Periodic algebras generated by groups

We consider algebras with basis numerated by elements of a group $G.$ We fix a function $f$ from $Gtimes G$ to a ground field and give a multiplication of the algebra which depends on $f$. We study the basic properties of such algebras. In particular, we find a condition on $f$ under which the corresponding algebra is a Leibniz algebra. Moreover, for a given subgroup $hat G$ of $G$ we define a $hat G$-periodic algebra, which corresponds to a $hat G$-periodic function $f,$ we establish a criterion for the right nilpotency of a $hat G$-periodic algebra. In addition, for $G=mathbb Z$ we describe all $2mathbb Z$- and $3mathbb Z$-periodic algebras. Some properties of $nmathbb Z$-periodic algebras are obtained.

Singular perturbations with boundary conditions and the Casimir effect in the half space

We study the self adjoint extensions of a class of non maximal multiplication operators with boundary conditions. We show that these extensions correspond to singular rank one perturbations (in the sense of cite{AK}) of the Laplace operator, namely the formal Laplacian with a singular delta potential, on the half space. This construction is the appropriate setting to describe the Casimir effect related to a massless scalar field in the flat space time with an infinite conducting plate and in the presence of a point like impurity. We use the relative zeta determinant (as defined in cite{Mul} and cite{SZ}) in order to regularize the partition function of this model. We study the analytic extension of the associated relative zeta function, and we present explicit results for the partition function, and for the Casimir force.

$J$-self-adjoint operators with $mathcal{C}$-symmetries: extension theory approach

A well known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac-Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian ($J$-self-adjoint) Hamiltonians with complex point-interactions. We demonstrate that the resulting Hamiltonians are bijectively related with so called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices $<n,n>$. General properties of the $cC$ operators for these Hamiltonians are derived. A detailed study of $cC$-operator parametrizations and Krein type resolvent formulas is provided for $J$-self-adjoint extensions of symmetric operators with deficiency indices $<2,2>$. The technique is exemplified on 1D pseudo-Hermitian Schrodinger and Dirac Hamiltonians with complex point-interaction potentials.

Bounds on variation of spectral subspaces under J-self-adjoint perturbations

Let $A$ be a self-adjoint operator on a Hilbert space $fH$. Assume that the spectrum of $A$ consists of two disjoint components $sigma_0$ and $sigma_1$. Let $V$ be a bounded operator on $fH$, off-diagonal and $J$-self-adjoint with respect to the orthogonal decomposition $fH=fH_0oplusfH_1$ where $fH_0$ and $fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $sigma_0$ and $sigma_1$, respectively. We find (optimal) conditions on $V$ guaranteeing that the perturbed operator $L=A+V$ is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on variation of the spectral subspaces of $A$ under the perturbation $V$. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed.

$p$-Adic multiresolution analyses

We study $p$-adic multiresolution analyses (MRAs). A complete characterisation of test functions generating a MRA (scaling functions) is given. We prove that only 1-periodic test functions may be taken as orthogonal scaling functions and that all such scaling functions generate Haar MRA. We also suggest a method of constructing sets of wavelet functions and prove that any set of wavelet functions generates a $p$-adic wavelet frame.

Frames of p-adic wavelets and orbits of the affine group

The general construction of frames of p-adic wavelets is described. We consider the orbit of a mean zero generic locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss relation of this result to the multiresolution wavelet analysis.

Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Given a von Neumann algebra $M$ denote by $S(M)$ and $LS(M)$ respectively the algebras of all measurable and locally measurable operators affiliated with $M.$ For a faithful normal semi-finite trace $tau$ on $M$ let $S(M, tau)$ (resp. $S_0(M, tau)$) be the algebra of all $tau$-measurable (resp. $tau$-compact) operators from $S(M).$ We give a complete description of all derivations on the above algebras of operators in the case of type I von Neumann algebra $M.$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $LS(M)$ (resp. $S(M)$ and $S(M,tau)$) is inner, and each derivation on $S_0(M, tau)$ is spatial and implemented by an element from $S(M, tau).$

Complete description of derivations on $tau$-compact operators for type I von Neumann algebras

Given a type I von Neumann algebra $M$ with a faithful normal semi-finite trace $tau,$ let $S_0(M, tau)$ be the algebra of all $tau$-compact operators affiliated with $M.$ We give a complete description of all derivations on the algebra $S_0(M, tau).$ In particular, we prove that if $M$ is of type I$_infty$ then every derivation on $S_0(M, tau)$ is spatial.

On the classification of complex Leibniz superalgebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and nilindex $n+m$

In this work we investigate the complex Leibniz superalgebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m.$ We prove that such superalgebras with the condition $m_2 eq0$ have nilindex less than $n+m$. Therefore the complete classification of Leibniz algebras with characteristic sequence $(n-1, 1 | m_1, ..., m_k)$ and with nilindex equal to $n+m$ is reduced to the classification of filiform Leibniz superalgebras of nilindex equal to $n+m,$ which was provided in cite{AOKh} and cite{GKh}.

Cartan Subalgebras of Leibniz $n$-Algebras

The present paper is devoted to the investigation of properties of Cartan subalgebras and regular elements in Leibniz $n$-algebras. The relationship between Cartan subalgebras and regular elements of given Leibniz $n$-algebra and Cartan subalgebras and regular elements of the corresponding factor $n$-Lie algebra is established.