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Yurii Savchuk

Publications and source records attributed to Yurii Savchuk.

8 recordsLinked to original sources
arXiv2014

Induced quadratic modules in $*$-algebras

Positivity in $\ast$-algebras can be defined either algebraically, by quadratic modules, or analytically, by $\ast$-representations. By the induction procedure for $\ast$-representations we can lift the analytical notion of positivity from a $\ast$-subalgebra to the entire $\ast$-algebra. The aim of this paper is to define and study the induction procedure for quadratic modules. The main question is when a given quadratic module on the $\ast$-algebra is induced from its intersection with the $\ast$-subalgebra. This question is very hard even for the smallest quadratic module (i.e. the set of all sums of hermitian squares) and will be answered only in very special cases.

math.AG↗
arXiv2012

Induced *-representations and $C^*$-envelopes of some quantum $*$-algebras

We consider three quantum algebras: the q-oscillator algebra, the Podles' sphere and the q-deformed enveloping algebra of $su(2).$ To each of these *-algebras we associate certain partial dynamical system and perform the "Mackey analysis" of *-representations developed in [SS]. As a result we get the description of "standard" irreducible *-representations. Further, for each of these examples we show the existence of a "$C^*$-envelope" which is canonically isomorphic to the covariance $C^*$-algebra of the partial dynamical system. Finally, for the q-oscillator algebra and the q-deformed $\cU(su(2))$ we show the existence of "bad" representations.

math.OA↗
arXiv2011

On structure of homogenenous Wick ideals in Wick $*$-algebras with braided coefficients

We study the structure of Wick homogenenous ideals of higher degrees in quadratic algebras allowing Wick ordering. We present a method how to construct a homogeneous Wick ideal $\mathcal{I}_{n+1}$ of degree $n+1$ out of a homogeneous Wick ideal $\mathcal{I}_n$ of degree $n$ so that $\mathcal{I}_{n+1}\subset\mathcal{I}_n$. We show that in some particular cases our procedure allows one to get a description of the largest homogeneous Wick ideals of higher degrees having generators of the largest quadratic Wick ideal only. Finally we study classes of $*$-representations of Wick version of CCR annihilating certain homogeneous Wick ideals of degree higher than $2$.

math.OA↗
arXiv2011

On a class of $C^*$-algebras determined uniquely by dual space

Enveloping $C^*$-algebras for some finitely generated $*$-algebras are considered. It is shown that all of the considered algebras are identically defined by their dual spaces. The description in terms of matrix-functions is given. Keywords : algebraic bundles, finitely represented $C^*$-algebras, dual spaces.

math.OA↗
arXiv2011

On $q$-normal operators and quantum complex plane

For $q>0$ let $\cA$ denote the unital $\ast$-algebra with generator $x$ and defining relation $xx^\ast=qxx^\ast$. Based on this algebra we study $q$-normal operators, the complex $q$-moment problem, positive elements and sums of squares.

math.OA↗
arXiv2010

Positivstellensätze for Algebras of Matrices

The paper is concerned with various types of noncommutative Positivstellensätze for the matrix algebra $M_n(\cA)$, where $\cA$ is an algebra of operators acting on a unitary space, a path algebra, a cyclic algebra or a formally real field. Some new types of Positivstellensätze are proposed and proved, it is shown by examples that they occur. There are a number of results stating that a type of Positivstellensatz is valid for $M_n(\cA)$ provided that it holds for $\cA$.

math.AG↗
arXiv2004

On $C^*$-algebras generated by some deformations of CAR relations

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 boundary conditions is given.

math.OA↗