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We consider the problem of extending the classical S-lemma from commutative case to noncommutative cases. We show that a symmetric quadratic homogeneous matrix-valued polynomial is positive semidefinite if and only if its coefficient matrix…
In this paper we define an equivalence relation on the set of all $x_{J}$ in order to form a basis for a new descent algebra of Weyl groups of type $A_{n}$. By means of this, we construct a new commutative and semi-simple descent algebra of…
This paper presents analogous results of Hua [7][8] on numbers of representations of quivers over finite fields which respect nilpotent relations under certain assumptions. A closed formula which counts isomorphism classes of absolutely…
We study a deformation of the Cuntz-Toeplitz $C^*$-algebra determined by the relations $a_i^*a_i=1+q a_ia_i^*, a_i^*a_j=0$. We define well-behaved unbounded *-representations of the *-algebra defined by relations above and classify all such…
Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero. Yet, very little is known over a field of positive characteristic. In this paper we prove some…
Nuclear $C^*$-algebras having a system of completely positive approximations formed with convex combinations of a uniformly bounded number of order zero summands are shown to be approximately finite dimensional.
We find a substantial class of pairs of $*$-homomorphisms between graph C*-algebras of the form $C^*(E)\hookrightarrow C^*(G)\twoheadleftarrow C^*(F)$ whose pullback C*-algebra is an AF graph C*-algebra. Our result can be interpreted as a…
In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal…
Metabelian algebras are introduced and it is shown that an algebra $A$ is metabelian if and only if $A$ is a nilpotent algebra having the index of nilpotency at most $3$, i.e. $x y z t = 0$, for all $x$, $y$, $z$, $t \in A$. We prove that…
The purpose of this note is to provide a family of explicit examples of $4$-dimensional operator systems contained in the Calkin algebra $\mathcal{Q}(\mathcal{H})$ on a separable infinite-dimensional Hilbert space $\mathcal{H}$ for which…
In this note we prove a selection of commutativity theorems for various classes of semigroups. For instance, if in a separative or completely regular semigroup $S$ we have $x^p y^p = y^p x^p$ and $x^q y^q = y^q x^q$ for all $x,y\in S$ where…
We prove that separable C*-algebras which are completely close in a natural uniform sense have isomorphic Cuntz semigroups, continuing a line of research developed by Kadison - Kastler, Christensen, and Khoshkam. This result has several…
We prove stability theorems in the Cuntz semigroup of a commutative C*-algebra which are analogues of classical stability theorems for topological vector bundles over compact Hausdorff spaces. Several applications to simple unital AH…
We classify all nonnilpotent, solvable Leibniz algebras with the property that all proper subalgebras are nilpotent. This generalizes the work of Stitzinger and Towers in Lie algebras. We show several examples which illustrate the…
Studies among other things, the question of whether a Lie algebra over Z/(p^k)Z lifts to one over Z/(p^(k+1))Z. An obstruction theory is developed and examples of Fp-Lie algebras which don't lift to Lie algebras over Z/p^2Z are discussed.…
Motivated by the classical theory of spin structures, we develop a theory for lifting free C$^*$-dynamical systems, a.k.a. noncommutative principal bundles, along central extensions. This theory extends the bundle-theoretic notion of spin…
In this paper we continue the analysis undertaken in a series of previous papers on structures arising as completions of C*-algebras under topologies coarser that their norm and we focus our attention on the so-called {\em locally convex…
We find a class of algebras A satisfying the following property: for every nontrivial noncommutative polynomial, the linear span of all its values in A equals A. This class includes the algebras of all bounded and all compact operators on…
In the context of commutative $C^*$-algebras we solve a problem related to a question of M. Rieffel by showing that the all-units rank and the norm-one rank coincide with the topological stable rank. We also introduce the notion of unitary…
We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. We utilize -- and reprove -- D. Anick's fundamental result on…