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At least 19 recordsLinked to original sources

arXiv2000

Characterization of Cyclic and Separating Vectors and Application to an Inverse Problem in Modular Theory, II. Semifinite Factors

This paper generalizes the results obtained in an earlier paper (math.OA/0003087) for finite factors to infinite but still semifinite factors. First we give a characterization of cyclic and separating vectors for infinite semifinite factors in terms of operators associated with this vector and being affiliated with the factor. Further we show how this operator generates the modular objects of the given cyclic and separating vector generalizing an idea of Kadison and Ringrose. With the help of these results we can show that the second simple class of solutions for the inverse problem constructed in math.OA/0003087 never exists in infinite semifinite factors. Finally we give a classification of the solutions of the inverse problem in the case of modular operators having pure point spectrum, completely analoguous to the finite case.

math.OA↗

arXiv2004

Tangential dimensions I. Metric spaces

Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a la Gromov, of X at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a set, namely which is able to detect the "oscillations" of the dimension at a given point. In particular we exhibit examples where upper and lower tangential dimensions differ, even when the local upper and lower box dimensions coincide. Tangential dimensions can be considered as the classical analogue of the tangential dimensions for spectral triples introduced in math.OA/0202108 and math.OA/0404295, in the framework of Alain Connes' noncommutative geometry.

math.FA↗

arXiv2004

Tangential dimensions II. Measures

Notions of (pointwise) tangential dimension are considered, for measures of R^n. Under regularity conditions (volume doubling), the upper resp. lower dimension at a point x of a measure can be defined as the supremum, resp. infimum, of local dimensions of the measures tangent to the given measure at x. Our main purpose is that of introducing a tool which is very sensitive to the "multifractal behaviour at a point" of a measure, namely which is able to detect the "oscillations" of the dimension at a given point, even when the local dimension exists, namely local upper and lower dimensions coincide. These definitions are tested on a class of fractals, which we call translation fractals, where they can be explicitly calculated for the canonical limit measure. In these cases the tangential dimensions of the limit measure coincide with the metric tangential dimensions of the fractal defined in math.FA/0305091, and they are constant, i.e. do not depend on the point. However, upper and lower dimensions may differ. Moreover, on these fractals, these quantities coincide with their noncommutative analogues, defined in math.OA/0202108 and math.OA/0404295, in the framework of Alain Connes' noncommutative geometry.

math.FA↗

arXiv2006

Complex powers of the contact Laplacian and the Baum-Connes conjecture for SU(n,1)

This paper is an extended version of math.OA/0601528 where we point out and remedy a gap in the proof by P. Julg and G. Kasparov of the Baum-Connes conjecture for discrete subgroups of SU(n,1). In particular, here we explain in details why the non-microlocality of the Heisenberg calculus prevents us from implementing into this framework the classical approach of Seeley to pseudodifferential complex powers, which was the main issue at stake in math.OA/0601528.

math.AP↗

arXiv2019

CCR and CAR flows over convex cones

Recently it is proved in arXiv:1906.05493v1 [math.OA] that CCR flows over convex cones are cocycle conjugate if and only if the associated isometric representations are conjugate. We provide a very short, simple and direct proof of that. Using the same idea we prove the analogous statement for CAR flows as well. Further we show that CCR flows are not cocycle conjugate to the CAR flows when the (multi-parameter) isometric representation is `proper', a condition which is satisfied by all known examples.

math.OA↗

arXiv2020

Extension of frames and bases -- II

Operator-valued frame ($G$-frame), as a generalization of frame is introduced by Kaftal, Larson, and Zhang in \textit{Trans. Amer. Math. Soc.}, 361(12):6349-6385, 2009 and by Sun in \textit{J. Math. Anal. Appl.}, 322(1):437-452, 2006. It has been further extended in the paper arXiv:1810.01629 [math.OA] 3 October 2018, so as to have a rich theory on operator-valued frames for Hilbert spaces as well as for Banach spaces. The continuous version has been studied in this paper when the indexing set is a measure space. We study duality, similarity, orthogonality and stability of this extension. Several characterizations are given including a notable characterization when the measure space is a locally compact group. Variation formula, dimension formula and trace formula are derived when the Hilbert space is finite dimensional.

math.FA↗

arXiv2001

Free stochastic measures via noncrossing partitions II

We show that for stochastic measures with freely independent increments, the partition-dependent stochastic measures of math.OA/9903084 can be expressed purely in terms of the higher stochastic measures and the higher diagonal measures of the original.

math.OA↗

arXiv2001

An asymptotic dimension for metric spaces, and the 0-th Novikov-Shubin invariant

A nonnegative number d_infinity, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0-th Novikov-Shubin number alpha_0 defined previously (math.OA/9802015, cf. also math.DG/0110294). Thus the dimensional interpretation of alpha_0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos.

math.DG↗

arXiv2005

Dimensions and spectral triples for fractals in R^N

Two spectral triples are introduced for a class of fractals in R^n. The definitions of noncommutative Hausdorff dimension and noncommutative tangential dimensions, as well as the corresponding Hausdorff and Hausdorff-Besicovitch functionals considered in math.OA/0202108, are studied for the mentioned fractals endowed with these spectral triples, showing in many cases their correspondence with classical objects. In particular, for any limit fractal, the Hausdorff-Besicovitch functionals do not depend on the generalized limit procedure.

math.OA↗

arXiv2007

A Schur-Horn type theorem for II$_1$ factors

Given a II$_1$ factor M and a masa A of M, we prove a version of the Schur-Horn Theorem, together with a contractive version. These results are inspired on a recent conjecture of Arveson and Kadison (math.OA/0508482).

math.OA↗

arXiv2010

Non-cyclotomic fusion categories

Etingof, Nikshych and Ostrik ask in arXiv:math.QA/0203060 if every fusion category can be completely defined over a cyclotomic field. We show that this is not the case: in particular one of the fusion categories coming from the Haagerup subfactor arXiv:math.OA/9803044 and one coming from the newly constructed extended Haagerup subfactor arXiv:0909.4099 can not be completely defined over a cyclotomic field. On the other hand, we show that the double of the even part of the Haagerup subfactor is completely defined over a cyclotomic field. We identify the minimal field of definition for each of these fusion categories, compute the Galois groups, and identify their Galois conjugates.

math.QA↗

arXiv2001

A semicontinuous trace for almost local operators on an open manifold

A semicontinuous semifinite trace is constructed on the C*-algebra generated by the finite propagation operators acting on the L^2-sections of a hermitian vector bundle on an amenable open manifold of bounded geometry. This trace is the semicontinuous regularization of a functional already considered by J. Roe. As an application, we show that, by means of this semicontinuous trace, Novikov-Shubin numbers for amenable manifolds can be defined (cf. math.OA/9802015 for an alternate definition).

math.DG↗

arXiv2002

On the domain of singular traces

The question whether an operator belongs to the domain of some singular trace is addressed, together with the dual question whether an operator does not belong to the domain of some singular trace. We show that the answers are positive in general, namely for any (compact, infinite rank) positive operator A we exhibit two singular traces, the first being zero and the second being infinite on A. However, if we assume that the singular traces are generated by a "regular" operator, the answers change, namely such traces always vanish on trace-class, non singularly traceable operators and are always infinite on non trace-class, non singularly traceable operators. These results are achieved on a general semifinite factor, and make use of a new characterization of singular traceability (cf. math.OA/0202108).

math.OA↗

arXiv2017

An alternative characterization of normed interpolation spaces between $\ell^{1}$ and $\ell^{q}$

Given a constant $q\in(1,\infty)$, we study the following property of a normed sequence space $E$: ===================== If $\left\{ x_{n}\right\}_{n\in\mathbb{N}}$ is an element of $E$ and if $\left\{ y_{n}\right\}_{n\in\mathbb{N}}$ is an element of $\ell^{q}$ such that $\sum_{n=1}^{\infty}\left|x_{n}\right|^{q}=\sum_{n=1}^\infty \left|y_{n}\right|^{q}$ and if the nonincreasing rearrangements of these two sequences satisfy $\sum_{n=1}^{N}\left|x_{n}^{*}\right|^{q}\le\sum_{n=1}^{N}\left|y_{n}^{*}\right|^{q}$ for all $N\in\mathbb{N}$, then $\left\{ y_{n}\right\}_{n\in\mathbb{N}}\in E$ and $\left\Vert \left\{ y_{n}\right\}_{n\in\mathbb{N}}\right\Vert_{E}\le C\left\Vert \left\{ x_{n}\right\}_{n\in\mathbb{N}}\right\Vert_{E}$ for some constant $C$ which depends only on $E$. ===================== We show that this property is very close to characterizing the normed interpolation spaces between $\ell^{1}$ and $\ell^{q}$. More specificially, we first show that every space which is a normed interpolation space with respect to the couple $\left(\ell^{p},\ell^{q}\right)$ for some $p\in[1,q]$ has the above mentioned property. Then we show, conversely, that if $E$ has the above mentioned property, and also has the Fatou property, and is contained in $\ell^{q}$, then it is a normed interpolation space with respect to the couple $\left(\ell^{1},\ell^{q}\right)$. These results are our response to a conjecture of Galina Levitina, Fedor Sukochev and Dmitriy Zanin in arXiv:1703.04254v1 [math.OA].

math.FA↗

arXiv2006

Pseudodifferential operators on manifolds with a Lie structure at infinity

Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V \subset Γ(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds with a Lie structure at infinity" there -- was studied from an axiomatic point of view. In this paper, we define and study the algebra $Ψ_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at infinity $V \subsetΓ(TM)$. We show that many of the properties of the usual algebra of pseudodifferential operators on a compact manifold extend to $Ψ_{1,0,V}^\infty(M_0)$. We also consider the algebra $\DiffV{*}(M_0)$ of differential operators on $M_0$ generated by $V$ and $\CI(M)$, and show that $Ψ_{1,0,V}^\infty(M_0)$ is a ``microlocalization'' of $\DiffV{*}(M_0)$. Finally, we introduce and study semi-classical and ``suspended'' versions of the algebra $Ψ_{1,0,V}^\infty(M_0)$. Our construction solves a problem posed by Melrose in his talk at the ICM in Kyoto.

math.AP↗

arXiv1999

On the index of equivariant Toeplitz operators

The goal is to understand the index-theoretic aspects of the recent preprint of R. Nest and F. Radulescu, math.OA/9911042. The basic observation (due to E. Guenter/N. Higson) is that the index of the Toeplitz operator is equal to the index of an associated Callias type operator, i.e. a Dirac operator with potential, the index of which is easy to compute. We show how to extend this idea to the equivariant case.

math.DG↗

arXiv2018

Simple maps, Hurwitz numbers, and Topological Recursion

We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder. We show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks. We also obtain an elegant formula for cylinders. These relations reproduce the relation between moments and free cumulants established by Collins et al. math.OA/0606431, and implement the symplectic transformation $x \leftrightarrow y$ on the spectral curve in the context of topological recursion. We conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of $x$ and $y$. We propose an argument to prove this statement conditionally to a mild version of symplectic invariance for the $1$-hermitian matrix model, which is believed to be true but has not been proved yet. Our argument relies on an (unconditional) matrix model interpretation of fully simple maps, via the formal hermitian matrix model with external field. We also deduce a universal relation between generating series of fully simple maps and of ordinary maps, which involves double monotone Hurwitz numbers. In particular, (ordinary) maps without internal faces -- which are generated by the Gaussian Unitary Ensemble -- and with boundary perimeters $(λ_1,\ldots,λ_n)$ are strictly monotone double Hurwitz numbers with ramifications $λ$ above $\infty$ and $(2,\ldots,2)$ above $0$. Combining with a recent result of Dubrovin et al. math-ph/1612.02333, this implies an ELSV-like formula for these Hurwitz numbers.

math-ph↗

arXiv2024

Complete orthogonality preservers between C$^*$-algebras

We introduce $n$-orthogonality (and completely orthogonality) preserving operators between C$^*$-algebras. Our main theorem states that every completely orthogonality preserving bounded linear mapping between C$^*$-algebras is a weighted TRO homomorphism. We also give several characterisations of TRO homomorphisms among triple homomorphisms.

math.OA↗