Related papers: Gr\"obner bases for *-algebars and faithful operat…
Motivated by deformation quantization we investigate the algebraic GNS construction of *-representations of deformed *-algebras over ordered rings and compute their classical limit. The question if a GNS representation can be deformed leads…
This article is devoted to the investigation of $B^*$-algebras, dual and annihilator ultranormed algebras. Their structure is studied in the paper. Extensions of algebras and fields are considered and using them core radicals and radicals…
We study the representation theory and enveloping $C^*$-algebras for Wick analogues of CAR and twisted CAR algebras. The realization of the $C^*$-algebras under consideration as algebras of continuous matrix-functions satisfying certain…
In this note, we extend modular techniques for computing Gr\"obner bases from the commutative setting to the vast class of noncommutative $G$-algebras. As in the commutative case, an effective verification test is only known to us in the…
We introduce the notion of the full quiver of a representation of an algebra, which is a cover of the (classical) quiver, but which captures properties of the representation itself. Gluing of vertices and of arrows enables one to study…
This paper deals with the notion of Gr\"obner $\delta$-base for some rings of linear differential operators by adapting the works of W. Trinks, A. Assi, M. Insa and F. Pauer. We compare this notion with the one of Gr\"obner base for such…
We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and…
Let $\mathcal A$ be a unital algebra equipped with an involution $(\cdot)^\dagger$, and suppose that the multiplicative set $\mathcal S\subseteq \mathcal A$ generated by the elements of the form $1 + a^\dagger a$ satisfies the Ore…
A $*$-bimodule for a unital $*$-algebra $A$ is an $A$-bimodule $X$ which is a vector space with involution $x\mapsto x^+$ satisfying $(a\cdot x\cdot b)^+=b^+\cdot x^+\cdot b^+$ for $x\in X$ and $a,b\in A$. An algebraic model for…
We give a sufficient condition for a $^*$-algebra with a specified basis to have an enveloping $C^*$-algebra. Particularizing to the setting of a Hecke algebra $\h(G,\gm)$, we show that under a suitable assumption not only we can assure…
We describe all irreducible conformal subalgebras of Cend_N. The classification of simple and semisimple associative conformal algebras with finite faithful representation follows from this description.
Strongly continuous one-parameter representations on C*-algebras and their extension to the multiplier algebra are investigated. We give also a proof of the Stone theorem on Hilbert C*-modules and look into some related problems.
We describe a new method to determine faithful representations of small dimension for a finite dimensional nilpotent Lie algebra. We give various applications of this method. In particular we find a new upper bound on the minimal dimension…
A strong link between information geometry and algebraic statistics is made by investigating statistical manifolds which are algebraic varieties. In particular it it shown how first and second order efficient estimators can be constructed,…
In this paper we study of *-representations for polynomial algebras on quantum matrix spaces. We deal with two special cases of the polynomial algebras, namely the algebra of polynomials on quantum complex matrices $\mathrm{Mat_2}$ and on…
We consider $C^*$-algebra $Isom_{q_{ij}}$ generated by $n$ isometries $a_1, \ldots, a_n$ satisfying the relations $a_i^* a_j = q_{ij} a_j a_i^*$ with $\max |q_{ij}| < 1$. This $C^*$-algebra is shown to be nuclear. We prove that the Fock…
In this paper we discuss some physical applications of topological *-algebras of unbounded operators. Our first example is a simple system of free bosons. Then we analyze different models which are related to this one. We also discuss the…
Let $\Lambda$ be a commutative Noetherian ring, and let $I$ be a proper ideal of $\Lambda$, $R=\Lambda /I$. Consider the polynomial rings $T=\Lambda [x_1,...x_n]$ and $A=R[x_1,...,x_n]$. Suppose that linear equations are solvable in…
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics.…
Suppose that H is a complex Hilbert space and that B(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C*-algebra. We do this by showing that if A is an abelian subalgebra of…