arXiv subjects

Raul Quiroga-Barranco

Publications and source records attributed to Raul Quiroga-Barranco.

At least 19 recordsLinked to original sources

arXiv2026

A Bargmann transform for translation invariant operators on weighted Bergman spaces of the complex half-plane

Let us denote with $\mathcal{A}^2_λ(\mathbb{C}_+)$ ($λ> -1$) a weighted Bergman space over the right half-plane $\mathbb{C}_+$, which admits a unitary representation of $\mathbb{R}$ given by the (imaginary) translations of $\mathbb{C}_+$. We study the von Neumann algebra $\mathfrak{T}\big(\mathcal{A}^2_λ(\mathbb{C}_+)\big)$ of bounded translation invariant operators. We prove that every element of $\mathfrak{T}\big(\mathcal{A}^2_λ(\mathbb{C}_+)\big)$ is a Toeplit operator $T^{(λ)}_A$ for some translation invariant operator $A$ of the ambient $L^2$-space of $\mathcal{A}^2_λ(\mathbb{C}_+)$. Furthermore, we prove that this can be achieved through a commutative von Neumann algebra $\mathfrak{A}_λ$ that yields an assignment $A \mapsto T^{(λ)}_A$ that turns out to be a $*$-isomorphism of $*$-algebras. Our main tool is a Bargmann transform $\mathcal{B}_λ$ for which we establish several operator and representation theoretic properties. We also describe the translation invariant subspaces of $\mathcal{A}^2_λ(\mathbb{C}_+)$ and obtain formulas for the diagonalizing spectral functions for translation invariant Toeplitz operators whose symbols are translation invariant operators. The latter generalize previously known results for function symbols.

math.FA↗

arXiv2025

Operator algebras and unitary representations of Polish groups

We study the relationship between operator algebras, $C^*$ and von Neumann, acting on a Hilbert space and unitary representations of topological groups on the same space. We obtain certain correspondences between both these families of objects. In particular, we prove that the study of Abelian subalgebras of a norm-separable $C^*$-algebra can be, in many aspects, reduced to the study of unitary representations of Polish groups. It is shown that our techniques have applications to the construction and classification of Abelian subalgebras of operator $C^*$-algebras on separable Hilbert spaces.

math.OA↗

arXiv2025

Geometry of the slice regular Möbius transformations of the quaternionic unit ball

For the quaternionic unit ball $\mathbb{B}$, let us denote by $\mathcal{M}(\mathbb{B})$ the set of slice regular Möbius transformations mapping $\mathbb{B}$ onto itself. We introduce a smooth manifold structure on $\mathcal{M}(\mathbb{B})$, for which the evaluation(-action) map of $\mathcal{M}(\mathbb{B})$ on $\mathbb{B}$ is smooth. The manifold structure considered on $\mathcal{M}(\mathbb{B})$ is obtained by realizing this set as a quotient of the Lie group $\mathrm{Sp}(1,1)$, Furthermore, it turns out that $\mathbb{B}$ is a quotient as well of both $\mathcal{M}(\mathbb{B})$ and $\mathrm{Sp}(1,1)$. These quotients are in the sense of principal fiber bundles. The manifold $\mathcal{M}(\mathbb{B})$ is diffeomorphic to $\mathbb{R}^4 \times S^3$.

math.CV↗

arXiv2025

Lie theory of the slice Riemannian geometry on the quaternionic unit ball

The quaternionic unit ball carries a Riemannian metric built using regular Möbius transformations: the slice Riemannian metric. We prove that the geometry induced by this metric is strongly related to the group $\mathrm{Sp}(1,1)$. We also develop the foundations for a Lie theoretic study of the slice Riemannian metric. In particular, we compute its isometry group and prove that it is built from symmetries of the Lie group $\mathrm{Sp}(1,1)$. We also compare the slice Riemannian geometry with the quaternionic Poincaré geometry, where the latter is considered within the setup of Riemannian symmetric spaces.

math.DG↗

arXiv2024

Geometric structures on the quaternionic unit ball and slice regular Möbius transformations

Building from ideas of hypercomplex analysis on the quaternionic unit ball, we introduce Hermitian, Riemannian and Kähler-like structures on the latter. These are built from the so-called regular Möbius transformations. Such geometric structures are shown to be natural generalizations of those from the complex setup. Our structures can be considered as more natural, from the hypercomplex viewpoint, than the usual quaternionic hyperbolic geometry. Furthermore, our constructions provide solutions to problems not achieved by hyper-Kähler and quaternion-Kähler geometries when applied to the quaternionic unit ball. We prove that the Riemannian metric obtained in this work yields the same tensor previously computed by Arcozzi-Sarfatti. However, our approach is completely geometric as opposed to the function theoretic methods of Arcozzi-Sarfatti.

math.CV↗

arXiv2024

The Heisenberg group action on the Siegel domain and the structure of Bergman spaces

We study the biholomorphic action of the Heisenberg group $\mathbb{H}_n$ on the Siegel domain $D_{n+1}$ ($n \geq 1$). Such $\mathbb{H}_n$-action allows us to obtain decompositions of both $D_{n+1}$ and the weighted Bergman spaces $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$). Through the use of symplectic geometry we construct a natural set of coordinates for $D_{n+1}$ adapted to $\mathbb{H}_n$. This yields a useful decomposition of the domain $D_{n+1}$. The latter is then used to compute a decomposition of the Bergman spaces $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$) as direct integrals of Fock spaces. This effectively shows the existence of an interplay between Bergman spaces and Fock spaces through the Heisenberg group $\mathbb{H}_n$. As an application, we consider $\mathcal{T}^{(λ)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ the $C^*$-algebra acting on the weighted Bergman space $\mathcal{A}^2_λ(D_{n+1})$ ($λ> -1$) generated by Toeplitz operators whose symbols belong to $L^\infty(D_{n+1})^{\mathbb{H}_n}$ (essentially bounded and $\mathbb{H}_n$-invariant). We prove that $\mathcal{T}^{(λ)}(L^\infty(D_{n+1})^{\mathbb{H}_n})$ is commutative and isomorphic to $\mathrm{VSO}(\mathbb{R}_+)$ (very slowly oscillating functions on $\mathbb{R}_+$), for every $λ> -1$ and $n \geq 1$.

math.FA↗

arXiv2024

Toeplitz operators and group-moment coordinates for quasi-elliptic and quasi-hyperbolic symbols

For $\mathbb{B}^n$ the $n$-dimensional unit ball and $D_n$ its Siegel unbounded realization, we consider Toeplitz operators acting on weighted Bergman spaces with symbols invariant under the actions of the maximal Abelian subgroups of biholomorphisms $\mathbb{T}^n$ (quasi-elliptic) and $\mathbb{T}^n \times \mathbb{R}_+$ (quasi-hyperbolic). Using geometric symplectic tools (Hamiltonian actions and moment maps) we obtain simple diagonalizing spectral integral formulas for such kinds of operators. Some consequences show how powerful the use of our differential geometric methods are.

math.FA↗

arXiv2024

Bicomplex Bergman spaces on bounded domains

The bicomplex Bergman spaces are studied for any bounded bicomplex domain. Its Bergman kernel is computed in terms of the kernels of the complex projections of the domain. We also introduce two additional reproducing kernel Hilbert spaces and relate its kernels to that of the bicomplex Bergman space.

math.FA↗

arXiv2023

Commuting Toeplitz operators and moment maps on Cartan domains of type III

Let $D^{III}_n$ and $\mathscr{S}_n$ be the Cartan domains of type III that consist of the symmetric $n \times n$ complex matrices $Z$ that satisfy $Z\overline{Z} < I_n$ and $\mathrm{Im}(Z) > 0$, respectively. For these domains, we study weighted Bergman spaces and Toeplitz operators acting on them. We consider the Abelian groups $\mathbb{T}$, $\mathbb{R}_+$ and $\mathrm{Symm}(n,\mathbb{R})$ (symmetric $n \times n$ real matrices), and their actions on the Cartan domains of type III. We call the corresponding actions Abelian Elliptic, Abelian Hyperbolic and Parabolic. The moment maps of these three actions are computed and functions of them (moment map symbols) are used to construct commutative $C^*$-algebras generated by Toeplitz operators. This leads to a natural generalization of known results for the unit disk. We also compute spectral integral formulas for the Toeplitz operators corresponding to the Abelian Elliptic and Parabolic cases.

math.CV↗

arXiv2022

Commuting Toeplitz operators on Cartan domains of type IV and moment maps

Let us consider, for $n \geq 3$, the Cartan domain $\mathrm{D}_n^{\mathrm{IV}}$ of type IV. On the weighted Bergman spaces $\mathcal{A}^2_λ(\mathrm{D}_n^{\mathrm{IV}})$ we study the problem of the existence of commutative $C^*$-algebras generated by Toeplitz operators with special symbols. We focus on the subgroup $\mathrm{SO}(n) \times \mathrm{SO}(2)$ of biholomorphisms of $\mathrm{D}_n^{\mathrm{IV}}$ that fix the origin. The $\mathrm{SO}(n) \times \mathrm{SO}(2)$-invariant symbols yield Toeplitz operators that generate commutative $C^*$-algebras, but commutativity is lost when we consider symbols invariant under a maximal torus or under $\mathrm{SO}(2)$. We compute the moment map $μ^{\mathrm{SO}(2)}$ for the $\mathrm{SO}(2)$-action on $\mathrm{D}_n^{\mathrm{IV}}$ considered as a symplectic manifold for the Bergman metric. We prove that the space of symbols of the form $a = f \circ μ^{\mathrm{SO}(2)}$, denoted by $L^\infty(\mathrm{D}_n^{\mathrm{IV}})^{μ^{\mathrm{SO}(2)}}$, yield Toeplitz operators that generate commutative $C^*$-algebras. Spectral integral formulas for these Toeplitz operators are also obtained.

math.FA↗

arXiv2022

Radial-like Toeplitz operators on Cartan domains of type I

Let $\mathrm{D}^\mathrm{I}_{n \times n}$ be the Cartan domain of type I which consists of the complex $n \times n$ matrices $Z$ that satisfy $Z^*Z < I_n$. For a symbol $a \in L^\infty(\mathrm{D}^\mathrm{I}_{n \times n})$ we consider three radial-like type conditions: 1) left (right) $\mathrm{U}(n)$-invariant symbols, which can be defined by the condition $a(Z) = a\big((Z^*Z)^\frac{1}{2}\big)$ ($a(Z) = a\big((ZZ^*)^\frac{1}{2}\big)$, respectively), and 2) $\mathrm{U}(n) \times \mathrm{U}(n)$-invariant symbols, which are defined by the condition $a(A^{-1}ZB) = a(Z)$ for every $A, B \in \mathrm{U}(n)$. We prove that, for $n \geq 2$, these yield different sets of symbols. If $a$ satisfies 1), either left or right, and $b$ satisfies 2), then we prove that the corresponding Toeplitz operators $T_a$ and $T_b$ commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for $n \geq 2$, symbols $a$ whose corresponding Toeplitz operators $T_a$ are non-normal. We use these facts to prove the existence, for $n \geq 2$, of commutative Banach non-$C^*$ algebras generated by Toeplitz operators.

math.FA↗

arXiv2021

Toeplitz operators, $\mathbb{T}^m$-invariance and quasi-homogeneous symbols

For a partition $\boldsymbol{k} = (k_1, \dots, k_m)$ of $n$ consider the group $\mathrm{U}(\boldsymbol{k}) = \mathrm{U}(k_1) \times \dots \times \mathrm{U}(k_m)$ block diagonally embedded in $\mathrm{U}(n)$ and the center $\mathbb{T}^m$ of $\mathrm{U}(\boldsymbol{k})$. We study the Toeplitz operators with $\mathbb{T}^m$-invariant symbols acting on the weighted Bergman spaces on the unit ball $\mathbb{B}^n$. We introduce the $(\boldsymbol{k},j)$-quasi-radial quasi-homogeneous symbols as those that are invariant under the group $\mathrm{U}(\boldsymbol{k},j,\mathbb{T})$ obtained from $\mathrm{U}(\boldsymbol{k})$ by replacing the factor $\mathrm{U}(k_j)$ with its center $\mathbb{T}$. These symbols are used to build commutative Banach non-$C^*$ algebras generated by Toeplitz operators. These algebras generalize those from the literature and show that they can be built using groups. We describe the action of such Toeplitz operators on monomials through explicit integral formulas involving the symbols. We prove that every Toeplitz operator with $\mathbb{T}^m$-invariant symbol has an associated Toeplitz operator with $\mathrm{U}(\boldsymbol{k})$-invariant symbol in terms of which we can describe some properties.

math.FA↗

arXiv2020

Moment maps of Abelian groups and commuting Toeplitz operators acting on the unit ball

We prove that to every connected Abelian subgroup $H$ of the biholomorphisms of the unit ball $\mathbb{B}^n$ we can associate a set of bounded symbols whose corresponding Toeplitz operators generate a commutative $C^*$-algebra on every weighted Bergman space. These symbols are of the form $a(z) = f(μ^H(z))$, where $μ^H$ is the moment map for the action of $H$ on $\mathbb{B}^n$. We show that, for this construction, if $H$ is a maximal Abelian subgroup, then the symbols introduced are precisely the $H$-invariant symbols. We provide the explicit computation of moment maps to obtain special sets of symbols described in terms of coordinates. In particular, it is proved that our symbol sets have as particular cases all symbol sets from the current literature that yield Toeplitz operators generating commutative $C^*$-algebras on all weighted Bergman spaces on the unit ball $\mathbb{B}^n$. Furthermore, we exhibit examples that show that some of the symbol sets introduced in this work have not been considered before. Finally, several explicit formulas for the corresponding spectra of the Toeplitz operators are presented. These include spectral integral expressions that simplify the known formulas for maximal Abelian subgroups for the unit ball.

math.CV↗

arXiv2020

Local and global rigidity for isometric actions of simple Lie groups on pseudo-Riemannian manifolds

Let $M$ be a finite volume analytic pseudo-Riemannian manifold that admits an isometric $G$-action with a dense orbit, where $G$ is a connected non-compact simple Lie group. For low-dimensional $M$, i.e. $\dim(M) < 2\dim(G)$, when the normal bundle to the $G$-orbits is non-integrable and for suitable conditions, we prove that $M$ has a $G$-invariant metric which is locally isometric to a Lie group with a bi-invariant metric (local rigidity theorem). The latter does not require $M$ to be complete as in previous works. We also prove a general result showing that $M$ is, up to a finite covering, of the form $H/Γ$ ($Γ$ a lattice in the group $H$) when we assume that $M$ is complete (global rigidity theorem). For both the local and the global rigidity theorems we provide cases that imply the rigidity of $G$-actions for $G$ given by $\mathrm{SO}_0(p,q)$, $G_{2(2)}$ or a non-compact simple Lie group of type $F_4$ over $\mathbb{R}$. We also survey the techniques and results related to this work.

math.DG↗

arXiv2019

Toeplitz operators on the domain $\{Z\in M_{2\times2}(\mathbb{C}) \mid Z Z^* < I\}$ with $\mathrm{U}(2)\times\mathbb{T}^2$-invariant symbols

Let $D$ be the irreducible bounded symmetric domain of $2\times2$ complex matrices that satisfy $ZZ^* < I_2$. The biholomorphism group of $D$ is realized by $\mathrm{U}(2,2)$ with isotropy at the origin given by $\mathrm{U}(2)\times\mathrm{U}(2)$. Denote by $\mathbb{T}^2$ the subgroup of diagonal matrices in $\mathrm{U}(2)$. We prove that the set of $\mathrm{U}(2)\times\mathbb{T}^2$-invariant essentially bounded symbols yield Toeplitz operators that generate commutative $C^*$-algebras on all weighted Bergman spaces over $D$. Using tools from representation theory, we also provide an integral formula for the spectra of these Toeplitz operators.

math.FA↗

arXiv2019

Information geometry and asymptotic geodesics on the space of normal distributions

The family $\mathcal{N}$ of $n$-variate normal distributions is parameterized by the cone of positive definite symmetric $n\times n$-matrices and the $n$-dimensional real vector space. Equipped with the Fisher information metric, $\mathcal{N}$ becomes a Riemannian manifold. As such, it is diffeomorphic, but not isometric, to the Riemannian symmetric space $Pos_1(n+1,\mathbb{R})$ of unimodular positive definite symmetric $(n+1)\times(n+1)$-matrices. As the computation of distances in the Fisher metric for $n>1$ presents some difficulties, Lovrič et al.~(2000) proposed to use the Killing metric on $Pos_1(n+1,\mathbb{R})$ as an alternative metric in which distances are easier to compute. In this work, we survey the geometric properties of the space $\mathcal{N}$ and provide a quantitative analysis of the defect of certain geodesics for the Killing metric to be geodesics for the Fisher metric. We find that for these geodesics the use of the Killing metric as an approximation for the Fisher metric is indeed justified for long distances.

cs.IT↗

arXiv2017

Radial Toeplitz operators on the weighted Bergman spaces of Cartan domains

Let $D$ be an irreducible bounded symmetric domain with biholomorphism group $G$ with maximal compact subgroup $K$. For the Toeplitz operators with $K$-invariant symbols we provide explicit simultaneous diagonalization formulas on every weighted Bergman space. The expressions are given in the general case, but are also worked out explicitly for every irreducible bounded symmetric domain including the exceptional ones.

math.FA↗

arXiv2016

Rigidity of an Isometric $SL(3,\mathbb{R})$-Action

We characterize the universal covering of connected analytic pseudo-Riemannian manifolds which admit a non-trivial and isometric action of the simple Lie group $SL(3,\mathbb{R})$ with a dense orbit preserving a finite volume. If such manifold is also weakly irreducible we prove that $M$ is isometric to, or a quotient space of, a simple Lie group containing $SL(3,\mathbb{R})$.

math.DG↗