem constructs a C*-hull for a certain class of integrable representations of a graded *-algebra, given a C*-hull for its unit fibre.

math.OA↗

arXiv2017

A hybrid algorithm for parallel molecular dynamics simulations

This article describes algorithms for the hybrid parallelization and SIMD vectorization of molecular dynamics simulations with short-range forces. The parallelization method combines domain decomposition with a thread-based parallelization approach. The goal of the work is to enable efficient simulations of very large (tens of millions of atoms) and inhomogeneous systems on many-core processors with hundreds or thousands of cores and SIMD units with large vector sizes. In order to test the efficiency of the method, simulations of a variety of configurations with up to 74 million atoms have been performed. Results are shown that were obtained on multi-core systems with Sandy Bridge and Haswell processors as well as systems with Xeon Phi many-core processors.

cond-mat.mtrl-sci↗

arXiv2017

Reduced C*-algebras of Fell bundles over inverse semigroups

We construct a weak conditional expectation from the section C*-algebra of a Fell bundle over a unital inverse semigroup to its unit fibre. We use this to define the reduced C*-algebra of the Fell bundle. We study when the reduced C*-algebra for an inverse semigroup action on a groupoid by partial equivalences coincides with the reduced groupoid C*-algebra of the transformation groupoid, giving both positive results and counterexamples.

math.OA↗

arXiv2017

Inverse semigroup actions on groupoids

We define inverse semigroup actions on topological groupoids by partial equivalences. From such actions, we construct saturated Fell bundles over inverse semigroups and non-Hausdorff étale groupoids. We interpret these as actions on C*-algebras by Hilbert bimodules and describe the section algebras of these Fell bundles. Our constructions give saturated Fell bundles over non-Hausdorff étale groupoids that model actions on locally Hausdorff spaces. We show that these Fell bundles are usually not Morita equivalent to an action by automorphisms. That is, the Packer-Raeburn Stabilisation Trick does not generalise to non-Hausdorff groupoids.

math.OA↗

arXiv2016

Noncommutative coarse geometry

We use compactifications of C*-algebras to introduce noncommutative coarse geometry. We transfer a noncommutative coarse structure on a C*-algebra with an action of a locally compact Abelian group by translations to Rieffel deformations and prove that the resulting noncommutative coarse spaces are coarsely equivalent. We construct a noncommutative coarse structure from a cocompact continuously square-integrable action of a group and show that this is coarsely equivalent to the standard coarse structure on the group in question. We define noncommutative coarse maps through certain completely positive maps that induce *-homomorphisms on the boundaries of the compactifications. We lift *-homomorphisms between separable, nuclear boundaries to noncommutative coarse maps and prove an analogous lifting theorem for maps between the metrisable boundaries of ordinary locally compact spaces.

math.OA↗

arXiv2016

Colimits in the correspondence bicategory

We interpret several constructions with C*-algebras as colimits in the bicategory of correspondences. This includes crossed products for actions of groups and crossed modules, Cuntz-Pimsner algebras of proper product systems, direct sums and inductive limits, and certain amalgamated free products.

math.OA↗

arXiv2016

Semidirect products of C*-quantum groups: multiplicative unitaries approach

C*-quantum groups with projection are the noncommutative analogues of semidirect products of groups. Radford's Theorem about Hopf algebras with projection suggests that any C*quantum group with projection decomposes uniquely into an ordinary C*-quantum group and a "braided" C*-quantum group. We establish this on the level of manageable multiplicative unitaries.

math.OA↗

arXiv2016

Iterated crossed products for groupoid fibrations

We define and study fibrations of topological groupoids. We interpret a groupoid fibration L->H with fibre G as an action of H on G by groupoid equivalences. Our main result shows that a crossed product for an action of L is isomorphic to an iterated crossed product first by G and then by H. Here groupoid action means a Fell bundle over the groupoid, and crossed product means the section C*-algebra.

math.OA↗

arXiv2016

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an étale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.

math.DS↗

arXiv2016

Characters and growth of admissible representations of reductive p-adic groups

We use coefficient systems on the affine Bruhat-Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.

math.RT↗

arXiv2016

Crossed module actions on continuous trace $C^*$-algebras

We lift an action of a torus $\mathbb{T}^n$ on the spectrum of a continuous trace algebra to an action of a certain crossed module of Lie groups that is an extension of $\mathbb{R}^n$. We compute equivariant Brauer and Picard groups for this crossed module and describe the obstruction to the existence of an action of $\mathbb{R}^n$ in our framework.

math.OA↗

arXiv2015

A more general method to classify up to equivariant KK-equivalence

Using a homological invariant together with an obstruction class in a certain Ext^2-group, we may classify objects in triangulated categories that have projective resolutions of length two. This invariant gives strong classification results for actions of the circle group on C*-algebras, C*-algebras over finite unique path spaces, and graph C*-algebras with finitely many ideals.

math.OA↗

arXiv2015

Product systems over Ore monoids

We interpret the Cuntz-Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz-Pimsner algebras over Ore monoids are constructed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz-Pimsner algebra coming from an action of an Ore monoid on a space by topological correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures.

math.OA↗