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Andreas Thom

Publications and source records attributed to Andreas Thom.

At least 109 records · Page 6Linked to original sources
arXiv2013

Can you compute the operator norm?

In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on Hilbert space. We show that the operator norm in the universal unitary representation is computable if the group is residually finite-dimensional or amenable with decidable word problem. Moreover, we relate the computability of the operator norm on the product of non-abelian free groups to Kirchberg's QWEP Conjecture, a fundamental open problem in the theory of operator algebras.

math.FA↗
arXiv2012

Convergent sequences in discrete groups

We prove that a finitely generated group contains a sequence of non-trivial elements which converge to the identity in every compact homomorphic image if and only if the group is not virtually abelian.

math.GR↗
arXiv2012

About Goto's method showing surjectivity of word maps

Let F be the free group on two letters. For ω \in F we study the associated word map ω: SU(n) \times SU(n) \rightarrow SU(n). Extending a method of Goto, we show that for ω not in the second derived subgroup F^(2) of F, there are infinitely many n \in N such that the associated word map ω : SU(n) \times SU(n) \rightarrow SU(n) is surjective.

math.GR↗
arXiv2012

Hyperbolic Polynomials and Generalized Clifford Algebras

We consider the problem of realizing hyperbolicity cones as spectrahedra, i.e. as linear slices of cones of positive semidefinite matrices. The generalized Lax conjecture states that this is always possible. We use generalized Clifford algebras for a new approach to the problem. Our main result is that if -1 is not a sum of hermitian squares in the Clifford algebra of a hyperbolic polynomial, then its hyperbolicity cone is spectrahedral. Our result also has computational applications, since this sufficient condition can be checked with a single semidefinite program.

math.AG↗
arXiv2012

Is an irng singly generated as an ideal?

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for perfect groups is discussed.

math.RA↗
arXiv2012

Real Closed Separation Theorems and Applications to Group Algebras

In this paper we prove a strong Hahn-Banach theorem: separation of disjoint convex sets by linear forms is possible without any further conditions, if the target field $\R$ is replaced by a more general real closed extension field. From this we deduce a general Positivstellensatz for *-algebras, involving representations over real closed fields. We investigate the class of group algebras in more detail. We show that the cone of sums of squares in the augmentation ideal has an interior point if and only if the first cohomology vanishes. For groups with Kazhdan's property (T) the result can be strengthened to interior points in the $\ell^1$-metric. We finally reprove some strong Positivstellensätze by Helton and Schmüdgen, using our separation method.

math.AG↗
arXiv2011

Detecting Quasars in Large-Scale Astronomical Surveys

We present a classification-based approach to identify quasi-stellar radio sources (quasars) in the Sloan Digital Sky Survey and evaluate its performance on a manually labeled training set. While reasonable results can already be obtained via approaches working only on photometric data, our experiments indicate that simple but problem-specific features extracted from spectroscopic data can significantly improve the classification performance. Since our approach works orthogonal to existing classification schemes used for building the spectroscopic catalogs, our classification results are well suited for a mutual assessment of the approaches' accuracies.

astro-ph.IM↗
arXiv2011

Determinantal Representations and the Hermite Matrix

We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the viewpoint of convex optimization. We relate the question to sums of squares decompositions of a certain Hermite matrix. If some power of a polynomial admits a definite determinantal representation, then its Hermite matrix is a sum of squares. Conversely, we show how a determinantal representation can sometimes be constructed from a sums-of-squares decomposition of the Hermite matrix. We finally show that definite determinantal representations always exist, if one allows for denominators.

math.AG↗
arXiv2011

Applications of Foelner's condition to quantum groups

Using the Foelner condition for coamenable quantum groups we derive information about the ring theoretical structure of the Hopf algebras arising from such quantum groups, as well as an approximation result concerning the Murray von Neumann dimension associated with the corresponding the von Neumann algebra.

math.OA↗
arXiv2011

Algebraic Geometry of Topological Spaces I

We use techniques from both real and complex algebraic geometry to study K-theoretic and related invariants of the algebra C(X) of continuous complex-valued functions on a compact Hausdorff topological space X. For example, we prove a parametrized version of a theorem of Joseph Gubeladze; we show that if M is a countable, abelian, cancellative, torsion-free, seminormal monoid, and X is contractible, then every finitely generated projective module over C(X)[M] is free. The particular case when M=N^n gives a parametrized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic K-theory of C(X) follows from the particular case when M=Z^n. We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on C*-algebras, and for a homology theory of commutative algebras to vanish on C*-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil-K-theory implies that commutative C*-algebras are K-regular. As another application, we show that the familiar formulas of Hochschild-Kostant-Rosenberg and Loday-Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of C(X). Applications to the conjectures of Beilinson-Soule and Farrell-Jones are also given.

math.RA↗
arXiv2011

Polynomials with and without determinantal representations

The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal representation. Brändén has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. So the generalized Lax conjecture fails badly. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation, improving upon Brändén's mostly unconstructive result. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence.

math.OC↗
arXiv2010

On Ulam stability

We study $ε$-representations of discrete groups by unitary operators on a Hilbert space. We define the notion of Ulam stability of a group which loosely means that finite-dimensional $ε$-represendations are uniformly close to unitary representations. One of our main results is that certain lattices in connected semi-simple Lie groups of higher rank are Ulam stable. For infinite-dimensional $ε$-representations, the similarly defined notion of strong Ulam stability is defined and it is shown that groups with free subgroups are not strongly Ulam stable. We also study deformation rigidity of unitary representations and show that groups containing a free subgroup are not deformation rigid.

math.FA↗
arXiv2010

Tracial algebras and an embedding theorem

We prove that every positive trace on a countably generated *-algebra can be approximated by positive traces on algebras of generic matrices. This implies that every countably generated tracial *-algebra can be embedded into a metric ultraproduct of generic matrix algebras. As a particular consequence, every finite von Neumann algebra with separable pre-dual can be embedded into an ultraproduct of tracial *-algebras, which as *-algebras embed into a matrix-ring over a commutative algebra.

math.OA↗
arXiv2010

Group cocycles and the ring of affiliated operators

In this article we study cocycles of discrete countable groups with values in l^2(G) and the ring of affiliated operators UG. We clarify properties of the first cohomology of a group G with coefficients in l^2(G) and answer several questions from [CTV]. Moreover, we obtain strong results about the existence of free subgroups and the subgroup structure, provided the group has a positive first l^2-Betti number. We give numerous applications and examples of groups which satisfy our assumptions.

math.GR↗
arXiv2009

A spectral sequence to compute L2-Betti numbers of groups and groupoids

We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations.

math.DS↗
arXiv2008

Low degree bounded cohomology and l^2-invariants for negatively curved groups

We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. We provide strong restrictions on the possible s-normal subgroups of a Gromov hyperbolic group, or more generally a 'negatively curved' group. Another result says that the image of a group, which is boundedly generated by a finite set of amenable subgroups, in a Gromov hyperbolic group has to be virtually cyclic. Moreover, we show that any homomorphic image of an analogue of a higher rank lattices in a Gromov hyperbolic group must be finite. These results extend to a certain class of randomorphisms in the sense of Monod. We study the class of groups which admit proper quasi-1-cocycles and show that it is closed under l2-orbit equivalence.

math.GR↗
arXiv2008

Examples of hyperlinear groups without factorization property

In this note we give an example of a group which is locally embeddable into finite groups (in particular it is initially subamenable, sofic and hence hyperlinear) but does not have Kirchberg's factorization property. This group provides also an example of a sofic Kazhdan group which is not residually finite, answering a question of Elek and Szabo. We also give an example of a group which is not initially subamenable but hyperlinear. Finally, we point out a mistake in an assertion of Kirchberg and provide an example of a group which does not have the factorization property and is still a subgroup of a connected finite-dimensional Lie group.

math.GR↗