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Daniel Beltita

Publications and source records attributed to Daniel Beltita.

At least 55 records · Page 3Linked to original sources
arXiv2014

Nonlinear completely positive maps and dilation theory for real involutive algebras

A real seminormed involutive algebra is a real associative algebra ${\mathcal A}$ endowed with an involutive antiautomorphism $*$ and a submultiplicative seminorm $p$ with $p(a^*) =p(a)$ for $a\in {\mathcal A}$. Then ${\mathop{\tt ball}\nolimits}({\mathcal A},p) := \{a \in {\mathcal A} \colon p(a) < 1\}$ is an involutive subsemigroup. For the case where ${\mathcal A}$ is unital, our main result asserts that a function $ϕ\colon{\mathop{\tt ball}\nolimits}({\mathcal A},p) \to B(V)$, $V$ a Hilbert space, is completely positive (defined suitably) if and only if it is positive definite and analytic for any locally convex topology for which ${\mathop{\tt ball}\nolimits}({\mathcal A},p)$ is open. If $η_{\mathcal A} \colon {\mathcal A} \to C^*({\mathcal A},p)$ is the enveloping $C^*$-algebra of $({\mathcal A},p)$ and $e^{C^*({\mathcal A},p)}$ is the $c_0$-direct sum of the symmetric tensor powers $S^n(C^*({\mathcal A},p))$, then the above two properties are equivalent to the existence of a factorization $ϕ= Φ\circ Γ$, where $Φ\colon e^{C^*({\mathcal A},p)} \to B(V)$ is linear completely positive and $Γ(a) = \sum_{n = 0}^\infty η_{\mathcal A}(a)^{\otimes n}$. We also obtain a suitable generalization to non-unital algebras. An important consequence of this result is a description of the unitary representations of ${\rm U}({\mathcal A})$ with bounded analytic extensions to ${\mathop{\tt ball}\nolimits}({\mathcal A},p)$ in terms of representations of the $C^*$-algebra $e^{C^*({\mathcal A},p)}$.

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arXiv2014

Weyl-Pedersen calculus for some semidirect products of nilpotent Lie groups

For certain nilpotent real Lie groups constructed as semidirect products, algebras of invariant differential operators on some coadjoint orbits are used in the study of boundedness properties of the Weyl-Pedersen calculus of their corresponding unitary irreducible representations. Our main result is applicable to all unitary irreducible representations of arbitrary 3-step nilpotent Lie groups.

math.RT↗
arXiv2014

Cartan subalgebras of operator ideals

Denote by $U_{\mathcal I}({\mathcal H})$ the group of all unitary operators in ${\bf 1}+{\mathcal I}$ where ${\mathcal H}$ is a separable infinite-dimensional complex Hilbert space and ${\mathcal I}$ is any two-sided ideal of ${\mathcal B}({\mathcal H})$. A Cartan subalgebra ${\mathcal C}$ of ${\mathcal I}$ is defined in this paper as a maximal abelian self-adjoint subalgebra of~${\mathcal I}$ and its conjugacy class is defined herein as the set of Cartan subalgebras $\{V{\mathcal C} V^*\mid V\in U_{\mathcal I}({\mathcal H})\}$. For nonzero proper ideals ${\mathcal I}$ we construct an uncountable family of Cartan subalgebras of ${\mathcal I}$ with distinct conjugacy classes. This is in contrast to the by now classical observation of P. de La Harpe who noted that when ${\mathcal I}$ is any of the Schatten ideals, there is precisely one conjugacy class under the action of the full group of unitary operators on~${\mathcal B}$. In the case when ${\mathcal I}$ is a symmetrically normed ideal and is the dual of some Banach space, we show how the conjugacy classes of the Cartan subalgebras of ${\mathcal I}$ become smooth manifolds modeled on suitable Banach spaces. These manifolds are endowed with groups of smooth transformations given by the action of the group $U_{\mathcal I}({\mathcal H})$ on the orbits, and are equivariantly diffeomorphic to each other. We then find that there exists a unique diffeomorphism class of full flag manifolds of $U_{\mathcal I}({\mathcal H})$ and we give its construction.

math.OA↗
arXiv2014

Inverse-closed algebras of integral operators on locally compact groups

We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make systematic use of covariance algebras associated to $C^*$-dynamical systems defined by the abelian $C^*$-algebras of right uniformly continuous functions with respect to the left regular representation.

math.FA↗
arXiv2014

Moment convexity of solvable locally compact groups

We introduce moment maps for continuous unitary representations of general topological groups. For solvable separable locally compact groups, we prove that the closure of the image of the moment map of any representation is convex.

math.RT↗
arXiv2014

Coadjoint orbits of stepwise square integrable representations

Nilpotent Lie groups with stepwise square integrable representations were recently investigated by J.A.~Wolf. We give an alternative approach to these representations by relating them to the stratifications of the duals of nilpotent Lie algebras, thus proving that they correspond to a subset with relative Hausdorff topology, dense interior, and total Plancherel measure in the unitary dual of the Lie group under consideration.

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arXiv2014

On universal enveloping algebras in a topological setting

We establish the exponential law for suitably topologies on spaces of vector-valued smooth functions on topological groups, where smoothness is defined by using differentiability along continuous one-parameter subgroups. As an application, we investigate the canonical correspondences between the universal enveloping algebra, the invariant local operators, and the convolution algebra of distributions supported at the unit element of any finite-dimensional Lie group, when one passes from finite-dimensional Lie groups to pre-Lie groups. The latter class includes for instance any locally compact groups, Banach-Lie groups, additive groups underlying locally convex vector spaces, and also mapping groups consisting of rapidly decreasing Lie group-valued functions.

math.FA↗
arXiv2014

Reproducing kernels and positivity of vector bundles in infinite dimensions

We investigate the interaction between the existence of reproducing kernels on infinite-dimensional Hermitian vector bundles and the positivity properties of the corresponding bundles. The positivity refers to the curvature form of certain covariant derivatives associated to reproducing kernels on the vector bundles under consideration. The values of the curvature form are Hilbert space operators, and its positivity is thus understood in the usual sense from operator theory.

math.FA↗
arXiv2013

Linear connections for reproducing kernels on vector bundles

We construct a canonical correspondence from a wide class of reproducing kernels on infinite-dimensional Hermitian vector bundles to linear connections on these bundles. The linear connection in question is obtained through a pull-back operation involving the tautological universal bundle and the classifying morphism of the input kernel. The aforementioned correspondence turns out to be a canonical functor between categories of kernels and linear connections. A number of examples of linear connections including the ones associated to classical kernels, homogeneous reproducing kernels and kernels occurring in the dilation theory for completely positive maps are given, together with their covariant derivatives.

math.RT↗
arXiv2013

Boundedness for Weyl-Pedersen calculus on flat coadjoint orbits

We describe boundedness and compactness properties for the operators obtained by the Weyl-Pedersen calculus in the case of the irreducible unitary representations of nilpotent Lie groups that are associated with flat coadjoint orbits. We use spaces of smooth symbols satisfying appropriate growth conditions expressed in terms of invariant differential operators on the coadjoint orbit under consideration. Our method also provides conditions for these operators to belong to one of the Schatten ideals of compact operators. In the special case of the Schrödinger representation of the Heisenberg group we recover some classical properties of the pseudo-differential Weyl calculus, as the Calderón-Vaillancourt theorem, and the Beals characterization in terms of commutators.

math.AP↗
arXiv2013

On the differentiable vectors for contragredient representations

We establish a few simple results on contragredient representations of Lie groups, with a view toward applications to the abstract characterization of some spaces of pseudo-differential operators. In particular, this method provides an abstract approach to J.Nourrigat's recent description of the norm closure of the pseudo-differential operators of order zero.

math.RT↗
arXiv2013

Interplay between Algebraic Groups, Lie Algebras and Operator Ideals

In the framework of operator theory, we investigate a close Lie theoretic relationship between all operator ideals and certain classical groups of invertible operators that can be described as the solution sets of certain algebraic equations, hence can be regarded as infinite-dimensional linear algebraic groups. Historically, this has already been done for only the complete-norm ideals; in that case one can work within the framework of the well-known Lie theory for Banach-Lie groups. That kind of Lie theory is not applicable for arbitrary operator ideals, so we needed to find a new approach for dealing with the general situation. The simplest instance of the aforementioned relationship is provided by the Lie algebra $\ug_{\Ic}(\Hc)=\{X\in\Ic\mid X^*=-X\}$ associated with the group $\U_{\Ic}(\Hc)=\U(\Hc)\cap(\1+\Ic)$ where $\Ic$ is an arbitrary operator ideal in $\Bc(\Hc)$ and $\U(\Hc)$ is the full group of unitary operators. We investigate the Cartan subalgebras (maximal abelian self-adjoint subalgebras) of $\ug_{\Ic}(\Hc)$ for $\{0\}\subsetneqq\Ic\subsetneqq\Bc(\Hc)$, and obtain an uncountably many $\U_{\Ic}(\Hc)$-conjugacy classes of these Cartan subalgebras. The cardinality proof will be given in a follow up paper \cite{BPW13} and stands in contrast to the $\U(\Hc)$-uniqueness work of de la Harpe \cite{dlH72}.

math.OA↗
arXiv2013

$B(H)$-Commutators: A Historical Survey II and recent advances on commutators of compact operators

A sequel to \cite{gW05}, we address again the single commutator problem \cite{PT71} of Pearcy and Topping: Is every compact operator a single commutator of compact operators? by focusing on a 35 year old test question for this posed in 1976 by the last named author and others: Are there any strictly positive operators that are single commutators of compact operators? The latter we settle here affirmatively with a modest modification of Anderson's fundamental construction \cite{jA77} constructing compact operators whose commutator is a rank one projection. Moreover we provide here a rich class of such strictly positive operators that are commutators of compact operators and pose a question for the rest. We explain also how these methods are related to the study of staircase matrix forms, their equivalent block tri-diagonal forms, and commutator problems. In particular, we present the original test question and solution that led to the negative solution of the Pearcy-Topping question on whether or not every trace class trace zero operator was a commutator (or linear combination of commutators) of Hilbert-Schmidt operators. And we show how this evolved from staircase form considerations along with a Larry Brown result on trace connections to ideals \cite{lB94} which itself is at the core of \cite[Section 7]{DFWW}. The omission in \cite{gW05} of this important 35 year old test question was inadvertent and we correct that in this paper. This sequel starts where [ibid] left off but can be read independently of [ibid]. The present paper also has a section on self-commutator equations $[X^*,X]=A$ within the framework of some classical operator Lie algebras. That problem was solved by Fan and Fong (1980) for the full algebra of compact operators, and we solve it here for the complex symplectic Lie algebra of compact operators and for complex semisimple Lie algebras.

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