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Marius Mantoiu

Publications and source records attributed to Marius Mantoiu.

At least 19 recordsLinked to original sources
arXiv2022

Topological Dynamics of Groupoid Actions

Some basic notions and results in Topological Dynamics are extended to continuous groupoid actions in topological spaces. We focus mainly on recurrence properties. Besides results that are analogous to the classical case of group actions, but which have to be put in the right setting, there are also new phenomena. Mostly for groupoids whose source map is not open (and there are many), some properties which were equivalent for group actions become distinct in this general framework; we illustrate this with various counterexamples.

math.DS↗
arXiv2021

Symmetry for algebras associated to Fell bundles over groups and groupoids

To every Fell bundle $\mathscr C$ over a locally compact group ${\sf G}$ one associates a Banach $^*$-algebra $L^1({\sf G}\,\vert\,\mathscr C)$. We prove that it is symmetric whenever ${\sf G}$ with the discrete topology is rigidly symmetric. This generalizes the known case of a global action without a twist. There is also a weighted version as well as a treatment of some classes of associated integral kernels. We also deal with the case of Fell bundles over discrete groupoids. We formulate a generalization of rigid symmetry in this case and show its equivalence with an a priori stronger concept. We also study the symmetry of transformation groupoids and some permanence properties.

math.OA↗
arXiv2021

Symmetry and Spectral Invariance for Topologically Graded C*-Algebras and Partial Action Systems

A discrete group $\G$ is called rigidly symmetric if the projective tensor product between the convolution algebra $\ell^1(\G)$ and any $C^*$-algebra $\A$ is symmetric. We show that in each topologically graded $C^*$-algebra over a rigidly symmetric group there is a $\ell^1$-type symmetric Banach $^*$-algebra, which is inverse closed in the $C^*$-algebra. This includes new general classes, as algebras admitting dual actions and partial crossed products. Results including convolution dominated kernels, inverse closedness with respect with ideals or weighted versions of the $\ell^1$-decay are included. Various concrete examples are presented.

math.OA↗
arXiv2021

On Persson's Formula; an Étale Groupoid Approach

Persson's formula expresses the infimum of the essential spectrum of a suitable self-adjoint Schrödinger operator in $R^n$ in terms of the lower spectral points of a family of restrictions of the operator to complements of relatively compact subsets. It has been extended to other situations. We present an approach based on $C^*$-algebras associated to étale groupoids. In such a setting there are intrinsic versions, referring to self-adjoint elements of the groupoid $C^*$-algebras. When representations in Hilbert spaces are considered, the results not always involve only the essential spectrum. The range of applications is quite distinct from the traditional one. We indicate examples related to symbolic dynamics and band dominated operators on discrete metric spaces. The treatment needs only a small amount of symmetry. Even when group actions are involved, restrictions to non-invariant subsets are needed and have to be treated carefully.

math.SP↗
arXiv2018

Spectral Theory in a Twisted Groupoid Setting: Spectral Decompositions, Localization and Fredholmness

We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary of this main orbit. As examples, we discuss various classes of magnetic pseudo-differential operators on nilpotent groups. We also prove localization and non-propagation properties associated to suitable parts of the essential spectrum. These are applied to twisted groupoids having a totally intransitive groupoid restriction at the boundary.

math.OA↗
arXiv2018

Quantum observables as magnetic pseudodifferential operators

In a series of papers we have argued that the 'basic' physical procedure of minimal coupling giving the quantum description of a Hamiltonian system interacting with a magnetic field, can be given a very satisfactory mathematical formulation as a twisted Weyl quantization \cite{MP2}. In this paper we shall present a review of some of these results with some modified proofs that allow a special focus on the dependence on the behavior of the magnetic field, having in view possible developments towards problems with unbounded magnetic fields. The main new result is contained in Theorem 2.9 and states that the the symbol of the evolution group of the self-adjoint operator defined by a real elliptic symbol of strictly positive order in a smooth bounded magnetic field is in the associated magnetic Moyal algebra, i.e. leaves invariant the space of Schwartz test functions and its dual

math-ph↗
arXiv2018

Modulation Spaces and Representations for Rieffel's Quantization

We define localized modulation maps and modulation spaces of symbols suited to the study of Rieffel's deformation quantization pseudodifferential calculus. They are used to generate Hilbert space representations for the quantized $C^*$-algebras, starting from covariant representations of the corresponding twisted $C^*$-dynamical system. In the case of an Abelian undeformed algebra, orthogonal relations and extra information about the representations are obtained.

math.FA↗
arXiv2016

Resolvent Estimates and Smoothing for Homogeneous Operators on Graded Lie Groups

By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators $H$ on graded groups, including Rockland operators, sublaplacians and many others. Left or right invariance is not required. Typically the globally smooth operator has the form $T=V|H|^{1/2}$, where $V$ only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the results involve refined estimates in terms of real interpolation spaces and are valid in an abstract setting. Even for the commutative group $\R^N$ some new classes of operators are treated.

math.FA↗
arXiv2016

Spectral Analysis for Perturbed Operators on Carnot Groups

Let $\G$ be a Carnot group of homogeneous dimension $M$ and $Δ$ its horizontal sublaplacian. For $α\in(0,M)$ we show that operators of the form $H_α:=(-Δ)^α+V$ have no singular spectrum, under generous assumptions on the multiplication operator $V$. The proof is based on commutator methods and Hardy inequalities.

math.FA↗
arXiv2015

A Positive Quantization on Type I Locally Compact Groups

Let $\G$ be a unimodular type I second countable locally compact group and $\wG$ its unitary dual. Motivated by a recent pseudo-differential calculus, we develop a positive Berezin-type quantization with operator-valued symbols defined on $\wG\times\G$.

math.RT↗
arXiv2015

Essential Spectrum and Fredholm Properties for Operators on Locally Compact Groups

We study the essential spectrum and Fredholm properties of integral and pseudodiferential operators associated to (maybe non-commutative) locally compact groups G. The techniques involve crossed product C*-algebras. We extend previous results on the structure of the essential spectrum to operators belonging (or affiliated) to the Schrödinger representation of certain crossed products. When the group G is unimodular and type I, we cover a new class of pseudo-differential differential operators with operator-valued symbols involving the unitary dual of G. We use recent results on the role of families of representations in spectral theory and the notion of quasi-regular dynamical system.

math.SP↗
arXiv2015

Pseudo-differential operators, Wigner transform and Weyl systems on type I locally compact groups

Let $G$ be a unimodular type I second countable locally compact group and $\hat G$ its unitary dual. We introduce and study a global pseudo-differential calculus for operator-valued symbols defined on $G\times\hat G$, and its relations to suitably defined Wigner transforms and Weyl systems. We also unveil its connections with crossed products $C^*$-algebras associated to certain $C^*$-dynamical systems, and apply it to the spectral analysis of covariant families of operators. Applications are given to nilpotent Lie groups, in which case we relate quantizations with operator-valued and scalar-valued symbols.

math.FA↗
arXiv2014

Symmetry and Inverse Closedness for Some Banach $^ *$-Algebras Associated to Discrete Groups

A discrete group $\G$ is called rigidly symmetric if for every $C^*$-algebra $\A$ the projective tensor product $\ell^1(\G)\widehat\otimes\A$ is a symmetric Banach $^*$-algebra. For such a group we show that the twisted crossed product $\ell^1_{α,ø}(\G;\A)$ is also a symmetric Banach $^*$-algebra, for every twisted action $(α,ø)$ of $\G$ in a $C^*$-algebra $\A$\,. We extend this property to other types of decay, replacing the $\ell^1$-condition. We also make the connection with certain classes of twisted kernels, used in a theory of integral operators involving group $2$-cocycles. The algebra of these kernels is studied, both in intrinsic and in represented version.

math.FA↗
arXiv2012

Covariant Fields of C*-Algebras under Rieffel Deformation

We show that Rieffel's deformation sends covariant C(T)-algebras into C(T)-algebras. We also treat the lower semi-continuity issue, proving that Rieffel's deformation transforms covariant continuous fields of C*-algebras into continuous fields of C*-algebras. Some examples are indicated, including certain quantum groups.

math.OA↗
arXiv2012

The multiplicative anomaly of three or more commuting elliptic operators

Zeta-regularized determinants are well-known to fail to be multiplicative. Hence one is lead to study the n-fold multiplicative anomaly M_n(A_1,...,A_n) :=\frac{\det_ζ\Big(\prod_{i=1}^n A_i\Big)}{\prod_{i=1}^n \det_ζ(A_i)} attached to n (suitable) operators A_1,...,A_n. We show that if the A_i are commuting pseudo-differential elliptic operators, then their joint multiplicative anomaly can be expressed in terms of the pairwise multiplicative anomalies. Namely M_n(A_1,...,A_n)^{m_1+...+m_n} =\prod_{1\le i<j\le n}M_2(A_i,A_j)^{m_i+m_j}, where m_j is the order of A_j. The proof relies on Wodzicki's 1987 formula for the pairwise multiplicative anomaly M_2(A,B) of two commuting elliptic operators.

math.FA↗