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Thomas Vils Pedersen - One of the best experts on this subject based on the ideXlab platform.
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compactness and weak star continuity of derivations on weighted convolution Algebras
Journal of Mathematical Analysis and Applications, 2013Co-Authors: Thomas Vils PedersenAbstract:Abstract Let ω be a continuous weight on R + and let L 1 ( ω ) be the corresponding convolution Algebra. By results of Gronbaek and Bade & Dales the continuous derivations from L 1 ( ω ) to its dual space L ∞ ( 1 / ω ) are exactly the maps of the form ( D φ f ) ( t ) = ∫ 0 ∞ f ( s ) s t + s φ ( t + s ) d s ( t ∈ R + and f ∈ L 1 ( ω ) ) for some φ ∈ L ∞ ( 1 / ω ) . Also, every D φ has a unique extension to a continuous derivation D ¯ φ : M ( ω ) → L ∞ ( 1 / ω ) from the corresponding Measure Algebra. We show that a certain condition on φ implies that D ¯ φ is weak-star continuous. The condition holds for instance if φ ∈ L 0 ∞ ( 1 / ω ) . We also provide examples of functions φ for which D ¯ φ is not weak-star continuous. Similarly, we show that D φ and D ¯ φ are compact under certain conditions on φ . For instance this holds if φ ∈ C 0 ( 1 / ω ) with φ ( 0 ) = 0 . Finally, we give various examples of functions φ for which D φ and D ¯ φ are not compact.
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compactness and weak star continuity of derivations on weighted convolution Algebras
arXiv: Functional Analysis, 2011Co-Authors: Thomas Vils PedersenAbstract:Let $\omega$ be a continuous weight on $\mathbb R^+$ and let $L^1(\omega)$ be the corresponding convolution Algebra. By results of Gr{\o}nb{\ae}k and Bade & Dales the continuous derivations from $L^1(\omega)$ to its dual space $L^{\infty}(1/\omega)$ are exactly the maps of the form $$(D_{\phi}f)(t)=\int_0^{\infty}f(s)\,\frac{s}{t+s}\,\phi(t+s)\,ds\qquad\text{($t\in\mathbb R^+$ and $f\in L^1(\omega)$)}$$ for some $\phi\in L^{\infty}(1/\omega)$. Also, every $D_{\phi}$ has a unique extension to a continuous derivation $\bar{D}_{\phi}:M(\omega)\to L^{\infty}(1/\omega)$ from the corresponding Measure Algebra. We show that a certain condition on $\phi$ implies that $\bar{D}_{\phi}$ is weak-star continuous. The condition holds for instance if $\phi\in L_0^{\infty}(1/\omega)$. We also provide examples of functions $\phi$ for which $\bar{D}_{\phi}$ is not weak-star continuous. Similarly, we show that $D_{\phi}$ and $\bar{D}_{\phi}$ are compact under certain conditions on $\phi$. For instance this holds if $\phi\in C_0(1/\omega)$ with $\phi(0)=0$. Finally, we give various examples of functions $\phi$ for which $D_{\phi}$ and $\bar{D}_{\phi}$ are not compact.
Matthias Neufang - One of the best experts on this subject based on the ideXlab platform.
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COMPLETELY BOUNDED MULTIPLIERS OVER LOCALLY COMPACT QUANTUM GROUPS
2016Co-Authors: Matthias Neufang, Zhong-jin RuanAbstract:Abstract. In this paper, we consider several interesting multiplier Algebras associated with a locally compact quantum group G. Firstly, we study the completely bounded right multiplier AlgebraMrcb(L1(G)). We show that Mrcb(L1(G)) is a dual Banach Algebra with a natural operator predual Qr(L1(G)), and the completely isometric representation ofMrcb(L1(G)) on B(L2(G)), studied recently by Junge, Neufang and Ruan, is actually weak*-weak * continuous. Secondly, we study the left uniformly continuous space LUC(G) and its Banach Algebra dual LUC(G)∗. We prove that LUC(G) is a unital C*-subAlgebra of L∞(G) if the quantum group G is semi-regular. We show the connection between LUC(G) ∗ and the quan-tum Measure Algebra M(G), as well as their representations on L∞(G) and B(L2(G)). Finally, we study the right uniformly complete qotient space UCQr(L1(G)) and its Banach Algebra dual UCQr(L1(G))∗. For co-amenable quanum groups G, we obtain the weak*-homeomorphic completely isometric Algebra iso-morphism Mrcb(L1(G)) ∼=M(G) and the completely isometric isomorphism UCQr(L1(G)) ∼ = LUC(G). 1
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uniform equicontinuity multiplier topology and continuity of convolution
Archiv der Mathematik, 2015Co-Authors: Matthias Neufang, Jan Pachl, Pekka SalmiAbstract:We characterise bounded uniformly equicontinuous sets of functions on locally compact groups in terms of uniform factorisation. We apply this result to study the continuity of the convolution product on the dual LUC(G)* of the space of bounded left uniformly continuous functions with the topology of uniform convergence on bounded uniformly equicontinuous sets. When restricted to the space of finite Radon Measures on a locally compact group, this is the right multiplier topology. For any topological group, the convolution is jointly continuous on bounded sets in the Measure Algebra. It is jointly continuous on all of LUC(G)* when G is a locally compact SIN group.
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a representation theorem for locally compact quantum groups
International Journal of Mathematics, 2009Co-Authors: Marius Junge, Matthias Neufang, Zhong-jin RuanAbstract:Recently, Neufang, Ruan and Spronk proved a completely isometric representation theorem for the Measure Algebra M(G) and for the completely bounded (Herz–Schur) multiplier Algebra McbA(G) on $\mathcal{B}(L_{2}(G))$, where G is a locally compact group. We unify and generalize both results by extending the representation to arbitrary locally compact quantum groups 𝔾 = (M, Γ, φ, ψ). More precisely, we introduce the Algebra $M_{\rm cb}^{r} (L_1(\mathbb{G}))$ of completely bounded right multipliers on L1(𝔾) and we show that $M^r_{\rm cb} (L_1(\mathbb{G}))$ can be identified with the Algebra of normal completely bounded $\hat{M}$-bimodule maps on $\mathcal{B}(L_2(\mathbb{G}))$ which leave the subAlgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L1(𝔾) is in fact implemented by an element of $M_{\rm cb}^r (L_1(\mathbb{G}))$. We also show that our representation framework allows us to express quantum group "Pontryagin" duality purely as a commutatio...
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COMPLETELY ISOMETRIC REPRESENTATIONS OF McbA(G) AND UCB ( ˆ G) ∗
2008Co-Authors: Matthias Neufang, Zhong-jin Ruan, Nico SpronkAbstract:Abstract. Let G be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier Algebra McbA(G), which is dual to the representation of the Measure Algebra M(G), on B(L2(G)). The image Algebras of M(G) and McbA(G) in CB σ (B(L2(G))) are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group G, there is a natural completely isometric representation of UCB ( ˆ G) ∗ on B(L2(G)), which can be regarded as a duality result of Neufang’s completely isometric representation theorem for LUC(G) ∗. 1. introduction In this paper we assume that G is a locally compact group with a fixed left Haar Measure µG. We will simply write dµG(t) = dt if there is no confusion. Ghahramani showed in [15, Theorem 2] that if G contains at least two elements, the convolution Algebra L1(G) (and thus the Measure Algebra M(G)) can not be isometrically isomorphic to a subAlgebra of operators on any Hilbert space. Therefore, the representation of the Measure Algebra M(G) has to be considered on some other spaces different from Hilbert spaces. The first such representation result was studied by Wendel [46], in which he showed that M(G) i
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on the topological centre of the Algebra luc g for general topological groups
Journal of Functional Analysis, 2007Co-Authors: Stefano Ferri, Matthias NeufangAbstract:We consider the Banach Algebra LUC(G) ∗ for a not necessarily locally compact topological group G .O ur goal is to characterize the topological centre Zt (LUC(G) ∗ ) of LUC(G) ∗ . For locally compact groups G ,i t is well known that Zt (LUC(G) ∗ ) equals the Measure Algebra M(G). We shall prove that for every second countable (not precompact) group G ,w e haveZt (LUC(G) ∗ ) = M( G) ,w here G denotes the completion of G with respect to its right uniform structure (if G is precompact, then Zt (LUC(G) ∗ ) = LUC(G) ∗ , of course). In fact, this will follow from our more general result stating that for any separable (or any precompact) group G, we have Zt (LUC(G) ∗ ) = Leb(G) ,w here Leb(G) denotes the Algebra of uniform Measures. The latter result also partially answers a conjecture made by I. Csiszar 35 years ago [I. Csiszar, On the weak ∗ continuity of convolution in a convolution Algebra over an arbitrary topological group, Studia Sci. Math. Hungar. 6 (1971) 27–40]. We shall give similar results for the topological centre Λ(G LUC ) of the LUC-compactification G LUC of G. In particular, we shall prove that for any second countable (not precompact) group G admitting a group completion, we have Λ(G LUC ) = G (if G is precompact, then Λ(G LUC ) = G LUC ). Finally, we shall show that every linear (left) LUC(G) ∗ -module map on LUC(G) is automatically continuous whenever G is,
Riccardo Camerlo - One of the best experts on this subject based on the ideXlab platform.
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the descriptive set theory of the lebesgue density theorem
Advances in Mathematics, 2013Co-Authors: Alessandro Andretta, Riccardo CamerloAbstract:Abstract Given an equivalence class [ A ] in the Measure Algebra of the Cantor space, let Φ ˆ ( [ A ] ) be the set of points having density 1 in A . Sets of the form Φ ˆ ( [ A ] ) are called T -regular. We establish several results about T -regular sets. Among these, we show that T -regular sets can have any complexity within Π 3 0 ( = F σ δ ), that is for any Π 3 0 subset X of the Cantor space there is a T -regular set that has the same topological complexity of X . Nevertheless, the generic T -regular set is Π 3 0 -complete, meaning that the classes [ A ] such that Φ ˆ ( [ A ] ) is Π 3 0 -complete form a comeager subset of the Measure Algebra. We prove that this set is also dense in the sense of forcing, as T -regular sets with empty interior turn out to be Π 3 0 -complete. Finally we show that the generic [ A ] does not contain a Δ 2 0 set, i.e., a set which is in F σ ∩ G δ .
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the descriptive set theory of the lebesgue density theorem
arXiv: Logic, 2011Co-Authors: Alessandro Andretta, Riccardo CamerloAbstract:Given an equivalence class $[A]$ in the Measure Algebra of the Cantor space, let $\hat\Phi([A])$ be the set of points having density 1 in $A$. Sets of the form $\hat\Phi([A])$ are called $\mathcal{T}$-regular. We establish several results about $\mathcal{T}$-regular sets. Among these, we show that $\mathcal{T}$-regular sets can have any complexity within $\Pi^{0}_{3}$ (=$ \mathbf{F}_{\sigma\delta}$), that is for any $\Pi^{0}_{3}$ subset $X$ of the Cantor space there is a $\mathcal{T}$-regular set that has the same topological complexity of $X$. Nevertheless, the generic $\mathcal{T}$-regular set is $\Pi^{0}_{3}$-complete, meaning that the classes $[A]$ such that $\hat{\Phi}([A]) $ is $\Pi^{0}_{3}$-complete form a comeagre subset of the Measure Algebra. We prove that this set is also dense in the sense of forcing, as $\mathcal{T}$-regular sets with empty interior turn out to be $\Pi^{0}_{3}$-complete. Finally we show that the generic $[A]$ does not contain a $\Delta^{0}_{2}$ set, i.e., a set which is in $\mathbf{F}_\sigma\cap\mathbf{G}_\delta$
Roginskaya Maria - One of the best experts on this subject based on the ideXlab platform.
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The Spectral Radius Formula for Fourier-Stieltjes Algebras
'Canadian Mathematical Society', 2020Co-Authors: Ohrysko Przemyslaw, Roginskaya MariaAbstract:In this short note we first extend the validity of the spectral radius formula, obtained by M. Anoussis and G. Gatzouras, for Fourier-Stieltjes Algebras. The second part is devoted to showing that, for the Measure Algebra on any locally compact non-discrete Abelian group, there are no nontrivial constraints among three quantities: the norm, the spectral radius, and the supremum of the Fourier-Stieltjes transform, even if we restrict our attention to Measures with all convolution powers singular with respect to the Haar Measure
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The spectral radius formula for Fourier-Stieltjes Algebras
2019Co-Authors: Ohrysko Przemysław, Roginskaya MariaAbstract:In this short note we first extend the validity of the spectral radius formula obtained in \cite{ag} to Fourier--Stieltjes Algebras. The second part is devoted to showing that for the Measure Algebra on any locally compact non-discrete Abelian group there are no non-trivial constraints between three quantities: the norm, the spectral radius and the supremum of the Fourier--Stieltjes transform even if we restrict our attention to Measures with all convolution powers singular with respect to Haar Measure
Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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the independence of mathsf gch gch and a combinatorial principle related to banach mazur games
Archive for Mathematical Logic, 2021Co-Authors: Will Brian, Alan Dow, Saharon ShelahAbstract:It was proved recently that Telgarsky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of $$\mathsf {GCH}+\square $$ . The proof introduces a combinatorial principle that is shown to follow from $$\mathsf {GCH}+\square $$ , namely: We prove this principle is independent of $$\mathsf {GCH}$$ and $$\mathsf {CH}$$ , in the sense that $$\bigtriangledown $$ does not imply $$\mathsf {CH}$$ , and $$\mathsf {GCH}$$ does not imply $$\bigtriangledown $$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $$\bigtriangledown $$ holds with $${\mathbb {P}}$$ equal to the weight- $$\aleph _\omega $$ Measure Algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $$\mathsf {ZFC}+\mathsf {GCH}$$ .
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The independence of GCH and a combinatorial principle related to Banach-Mazur games
arXiv: Logic, 2020Co-Authors: Will Brian, Alan Dow, Saharon ShelahAbstract:It was proved recently that Telg\'arsky's conjecture, which concerns partial information strategies in the Banach-Mazur game, fails in models of $\mathsf{GCH}+\square$. The proof introduces a combinatorial principle that is shown to follow from $\mathsf{GCH}+\square$, namely: $\triangledown$: Every separative poset $\mathbb P$ with the $\kappa$-cc contains a dense sub-poset $\mathbb D$ such that $|\{ q \in \mathbb D \,:\, p \text{ extends } q \}| < \kappa$ for every $p \in \mathbb P$. We prove this principle is independent of $\mathsf{GCH}$ and $\mathsf{CH}$, in the sense that $\triangledown$ does not imply $\mathsf{CH}$, and $\mathsf{GCH}$ does not imply $\triangledown$ assuming the consistency of a huge cardinal. We also consider the more specific question of whether $\triangledown$ holds with $\mathbb P$ equal to the weight-$\aleph_\omega$ Measure Algebra. We prove, again assuming the consistency of a huge cardinal, that the answer to this question is independent of $\mathsf{ZFC}+\mathsf{GCH}$.