arXiv subjects

Raul Quiroga-Barranco

Publications and source records attributed to Raul Quiroga-Barranco.

27 records · Page 2Linked to original sources

arXiv2016

Separately Radial and Radial Toeplitz Operators on the Unit Ball and Representation Theory

We study Toeplitz operators with separately radial and radial symbols on the weighted Bergman spaces on the unit ball. The unitary equivalence of such operators with multiplication operators on $\ell^2$ spaces was previously obtained by analytic methods elsewhere (see the references). We prove that the same constructions can be performed with a purely representation theoretic approach to obtain the same conclusions and formulas. However, our method is shorter, more elementary and more elucidating.

math.FA↗

arXiv2015

On low-dimensional manifolds with isometric $\widetilde{\mathrm{U}}(p,q)$-actions

Denote by $\widetilde{\mathrm{U}}(p,q)$ the universal covering group of $\mathrm{U}(p,q)$, the linear group of isometries of the pseudo-Hermitian space $\mathbb{C}^{p,q}$ of signature $p,q$. Let $M$ be a connected analytic complete pseudo-Riemannian manifold that admits an isometric $\widetilde{\mathrm{U}}(p,q)$-action and that satisfies $\dim M \leq n(n+2)$ where $n = p+q$. We prove that if the action of $\widetilde{\mathrm{SU}}(p,q)$ (the connected derived group of $\widetilde{\mathrm{U}}(p,q)$) has a dense orbit and the center of $\widetilde{\mathrm{U}}(p,q)$ acts non-trivially, then $M$ is an isometric quotient of manifolds involving simple Lie groups with bi-invariant metrics. Furthermore, the $\widetilde{\mathrm{U}}(p,q)$-action is lifted to $\widetilde{M}$ to natural actions on the groups involved. As a particular case, we prove that when $\widetilde{M}$ is not a pseudo-Riemannian product, then its geometry and $\widetilde{\mathrm{U}}(p,q)$-action are obtained from one of the symmetric pairs $(\mathfrak{su}(p,q+1), \mathfrak{u}(p,q))$ or $(\mathfrak{su}(p+1,q), \mathfrak{u}(p,q))$.

math.DG↗

arXiv2014

Toeplitz Operators with Quasi-Homogeneuos Quasi-Radial Symbols on some Weakly Pseudoconvex Domains

On the weakly pseudo-convex domains $Ω_p^n$ we introduce quasi-homogeneous quasi-radial symbols. These are used to prove the existence of a commutative Banach algebra of Toeplitz operators on Bergman space of $Ω_p^n$. We also show that group theoretic and geometric properties for our symbols are satisfied. The results presented here contain the geometric description of the symbols introduced by N. Vasilevski for the unit ball $\mathbb{B}^n$.

math.FA↗

arXiv2014

Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of Lagrangian frames

We prove that the quasi-homogenous symbols on the projective space $\mathbb{P}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball $\mathbb{B}^n$. These algebras are Banach but not $C^*$. We prove the existence of a strong link between such symbols and algebras with the geometry of $\mathbb{P}^n(\mathbb{C})$. The latter is also proved for the corresponding symbols and algebras on $\mathbb{B}^n$.

math.OA↗

arXiv2012

Commutative $C^*$-algebras of Toeplitz operators on complex projective spaces

We prove the existence of commutative $C^*$-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space $\mathbb{P}^n(\mathbb{C})$. The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an associated pair of Lagrangian foliations with distinguished geometric properties and are closely related to the geometry of $\mathbb{P}^n(\mathbb{C})$.

math.OA↗

arXiv2011

On low-dimensional manifolds with isometric $\mathrm{SO}_0(p,q)$-actions

Let $G$ be a non-compact simple Lie group with Lie algebra $\mathfrak{g}$. Denote with $m(\mathfrak{g})$ the dimension of the smallest non-trivial $\mathfrak{g}$-module with an invariant non-degenerate symmetric bilinear form. For an irreducible finite volume pseudo-Riemannian analytic manifold $M$ it is observed that $\dim(M) \geq \dim(G) + m(\mathfrak{g})$ when $M$ admits an isometric $G$-action with a dense orbit. The Main Theorem considers the case $G = \widetilde{\mathrm{SO}}_0(p,q)$ providing an explicit description of $M$ when the bound is achieved. In such case, $M$ is (up to a finite covering) the quotient by a lattice of either $\widetilde{\mathrm{SO}}_0(p+1,q)$ or $\widetilde{\mathrm{SO}}_0(p,q+1)$.

math.DG↗

arXiv2010

Isometric actions of simple groups and transverse structures: The integrable normal case

For actions with a dense orbit of a connected noncompact simple Lie group $G$, we obtain some global rigidity results when the actions preserve certain geometric structures. In particular, we prove that for a $G$-action to be equivalent to one on a space of the form $(G\times K\backslash H)/Γ$, it is necessary and sufficient for the $G$-action to preserve a pseudo-Riemannian metric and a transverse Riemannian metric to the orbits. A similar result proves that the $G$-actions on spaces of the form $(G\times H)/Γ$ are characterized by preserving transverse parallelisms. By relating our techniques to the notion of the algebraic hull of an action, we obtain infinitesimal Lie algebra structures on certain geometric manifolds acted upon by $G$.

math.DG↗

arXiv2007

Isometric actions of simple Lie groups on pseudoRiemannian manifolds

Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m_0, n_0 are the dimensions of the maximal lightlike subspaces tangent to M and G, respectively, where G carries any bi-invariant metric, then we have n_0 \leq m_0. We study G-actions that satisfy the condition n_0 = m_0. With no rank restrictions on G, we prove that M has a finite covering \hat{M} to which the G-action lifts so that \hat{M} is G-equivariantly diffeomorphic to an action on a double coset K\backslash L/Γ, as considered in Zimmer's program, with G normal in L (Theorem A). If G has finite center and \rank_\R(G)\geq 2, then we prove that we can choose \hat{M} for which L is semisimple and Γis an irreducible lattice (Theorem B). We also prove that our condition n_0 = m_0 completely characterizes, up to a finite covering, such double coset G-actions (Theorem C). This describes a large family of double coset G-actions and provides a partial positive answer to the conjecture proposed in Zimmer's program.

math.DG↗

arXiv2007

Arithmeticity of totally geodesic Lie foliations with locally symmetric leaves

R. Zimmer proved that, on a compact manifold, a foliation with a dense leaf, a suitable leafwise Riemannian symmetric metric and a transverse Lie structure has arithmetic holonomy group. In this work we improve such result for totally geodesic foliations by showing that the manifold itself is arithmetic. This also gives a positive answer, for some special cases, to a conjecture of E. Ghys.

math.DG↗