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We study nonunital $C^*$-algebras such that for any element there exists a local unit and prove that in such algebras there are no frames. This fact was previously known only for commutative algebras. Among other results, we establish some…

Gelfand-Naimark duality (Commutative $C^*$-algebras $\equiv$ Locally compact Hausdorff spaces) is extended to $C^*$-algebras $\equiv$ Quotient maps on locally compact Hausdorff spaces. Using this duality, we give for an \emph{arbitrary}…

Let $L$ be a restricted Cartan type Lie algebra over an algebraically closed field $k$ of characteristic $p>3$, and let $G$ denote the automorphism group of $L$. We prove that there are no nontrivial invariants of $L^*$ under the coadjoint…

The topic of the paper is the study of germs of local holomorphisms $f$ between $C^n$ and $C^{n'}$ such that $f(M)\subset M'$ and $df(T^cM)=T^cM'$ for $M\subset C^n$ and $M'\subset C^{n'}$ generic real-analytic CR submanifolds of arbitrary…

We discuss the links between stationary discs, the defect of analytic discs, and 2-jet determination of CR automorphisms of generic nondegenerate real submanifolds of C^N of class C^4.

In this article, we study and review some aspects of twisted cohomologies on algebraic and analytic varieties. We compared such cohomologies in both the algebraic and analytic categories and defined two types of twisting parameters in the…

We construct a deformed $C_{\lambda}$-extended Heisenberg algebra in two-dimensional space using non-commuting coordinates which close an algebra depends on statistical parameter characterizing exotic particles. The obtained symmetry is…

In this paper we study subalgebras of complex finite dimensional evolution algebras. We obtain the classification of nilpotent evolution algebras whose any subalgebra is an evolution subalgebra with a basis which can be extended to a…

The C*-algebra of bounded operators on the separable infinite-dimensional Hilbert space cannot be mapped to a W*-algebra in such a way that each unital commutative C*-subalgebra C(X) factors normally through $\ell^\infty(X)$. Consequently,…

We study automorphism groups of formal matrix algebras. We also consider automorphisms of ordinary matrix algebras (in particular, triangular matrix algebras).

In this paper, we first introduce the non-abelian cohomology group of a Nijenhuis Lie algebra with values in another Nijenhuis Lie algebra and show that it parametrizes the isomorphism classes of all non-abelian extensions. In particular,…

We consider various notions of holomorphic extendability of complex valued functions defined on subsets of $\mathbf C^n$, including one-sided extendability. We show that in the relevant function spaces, these phenomena of holomorphic…

We give a simple example of a polynomial contraction automorphism of $\mathbb C^d$, $ d\ge 3 $, with unbounded degree growth. Combined with Poincar\'e-Dulac theorem it provides an algebraic automorphism of $\mathbb C^d$, $ d\ge 3 $, which…

We study the compactification of nonautonomous systems with autonomous limits and related dynamics. Although the $C^{1}$ extension of the compactification was well established, a great number of problems arising in bifurcation and stability…

Quasi-free endomorphisms of the CAR and CCR algebras are studied from the point of view of the theory of superselection sectors.

We study the existence of universal autohomeomorphisms of $\mathbb{N}^*$. We prove that $\mathsf{CH}$ implies there is such an autohomeomorphism and show that there are none in any model where all autohomeomorphisms of $\mathbb{N}^*$ are…

We study the class of overconvergent subanalytic subsets of a $k$-affinoid space $X$ when $k$ is a non-archimedean field. These are the images along the projection $X \times B^n \to X$ of subsets defined with inequalities between functions…

We prove that every weak-local derivation on a C$^*$-algebra is continuous, and the same conclusion remains valid for weak$^*$-local derivations on von Neumann algebras. We further show that weak-local derivations on C$^*$-algebras and…

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of…

Let X be a flexible variety of F be an isomorphism of closed one-dimensional subschemes of $X$. We develop criteria which guarantee that F extends to au automorphism of X.