arXiv subjects

Costel Peligrad

Publications and source records attributed to Costel Peligrad.

At least 19 recordsLinked to original sources

arXiv2024

Outer actions of finite groups on prime C*-algebras

An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra and strictly outer if the commutant of the algebra in the algebra of local mutipliers of the cross product consists of scalars [11]. In [11, Theorem 11] I proved that for finite groups and prime C*-algebras (not necessarily separable), the two notions are equivalent. I also proved that for finite abelian groups this is equivalent to other relevant properties of the action [11 Theorem 14]. In this paper I add other properties to the list in [11, Theorem 14].

math.OA↗

arXiv2024

Properly Outer and Strictly Outer Actions of Finite Groups on Prime C*-algebras

An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra. In this paper I define the notion of strictly outer action (similar to the definition for von Neumann factors in [11]) and prove that for finite groups it is equivalent with proper outerness of the action. For finite abelian groups this is equivalent with other relevant properities of the action.

math.OA↗

arXiv2022

Convergence of series of conditional expectations

This paper deals with rates of convergence in the strong law of large numbers, in the Baum-Katz form, for partial sums of Banach space valued random variables. The results are then applied to solve similar problems for weighted partial sums of conditional expectations. They are further used to treat partial sums of powers of a reversible Markov chain operator. The method of proof is based on martingale approximation. The conditions are expressed in terms moments of the individual summands.

math.PR↗

arXiv2022

On the local limit theorems for lower psi-mixing Markov chains

In this paper we investigate the local limit theorem for additive functionals of nonstationary Markov chains that converge in distribution. We consider both the lattice and the non-lattice cases. The results are also new in the stationary setting and lead to local limit theorems linked to convergence to stable distributions. The conditions are imposed to individual summands and are expressed in terms of lower psi-mixing coefficients.

math.PR↗

arXiv2020

On the local limit theorems for psi-mixing Markov chains

In this paper we investigate the local limit theorem for additive functionals of a nonstationary Markov chain with finite or infinite second moment. The moment conditions are imposed on the individual summands and the weak dependence structure is expressed in terms of some uniformly mixing coefficients.

math.PR↗

arXiv2020

Duality for compact group actions on operator algebras and applications: irreducible inclusions and Galois correspondence

We consider compact group actions on C*- and W*- algebras. We prove results that relate the duality property of the action (as defined in the Introduction) with other relevant properties of the system such as the relative commutant of the fixed point algebras being trivial (called the irreducibility of the inclusion) and also to the Galois correspondence between invariant C*-subalgebras containing the fixed point algebra and the class of closed normal subgroups of the compact group.

math.OA↗

arXiv2019

Invariant subspaces of generalized Hardy algebras associated with compact abelian group actions on W*-algebras

We consider an action of a compact group whose dual is archimedean linearly ordered or a direct product (or sum) of such groups on a von Neumann algebra, M. We define the generalized Hardy subspace of the Hilbert space of a standard representation the algebra, and the Hardy subalgebra of analytic elements of M with respect to the action. We find conditions in order that the Hardy algebra is a hereditarily reflexive algebra of operators. In particular if every non zero spectral subspace, contains a unitary operator, the condition is satisfied and therefore the Hardy algebra is hereditarily reflexive. This is the case if the action is the dual action on a crossed product, or an ergodic action, or, if, in some situations, the fixed point algebra is a factor.

math.OA↗

arXiv2018

Compact Group Actions On Operator Algebras and Their Spectra

We consider a class of dynamical systems with compact non abelian groups that include C*-, W*- and multiplier dynamical systems. We prove results that relate the algebraic properties such as simplicity or primeness of the fixed point algebras as defned in Section 3., to the spectral properties of the action, including the Connes and strong Connes spectra.

math.OA↗

arXiv2018

A solution of the maximality problem for one-parameter dynamical systems

We prove a maximality theorem for one-parameter dynamical systems including multiplier one-parameter dynamical systems. Our main result is new even for one-parameter actions on commutative multiplier algebras including the algebra of bounded continuous functions on the set of real numbers acted upon by translations. The methods we develop and use in our characterization of maximality include harmonic analysis, topological vector spaces and operator algebra techniques.

math.FA↗

arXiv2016

The limiting spectral distribution in terms of spectral density

For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists and we describe it via an equation satisfied by its Stieltjes transform. No rate of convergence to zero of correlations is imposed, therefore the process is allowed to have long memory. In particular, if the symmetrized matrices are constructed from stationary Gaussian random fields which have spectral density, the result of this paper gives a complete solution to the limiting eigenvalue distribution. More generally, for matrices whose entries are functions of independent identically distributed random variables the result also holds.

math.PR↗

arXiv2016

On the functional CLT for stationary Markov Chains started at a point

We present a general functional central limit theorem started at a point also known under the name of quenched. As a consequence, we point out several new classes of stationary processes, defined via projection conditions, which satisfy this type of asymptotic result. One of the theorems shows that if a Markov chain is stationary ergodic and reversible, this result holds for bounded additive functionals of the chain which have a martingale coboundary in L_1 representation. Our results are also well adapted for strongly mixing sequences providing for this case an alternative, shorter approach to some recent results in the literature.

math.PR↗

arXiv2014

Invariant subspaces of algebras of analytic elements associated with periodic flows on W*-algebras

We consider an action of the circle group, T on a von Neumann algebra, M. Similarly to the case when the algebra of essentially bounded functions on T is acted upon by translations, we define the generalized Hardy subspace of H,where H is the Hilbert space of a standard representation of M and the subalgebra of analytic elements of M with respect to the action. We prove that the subalgebra of analytic elements is a reflexive algebra of operators if the Arveson spectrum is finite or, if the spectrum is infinite, the spectral subspace corresponding to the least positive element contains an unitary operator. We also prove that the analytic algebra is reflexive if M is an abelian W*-algebra. Examples include the algebra of analytic Toeplitz operators, crossed products, reduced semicrossed products and some reflexive nest subalgebras of von Neumann algebras.

math.OA↗

arXiv2014

On the universality of spectral limit for random matrices with martingale differences entries

For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral distribution behaves asymptotically similarly to a corresponding matrix with independent centered Gaussian entries having the same variances. Under a slightly reinforced condition, the approximation holds in the almost sure sense. We also point out several sufficient regularity conditions imposed to the variance structure for convergence to the semicircle law or the Marchenko-Pastur law and other convergence results. In the stationary case we obtain a full extension from the i.i.d. case to the martingale case of the convergence to the semicircle law as well as to the Marchenko-Pastur one. Our results are well adapted to study several examples including non linear ARCH infinite random fields.

math.PR↗

arXiv2013

Reflexive Operator Algebras on Banach Spaces

In this paper we study the reflexivity of a unital strongly closed algebra of operators with complemented invariant subspace lattice on a Banach space. We prove that if such an algebra contains a complete Boolean algebra of projections of finite uniform multiplicity and with the direct sum property, then it is reflexive, i.e. it contains every operator that leaves invariant every closed subspace in the invariant subspace lattice of the algebra. In particular, such algebras coincide with their bicommutant.

math.FA↗

arXiv2013

On Functional CLT for Reversible Markov Chains with nonlinear growth of the Variance

In this paper we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n; and establish that conditional convergence in distribution of partial sums implies functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion.

math.PR↗

arXiv2011

Maximal subalgebras of C*-algebras associated with periodic flows

We find necessary and sufficient conditions for the subalgebra of analytic elements associated with a periodic C*-dynamical system to be a maximal norm-closed subalgebra. Our conditions are in terms of the Arveson spectrum of the action. We also describe equivalent properties of the system in terms of the strong Connes spectrum and the simplicity of the crossed product.

math.OA↗

arXiv2011

Almost Sure Invariance Principles via Martingale Approximation

In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale with stationary differences. The results are exploited to further investigate the central limit theorem and its invariance principle started at a point, as well as the law of the iterated logarithm via almost sure approximation with a Brownian motion, improving the results available in the literature. The conditions are well suited for a variety of examples; they are easy to verify, for instance, for linear processes and functions of Bernoulli shifts.

math.PR↗

arXiv2011

A Galois correspondence for compact group actions on C*-algebras

In this paper, we prove a Galois correspondence for compact group actions on C*-algebras in the presence of a commuting minimal action. Namely, we show that there is a one to one correspondence between the C*-subalgebras that are globally invariant under the compact action and the commuting minimal action, that in addition contain the fixed point algebra of the compact action and the closed, normal subgroups of the compact group.

math.OA↗