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Register a new userVietoris topology on hyperspaces associated to a noncommutative compact space
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Maysam Maysami Sadr
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We study some topological spaces that can be considered as hyperspaces associated to noncommutative spaces. More precisely, for a NC compact space associated to a unital C*-algebra, we consider the set of closed projections of the second dual of the C*-algebra as the hyperspace of closed subsets of the NC space. We endow this hyperspace with an analog of Vietoris topology. In the case that the NC space has a quantum metric space structure in the sense of Rieffel we study the analogs of Hausdorff and infimum distances on the hyperspace. We also formulate some interesting problems about distances between sub-circles of a quantum torus.
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For every tuple $d_1,dots, d_lgeq 2,$ let $mathbb{R}^{d_1}otimescdotsotimesmathbb{R}^{d_l}$ denote the tensor product of $mathbb{R}^{d_i},$ $i=1,dots,l.$ Let us denote by $mathcal{B}(d)$ the hyperspace of centrally symmetric convex bodies in $mathbb{R}^d,$ $d=d_1cdots d_l,$ endowed with the Hausdorff distance, and by $mathcal{B}_otimes(d_1,dots,d_l)$ the subset of $mathcal{B}(d)$ consisting of the convex bodies that are closed unit balls of reasonable crossnorms on $mathbb{R}^{d_1}otimescdotsotimesmathbb{R}^{d_l}.$ It is known that $mathcal{B}_otimes(d_1,dots,d_l)$ is a closed, contractible and locally compact subset of $mathcal{B}(d).$ The hyperspace $mathcal{B}_otimes(d_1,dots,d_l)$ is called the space of tensorial bodies. In this work we determine the homeomorphism type of $mathcal{B}_otimes(d_1,dots,d_l).$ We show that even if $mathcal{B}_otimes(d_1,dots,d_l)$ is not convex with respect to the Minkowski sum, it is an Absolute Retract homeomorphic to $mathcal{Q}timesmathbb{R}^p,$ where $mathcal{Q}$ is the Hilbert cube and $p=frac{d_1(d_1+1)+cdots+d_l(d_l+1)}{2}.$ Among other results, the relation between the Banach-Mazur compactum and the Banach-Mazur type compactum associated to $mathcal{B}_otimes(d_1,dots,d_l)$ is examined.
We consider the Type 1 and Type 2 noncommutative Borsuk-Ulam conjectures of Baum, D$k{a}$browski, and Hajac: there are no equivariant morphisms $A to A circledast_delta H$ or $H to A circledast_delta H$, respectively, when $H$ is a nontrivial compact quantum group acting freely on a unital $C^*$-algebra $A$. Here $A circledast_delta H$ denotes the equivariant noncommutative join of $A$ and $H$; this join procedure is a modification of the topological join that allows a free action of $H$ on $A$ to produce a free action of $H$ on $A circledast_delta H$. For the classical case $H = mathcal{C}(G)$, $G$ a compact group, we present a reduction of the Type 1 conjecture and counterexamples to the Type 2 conjecture. We also present some examples and conditions under which the Type 2 conjecture does hold.
We extend a result about the gauge action on noncommutative solitons by showing that a family of functions can be gauged away to a Gaussian using the quantification condition given in On a gauge action on sigma model solitons IDAQP(2018).
We study the topic of quantum differentiability on quantum Euclidean $d$-dimensional spaces (otherwise known as Moyal $d$-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differential to have decay $O(n^{-alpha})$ for $0 < alpha leq frac{1}{d}$. This result is substantially more difficult than the analogous problems for Euclidean space and for quantum $d$-tori.
We introduce noncommutative weak Orlicz spaces associated with a weight and study their properties. We also define noncommutative weak Orlicz-Hardy spaces and characterize their dual spaces.
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