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Register a new userQuantum differentiability on noncommutative Euclidean spaces
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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.
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We provide a full characterisation of quantum differentiability (in the sense of Connes) on quantum tori. We also prove a quantum integration formula which differs substantially from the commutative case.
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.
For a closed cocompact subgroup $Gamma$ of a locally compact group $G$, given a compact abelian subgroup $K$ of $G$ and a homomorphism $rho:hat{K}to G$ satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations $C^*(hat{G}/Gamma, rho)$ of the homogeneous space $G/Gamma$, generalizing Rieffels construction of quantum Heisenberg manifolds. We show that when $G$ is a Lie group and $G/Gamma$ is connected, given any norm on the Lie algebra of $G$, the seminorm on $C^*(hat{G}/Gamma, rho)$ induced by the derivation map of the canonical $G$-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on $rho$ continuously, with respect to quantum Gromov-Hausdorff distances.
We consider the reduction of problems on general noncommutative $L_p$-spaces to the corresponding ones on those associated with finite von Neumann algebras. The main tool is a unpublished result of the first named author which approximates any noncommutative $L_p$-space by tracial ones. We show that under some natural conditions a map between two von Neumann algebras extends to their crossed products by a locally compact abelian group or to their noncommutative $L_p$-spaces. We present applications of these results to the theory of noncommutative martingale inequalities by reducing most recent general noncommutative martingale/ergodic inequalities to those in the tracial case.
Let $A$ be a finite subdiagonal algebra in Arvesons sense. Let $H^p(A)$ be the associated noncommutative Hardy spaces, $0<ple8$. We extend to the case of all positive indices most recent results about these spaces, which include notably the Riesz, Szego and inner-outer type factorizations. One new tool of the paper is the contractivity of the underlying conditional expectation on $H^p(A)$ for $p<1$.
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