arXiv subjects
Publications and source records attributed to Mehrdad Kalantar.
For a locally compact quantum group $\G$ with tracial Haar weight $φ$, and a quantum measure $μ$ on $\G$, we study the space ${H}_μ^p$ of $μ$-harmonic operators in the non-commutative $L^p$-space ${L}^p(\G)$ associated to the Haar weight $φ$. The main result states that if $μ$ is non-degenerate, then ${H}_μ^p$ is trivial for all $1 \leq p < \infty$.
In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign, to each locally compact quantum group $\mathbb{G}$, a locally compact group $\tilde \mathbb{G}$ which is the quantum version of point-masses, and is an invariant for the latter. We show that "quantum point-masses" can be identified with several other locally compact groups that can be naturally assigned to the quantum group $\mathbb{G}$. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of $\mathbb{G}$ are encoded by $\tilde \mathbb{G}$: the latter, despite being a simpler object, can carry very important information about $\mathbb{G}$.
In this paper we study various convolution-type algebras associated with a locally compact quantum group from cohomological and geometrical points of view. The quantum group duality endows the space of trace class operators over a locally compact quantum group with two products which are operator versions of convolution and pointwise multiplication, respectively; we investigate the relation between these two products, and derive a formula linking them. Furthermore, we define some canonical module structures on these convolution algebras, and prove that certain topological properties of a quantum group, can be completely characterized in terms of cohomological properties of these modules. We also prove a quantum group version of a theorem of Hulanicki characterizing group amenability. Finally, we study the Radon--Nikodym property of the $L^1$-algebra of locally compact quantum groups. In particular, we obtain a criterion that distinguishes discreteness from the Radon--Nikodym property in this setting.
The Hidden Subgroup Problem is used in many quantum algorithms such as Simon's algorithm and Shor's factoring and discrete log algorithms. A polynomial time solution is known in case of abelian groups, and normal subgroups of arbitrary finite groups. The general case is still open. An efficient solution of the problem for symmetric group $S_n$ would give rise to an efficient quantum algorithm for Graph Isomorphism Problem. We formulate a hidden sub-hypergroup problem for finite hypergroups and solve it for finite commutative hypergroups. The given algorithm is efficient if the corresponding QFT could be calculated efficiently.