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Gabriel Larotonda

Publications and source records attributed to Gabriel Larotonda.

At least 19 recordsLinked to original sources
arXiv2025

Lie groups with a bi-invariant distance

We show that a Lie group $G$ admitting a bi-invariant distance must be the product $G=H\times K$ of an abelian group $H$ and a compact group $K$ with discrete center. Moreover, the distance in $G$ must come from the infima of lengths of paths for a unique infinitesimal metric (a Finsler norm) defined in the Lie algebra of $G$. From this we derive the distance minimizing paths which are left or right translations of one-parameter groups (though these are not the unique minizing paths if the norm is not smooth or strictly convex). Then we introduce a notion of sectional curvature $sec(π)$ for a bi-invariant distance, following Milnor's ideas, and we show that this curvature is bounded and non-negative, and it is null when the $2$-plane $π$ is an abelian Lie subalgebra of $Lie(G)$. We show that when the distance is strictly convex, our sectional curvature vanishes if and only if the $2$-plane is abelian. We give finer characterizations for the case of vanishing curvature, for the case of non-strictly convex norms

math.DG↗
arXiv2024

Nijenhuis operators on homogeneous spaces related to $C^*$-algebras

For a unital non-simple $C^*$-algebra $\mathcal A$ we consider its Banach--Lie group $G$ of invertible elements. For a given closed ideal $\mathfrak k$ in $\mathcal A$, we consider the embedded Banach--Lie subgroup $K$ of $G$ of elements differing from the unit element by an element in $\mathfrak k$. We study vector bundle maps of the tangent space of the homogeneous space $G/K$, induced by an admissible bounded operator on $\mathcal A$. In particular, we discuss when this vector bundle map is a Nijenhuis operator in $G/K$. The special case of almost complex structures in $G/K$ is also addressed. Examples for particular classes of $C^*$-algebras are presented, including the Toeplitz algebra and crossed products by $\mathbb Z$.

math.DG↗
arXiv2024

Nijenhuis operators on Banach homogeneous spaces

For a Banach--Lie group $G$ and an embedded Lie subgroup $K$ we consider the homogeneous Banach manifold $\mathcal M=G/K$. In this context we establish the most general conditions for a bounded operator $N$ acting on $Lie(G)$ to define a homogeneous vector bundle map $\mathcal N:T\mathcal M\to T\mathcal M$. In particular our considerations extend all previous settings on the matter and are well-suited for the case where $Lie(K)$ is not complemented in $Lie(G)$. We show that the vanishing of the Nijenhuis torsion for a homogeneous vector bundle map $\mathcal N:T\mathcal M\to T\mathcal M$ (defined by an admissible bounded operator $N$ on $Lie(G)$) is equivalent to the Nijenhuis torsion of $N$ having values in $Lie(K)$. As an application, we consider the question of integrability of an almost complex structure $\mathcal J$ on $\mathcal M$ induced by an admissible bounded operator $J$, and we give a simple characterization of integrability in terms of certain subspaces of the complexification of $Lie(G)$ (which are not eigenspaces of the complex extension of $J$).

math.DG↗
arXiv2024

Connections and Finsler geometry of the structure group of a JB-algebra

We endow the Banach-Lie structure group $Str(V)$ of an infinite dimensional JB-algebra $V$ with a left-invariant connection and Finsler metric, and we compute all the quantities of its connection. We show how this connection reduces to $G(Ω)$, the group of transformations that preserve the positive cone $Ω$ of the algebra $V$, and to $Aut(V)$, the group of Jordan automorphisms of the algebra. We present the cone $Ω$ as an homogeneous space for the action of $G(Ω)$, therefore inducing a quotient Finsler metric and distance. With the techniques introduced, we prove the minimality of the one-parameter groups in $Ω$ for any symmetric gauge norm in $V$. We establish that the two presentations of the Finsler metric in $Ω$ give the same distance there, which helps us prove the minimality of certain paths in $G(Ω)$ for its left-invariant Finsler metric.

math.DG↗
arXiv2024

Convexity of sums of eigenvalues of a segment of unitaries

For a $n\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\le m\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path $t\mapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided the path stays in a vecinity of the identity matrix). This vecinity is described in terms of the opertor norm of matrices, and it is optimal. We show that the when all the maps $t\mapsto s_m(t)$ are linear, then $x$ commutes with $y$. Several application to unitarily invariant norms in the unitary group are given. Then we extend these applications to $Ad$-invariant Finsler norms in the special unitary group of matrices. This last result is obtained by proving that any $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the supremum of a family of what we call orbit norms, induced by the Killing form of $K$.

math.FA↗
arXiv2024

Totally geodesic submanifolds in the manifold SPD of symmetric positive-definite real matrices

This paper is a self-contained exposition of the geometry of symmetric positive-definite real $n\times n$ matrices $\operatorname{SPD}(n)$, including necessary and sufficent conditions for a submanifold $\mathcal{N} \subset\operatorname{SPD}(n)$ to be totally geodesic for the affine-invariant Riemannian metric. A non-linear projection $x\mapsto π(x)$ on a totally geodesic submanifold is defined. This projection has the minimizing property with respect to the Riemannian metric: it maps an arbitrary point $x \in\operatorname{SPD}(n)$ to the unique closest element $π(x)$ in the totally geodesic submanifold for the distance defined by the affine-invariant Riemannian metric. Decompositions of the space $\operatorname{SPD}(n)$ follow, as well as variants of the polar decomposition of non-singular matrices known as Mostow's decompositions. Applications to decompositions of covariant matrices are mentioned.

math.DG↗
arXiv2023

Conjugate points in the Grassmann manifold of a $C^*$-algebra

Let $Gr$ be a component of the Grassmann manifold of a $C^*$-algebra, presented as the unitary orbit of a given orthogonal projection $Gr=Gr(P)$. There are several natural connections in this manifold, and we first show that they all agree (in the presence of a finite trace in $\mathcal A$, when we give $Gr$ the Riemannian metric induced by the Killing form, this is the Levi-Civita connection of the metric). We study the cut locus of $P\in Gr$ for the spectral rectifiable distance, and also the conjugate tangent locus of $P\in Gr$ along a geodesic. Furthermore, for each tangent vector $V$ at $P$, we compute the kernel of the differential of the exponential map of the connection. We exhibit examples where points that are tangent conjugate in the classical setting, fail to be conjugate: in some cases they are not monoconjugate but epinconjugate, and in other cases they are not conjugate at all.

math.FA↗
arXiv2023

Hofer's metric in compact Lie groups

In this article we study the Hofer geometry of a compact Lie group $K$ which acts by Hamiltonian diffeomorphisms on a symplectic manifold $M$. Generalized Hofer norms on the Lie algebra of $K$ are introduced and analyzed with tools from group invariant convex geometry, functional and matrix analysis. Several global results on the existence of geodesics and their characterization in finite dimensional Lie groups $K$ endowed with bi-invariant Finsler metrics are proved. We relate the conditions for being a geodesic in the group $K$ and in the group of Hamiltonian diffeomorphisms. These results are applied to obtain necessary and sufficient conditions on the moment polytope of the momentum map, for the commutativity of the Hamiltonians of geodesics. Particular cases are studied, where a generalized non-crossing of eigenvalues property of the Hamiltonians hold.

math.MG↗
arXiv2023

Weakly invariant norms: geometry of spheres in the space of skew-Hermitian matrices

Let $N$ be a weakly unitarily invariant norm (i.e. invariant for the coadjoint action of the unitary group) in the space of skew-Hermitian matrices $\mathfrak{u}_n(\mathbb C)$. In this paper we study the geometry of the unit sphere of such a norm, and we show how its geometric properties are encoded by the majorization properties of the eigenvalues of the matrices. We give a detailed characterization of norming functionals of elements for a given norm, and we then prove a sharp criterion for the commutator $[X,[X,V]]$ to be in the hyperplane that supports $V$ in the unit sphere. We show that the adjoint action $V\mapsto V+[X,V]$ of $\mathfrak{u}_n(\mathbb C)$ on itself pushes vectors away from the unit sphere. As an application of the previous results, for a strictly convex norm, we prove that the norm is preserved by this last action if and only if $X$ commutes with $V$. We give a more detailed description in the case of any weakly $Ad$-invariant norm.

math.MG↗
arXiv2022

On the structure group of an infinite dimensional JB-algebra

We extend several results for the structure group of a real Jordan algebra $V$, to the setting of infinite dimensional JB-algebras. We prove that the structure group $Str(V)$, the cone preserving group $G(Ω)$ and the automorphism group $Aut(V)$ of the algebra $V$ are embedded Banach-Lie groups of $GL(V)$, and that each of the inclusions $Aut(V)\subset G(Ω)\subset Str(V)$ are of embedded Banach-Lie subgroups. We give a full description of the components of $Str(V)$ via cones, isotopes and central projections. We apply these results to $V=B(H)_{sa}$ the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups $Str(V), G(Ω), Aut(V)$, their Banach-Lie algebras and their connected components. We show that the action of the unitary group of $H$ on $Aut(V)$ has smooth local cross sections, thus $Aut(V)$ is a smooth principal bundle over the unitary group, with circle structure group.

math.FA↗
arXiv2021

Unitary group orbits versus groupoid orbits of normal operators

We study the unitary orbit of a normal operator $a\in \mathcal B(\mathcal H)$, regarded as a homogeneous space for the action of unitary groups associated with symmetrically normed ideals of compact operators. We show with an unified treatment that the orbit is a submanifold of the differing ambient spaces if and only if the spectrum of $a$ is finite, and in that case it is a closed submanifold. For arithmetically mean closed ideals, we show that nevertheless the orbit always has a natural manifold structure, modeled by the kernel of a suitable conditional expectation. When the spectrum of $a$ is not finite, we describe the closure of the orbits of $a$ for the different norm topologies involved. We relate these results to the action of the groupoid of the partial isometries via the moment map given by the range projection of normal operators. We show that all these groupoid orbits also have differentiable structures for which the target map is a smooth submersion. For any normal operator $a$ we also describe the norm closure of its groupoid orbit ${\mathcal O}_a$, which leads to necessary and sufficient spectral conditions on $a$ ensuring that ${\mathcal O}_a$ is norm closed and that ${\mathcal O}_a$ is a closed embedded submanifold of $\mathcal B(\mathcal H)$.

math.FA↗
arXiv2019

The metric geometry of infinite dimensional Lie groups and their homogeneous spaces

We study the geometry of Lie groups $G$ with a continuous Finsler metric, assuming the existence of a subgroup $K$ such that the metric is right-invariant for the action of $K$. We present a systematic study of the metric and geodesic structure of homogeneous spaces $M$ obtained by the quotient $M\simeq G/K$. Of particular interest are left-invariant metrics of $G$ which are then bi-invariant for the action of $K$. We then focus on the geodesic structure of groups $K$ that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds.

math.DG↗
arXiv2018

Canonical sphere bundles of the Grassmann manifold

For a given Hilbert space $\mathcal H$, consider the space of self-adjoint projections $\mathcal P(\mathcal H)$. In this paper we study the differentiable structure of a canonical sphere bundle over $\mathcal P(\mathcal H)$ given by $$ \mathcal R=\{\, (P,f)\in \mathcal P(\mathcal H)\times \mathcal H \, : \, Pf=f , \, \|f\|=1\, \}. $$ We establish the smooth action on $\mathcal R$ of the group of unitary operators of $\mathcal H$, therefore $\mathcal R$ is an homogeneous space. Then we study the metric structure of $\mathcal R$ by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into $\mathcal R$ by the natural action of the unitary group. Then we study the restricted bundle $\mathcal R_2^+$ given by considering only the projections in the restricted Grassmannian, locally modelled by Hilbert-Schmidt operators. Therefore we endow $\mathcal R_2^+$ with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi-Civita connection of this metric and establish a Hopf-Rinow theorem for $\mathcal R_2^+$, again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds.

math.DG↗
arXiv2016

The case of equality in Hölder's inequality for matrices and operators

Let $p>1$ and $1/p+1/q=1$. Consider Hölder's inequality $$ \|ab^*\|_1\le \|a\|_p\|b\|_q $$ for the $p$-norms of some trace ($a,b$ are matrices, compact operators, elements of a finite $C^*$-algebra or a semi-finite von Neumann algebra). This note contains a simple proof (based on the case $p=2$) of the fact that equality holds iff $|a|^p=λ|b|^q$ for some $λ\ge 0$.

math.OA↗
arXiv2016

Geometric significance of Toeplitz kernels

Let $L^2$ be the Lebesgue space of square-integrable functions on the unit circle. We show that the injectivity problem for Toeplitz operators is linked to the existence of geodesics in the Grassmann manifold of $L^2$. We also investigate this connection in the context of restricted Grassmann manifolds associated to $p$-Schatten ideals and essentially commuting projections.

math.FA↗
arXiv2015

Young's (in)equality for compact operators

If $a,b$ are $n\times n$ matrices, Ando proved that Young's inequality is valid for their singular values: if $p>1$ and $1/p+1/q=1$, then $$ λ_k|ab^*|\le λ_k( \frac1p |a|^p+\frac 1q |b|^q ) \, \textit{ for all }k. $$ Later, this result was extended for the singular values of a pair of compact operators acting on a Hilbert space by Erlijman, Farenick and Zeng. In this paper we prove that if $a,b$ are compact operators, then equality holds in Young's inequality if and only if $|a|^p=|b|^q$, obtaining a complete characterization of such $a,b$ in relation to other (operator norm) Young inequalities.

math.FA↗
arXiv2014

Inequalities related to Bourin and Heinz means with a complex parameter

A conjecture posed by S. Hayajneh and F. Kittaneh claims that given $A,B$ positive matrices, $0\le t\le 1$, and any unitarily invariant norm it holds $|||A^tB^{1-t}+B^tA^{1-t}|||\le|||A^tB^{1-t}+A^{1-t}B^t|||$. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for $t\in [1/4;3/4]$. In this paper, using complex methods we extend this result to complex values of the parameter $t=z$ in the strip $\{z \in {\mathbb C}: Re(z) \in [1/4;3/4]\}$. We give an elementary proof of the fact that equality holds for some $z$ in the strip if and only if $A$ and $B$ commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by $s_j(A^tB^{1-t}+B^tA^{1-t})\le s_j(A+B)$, answering in the negative a question made by K. Audenaert and F. Kittaneh.

math.FA↗