arXiv subjects
Publications and source records attributed to Yulia N. Kuznetsova.
Let $S$ be a subsemigroup of a second countable locally compact group $G$, such that $S^{-1}S=G$. We consider the $C^*$-algebra $C^*_δ(S)$ generated by the operators of translation by all elements of $S$ in $L^2(S)$. We show that this algebra admits a comultiplication which turns it into a compact quantum semigroup. The same is proved for the von Neumann algebra $VN(S)$ generated by $C^*_δ(S)$.
We define a category $\mathcal{QSI}$ of quantum semigroups with involution which carries a corepresentation-based duality map $M\mapsto \widehat M$. Objects in $\mathcal{QSI}$ are von Neumann algebras with comultiplication and coinvolution, we do not suppose the existence of a Haar weight or of a distinguished spatial realisation. In the case of a locally compact quantum group $\mathbb G$, the duality $\;\widehat{\ }\;$ in $\mathcal{QSI}$ recovers the universal duality of Kustermans: $\widehat{L^\infty(\mathbb G)} = C_0^u(\hat {\mathbb G})^{**}= \widehat{ C_0^u(\mathbb G)^{**}}$, and $\widehat{L^\infty(\hat{\mathbb G})} = C_0^u(\mathbb G)^{**} = \widehat{ C_0^u(\hat{\mathbb G})^{**}}$. Other various examples are given.
Let $G$ be a locally compact, compactly generated group of polynomial growth and let $ω$ be a weight on $G$. Under proper assumptions on the weight $ω$, the Banach space $L^p(G,ω)$ is a Banach \ast-algebra. In this paper we give examples of such weighted $L^p$-algebras and we study some of their harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, weak Wiener property, Wiener property, existence of minimal ideals of a given hull.
The paper deals with weighted spaces $L_p^w(G)$ on a locally compact group G. If w is a positive measurable function on G then we define the space $L_p^w(G)$, $p\ge1$, as $L_p^w(G)=\{f:fw\in L_p(G)\}$. We consider weights such that these weighted spaces are algebras with respect to usual convolution. It is shown that for p>1 such weights exists on any sigma-compact group. We prove also a criterion known earlier in special cases: $L_1^w(G)$ is an algebra if and only if w is submultiplicative. It is proved that invariant algebras $L_p^w(G)$, $p>1$, have approximate units of standard form, but this may not be true for a non-invariant algebra.