. It is proved that for any 0 < β < α , any bounded Ahlfors α -regular space contains a β -regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most O ( α/ ( α − β )). The bound on the distortion is asymptotically tight when β → α . The main tool used in the proof is a regular form of the ultrametric skeleton theorem.
{"title":"Dvoretzky-type theorem for Ahlfors regular spaces","authors":"M. Mendel","doi":"10.4064/sm210629-2-2","DOIUrl":"https://doi.org/10.4064/sm210629-2-2","url":null,"abstract":". It is proved that for any 0 < β < α , any bounded Ahlfors α -regular space contains a β -regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most O ( α/ ( α − β )). The bound on the distortion is asymptotically tight when β → α . The main tool used in the proof is a regular form of the ultrametric skeleton theorem.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47758939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this article, we obtain a version of the noncommutative Banach Principle suitable to prove Wiener-Wintner type results for weights in W 1 space. This is used to obtain noncommutative Wiener-Wintner type ergodic theorems for various types of weights for certain types of positive Dunford-Schwartz operators. We also study the b.a.u. (a.u.) convergence of some subsequential averages and moving averages of such operators.
{"title":"Noncommutative Wiener–Wintner type ergodic theorems","authors":"Morgan O'Brien","doi":"10.4064/sm211209-26-8","DOIUrl":"https://doi.org/10.4064/sm211209-26-8","url":null,"abstract":". In this article, we obtain a version of the noncommutative Banach Principle suitable to prove Wiener-Wintner type results for weights in W 1 space. This is used to obtain noncommutative Wiener-Wintner type ergodic theorems for various types of weights for certain types of positive Dunford-Schwartz operators. We also study the b.a.u. (a.u.) convergence of some subsequential averages and moving averages of such operators.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45118761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
F. Albiac, J. L. Ansorena, M. Berasategui, P. M. Berná, S. Lassalle
From the abstract perspective of Banach spaces, the theory of (nonlinear) greedy approximation using bases sprang from the seminal characterization of greedy bases by Konyagin and Temlyakov in 1999 as those bases that are simultaneously unconditional and democratic [16]. These two properties are, a priori, independent of each other and we find examples of unconditional bases which are not democratic and the other way around already in the very early stages of the theory (see, e.g., [7, Example 10.4.4]). However, the geometry of some spaces X can make these properties intertwine, to the extent that the unconditional semi-normalized bases in X end up being democratic (hence greedy). This is the case of unconditional bases in Hilbert spaces, and also in the spaces l1 and c0 for instance (see [12, Theorem 4.1], [21, Theorem 3] and [10, Corollary 8.6]).
{"title":"Weak forms of unconditionality of\u0000bases in greedy approximation","authors":"F. Albiac, J. L. Ansorena, M. Berasategui, P. M. Berná, S. Lassalle","doi":"10.4064/sm210601-2-2","DOIUrl":"https://doi.org/10.4064/sm210601-2-2","url":null,"abstract":"From the abstract perspective of Banach spaces, the theory of (nonlinear) greedy approximation using bases sprang from the seminal characterization of greedy bases by Konyagin and Temlyakov in 1999 as those bases that are simultaneously unconditional and democratic [16]. These two properties are, a priori, independent of each other and we find examples of unconditional bases which are not democratic and the other way around already in the very early stages of the theory (see, e.g., [7, Example 10.4.4]). However, the geometry of some spaces X can make these properties intertwine, to the extent that the unconditional semi-normalized bases in X end up being democratic (hence greedy). This is the case of unconditional bases in Hilbert spaces, and also in the spaces l1 and c0 for instance (see [12, Theorem 4.1], [21, Theorem 3] and [10, Corollary 8.6]).","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42058869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A commuting tuple (T1, . . . , Tm) of operators, also called a multioperator, on a Hilbert space H is called essentially normal if all of the commutators [Tj , T ∗ k ], j, k = 1, . . . ,m are compact. Alternatively, essential normality can be attributed to the Hilbert C[z1, . . . , zm]module generated by (T1, . . . , Tm), namely, H equipped with the module action P (z1, . . . , zm)· f , P ∈ C[z1, . . . , zm], f ∈ H given by P (T1, . . . , Tm)f . Brown, Douglas and Fillmore [12, 13, 19] classified essentially normal multioperators up to unitary equivalence. The complete classifier here is the odd K-homology functor K1 from the category of compact metrizable spaces to the category of abelian groups. More precisely, for any compact subspace X ⊆ Cm, the abelian group K1(X) classifies essentially normal multioperators with essential Taylor spectrum X up to unitary equivalence; the elements of K1(X) are equivalence classes of C*monomorphisms from C(X) to the algebra of bounded operators on H modulo the ideal of
{"title":"Essential normality of Bergman modules\u0000over intersections of complex ellipsoids","authors":"M. Jabbari","doi":"10.4064/sm211201-11-3","DOIUrl":"https://doi.org/10.4064/sm211201-11-3","url":null,"abstract":"A commuting tuple (T1, . . . , Tm) of operators, also called a multioperator, on a Hilbert space H is called essentially normal if all of the commutators [Tj , T ∗ k ], j, k = 1, . . . ,m are compact. Alternatively, essential normality can be attributed to the Hilbert C[z1, . . . , zm]module generated by (T1, . . . , Tm), namely, H equipped with the module action P (z1, . . . , zm)· f , P ∈ C[z1, . . . , zm], f ∈ H given by P (T1, . . . , Tm)f . Brown, Douglas and Fillmore [12, 13, 19] classified essentially normal multioperators up to unitary equivalence. The complete classifier here is the odd K-homology functor K1 from the category of compact metrizable spaces to the category of abelian groups. More precisely, for any compact subspace X ⊆ Cm, the abelian group K1(X) classifies essentially normal multioperators with essential Taylor spectrum X up to unitary equivalence; the elements of K1(X) are equivalence classes of C*monomorphisms from C(X) to the algebra of bounded operators on H modulo the ideal of","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47867569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a commutative unital ring $R$, and $nin mathbb{N}$, let $textrm{SL}_n(R)$ denote the special linear group over $R$, and $textrm{E}_n(R)$ the subgroup of elementary matrices. Let ${mathcal{M}}^+$ be the Banach algebra of all complex Borel measures on $[0,+infty)$ with the norm given by the total variation, the usual operations of addition and scalar multiplication, and with convolution. It is shown that $textrm{SL}_n(A)=textrm{E}_n(A)$ for Banach subalgebras $A$ of ${mathcal{M}}^+$ that are closed under the operation ${mathcal{M}}^+owns mu mapsto mu_t$, $tin [0,1]$, where $mu_t(E):=int_E (1-t)^x dmu(x)$ for $tin [0,1)$, and Borel subsets $E$ of $[0,+infty)$, and $mu_1:=mu({0})delta$, where $deltain {mathcal{M}}^+$ is the Dirac measure. Many illustrative examples of such Banach algebras $A$ are given. An example of a Banach subalgebra $Asubset {mathcal{M}}^+$, that does not possess the closure property above, but for which $textrm{SL}_n(A)=textrm{E}_n(A)$ neverthess holds, is also given.
{"title":"Generation of the special linear group by elementary matrices in some measure Banach algebras","authors":"A. Sasane","doi":"10.4064/sm210825-24-2","DOIUrl":"https://doi.org/10.4064/sm210825-24-2","url":null,"abstract":"For a commutative unital ring $R$, and $nin mathbb{N}$, let $textrm{SL}_n(R)$ denote the special linear group over $R$, and $textrm{E}_n(R)$ the subgroup of elementary matrices. Let ${mathcal{M}}^+$ be the Banach algebra of all complex Borel measures on $[0,+infty)$ with the norm given by the total variation, the usual operations of addition and scalar multiplication, and with convolution. It is shown that $textrm{SL}_n(A)=textrm{E}_n(A)$ for Banach subalgebras $A$ of ${mathcal{M}}^+$ that are closed under the operation ${mathcal{M}}^+owns mu mapsto mu_t$, $tin [0,1]$, where $mu_t(E):=int_E (1-t)^x dmu(x)$ for $tin [0,1)$, and Borel subsets $E$ of $[0,+infty)$, and $mu_1:=mu({0})delta$, where $deltain {mathcal{M}}^+$ is the Dirac measure. Many illustrative examples of such Banach algebras $A$ are given. An example of a Banach subalgebra $Asubset {mathcal{M}}^+$, that does not possess the closure property above, but for which $textrm{SL}_n(A)=textrm{E}_n(A)$ neverthess holds, is also given.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41834629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We establish a linear variational principle extending Deville–Godefroy– Zizler’s one. We use this variational principle to prove that if X is a Banach space having property ( α ) of Schachermayer and Y is any Banach space, then the set of all strongly norm attaining linear operators from X into Y is the complement of a σ -porous set. Moreover, we apply our results to an abstract class of (linear and nonlinear) operator spaces.
{"title":"Norm attaining operators and variational principle","authors":"M. Bachir","doi":"10.4064/sm210628-6-9","DOIUrl":"https://doi.org/10.4064/sm210628-6-9","url":null,"abstract":". We establish a linear variational principle extending Deville–Godefroy– Zizler’s one. We use this variational principle to prove that if X is a Banach space having property ( α ) of Schachermayer and Y is any Banach space, then the set of all strongly norm attaining linear operators from X into Y is the complement of a σ -porous set. Moreover, we apply our results to an abstract class of (linear and nonlinear) operator spaces.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42282445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the $H^3$ invariant of a group homomorphism $phi:G rightarrow mathrm{Out}(A)$, where $A$ is a classifiable C$^*$-algebra. We show the existence of an obstruction to possible $H^3$ invariants arising from considering the unitary algebraic $K_1$ group. In particular, we prove that when $A$ is the Jiang--Su algebra $mathcal{Z}$ this invariant must vanish. We deduce that the unitary fusion categories $mathrm{Hilb}(G, omega)$ for non-trivial $omega in H^3(G, mathbb{T})$ cannot act on $mathcal{Z}$.
{"title":"Anomalous symmetries of classifiable C*-algebras","authors":"Samuel Evington, Sergio Gir'on Pacheco","doi":"10.4064/sm220117-25-6","DOIUrl":"https://doi.org/10.4064/sm220117-25-6","url":null,"abstract":"We study the $H^3$ invariant of a group homomorphism $phi:G rightarrow mathrm{Out}(A)$, where $A$ is a classifiable C$^*$-algebra. We show the existence of an obstruction to possible $H^3$ invariants arising from considering the unitary algebraic $K_1$ group. In particular, we prove that when $A$ is the Jiang--Su algebra $mathcal{Z}$ this invariant must vanish. We deduce that the unitary fusion categories $mathrm{Hilb}(G, omega)$ for non-trivial $omega in H^3(G, mathbb{T})$ cannot act on $mathcal{Z}$.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41753975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the effect of a partial transpose on the limit ∗-distribution of a Haar distributed random unitary matrix. If we fix the number of blocks, b, we show that the partial transpose can be decomposed along diagonals into a sum of b matrices which are asymptotically free and identically distributed. We then consider the joint effect of different block decompositions and show that under some mild assumptions we also get asymptotic freeness.
{"title":"On the partial transpose of a Haar unitary matrix","authors":"J. Mingo, M. Popa, K. Szpojankowski","doi":"10.4064/sm210517-13-12","DOIUrl":"https://doi.org/10.4064/sm210517-13-12","url":null,"abstract":"We consider the effect of a partial transpose on the limit ∗-distribution of a Haar distributed random unitary matrix. If we fix the number of blocks, b, we show that the partial transpose can be decomposed along diagonals into a sum of b matrices which are asymptotically free and identically distributed. We then consider the joint effect of different block decompositions and show that under some mild assumptions we also get asymptotic freeness.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45539190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the absolute extendability constant of a finite metric space may be determined by computing relative projection constants of certain Lipschitz-free spaces. As an application, we show that $mbox{ae}(3)=4/3$ and $mbox{ae}(4)geq (5+4sqrt{2})/7$. Moreover, we discuss how to compute relative projection constants by solving linear programming problems.
{"title":"Absolute Lipschitz extendability and linear projection constants","authors":"Giuliano Basso","doi":"10.4064/sm210708-21-9","DOIUrl":"https://doi.org/10.4064/sm210708-21-9","url":null,"abstract":"We prove that the absolute extendability constant of a finite metric space may be determined by computing relative projection constants of certain Lipschitz-free spaces. As an application, we show that $mbox{ae}(3)=4/3$ and $mbox{ae}(4)geq (5+4sqrt{2})/7$. Moreover, we discuss how to compute relative projection constants by solving linear programming problems.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43283566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a Z-random walk in different frameworks: probabilistic (when the r.f. is i.i.d. or a moving average of i.i.d. random variables) and algebraic (when the r.f. is generated by commuting automorphisms of a torus or by commuting hyperbolic flows on homogeneous spaces).
{"title":"On the quenched functional CLT in random sceneries","authors":"J. Conze","doi":"10.4064/sm210421-13-10","DOIUrl":"https://doi.org/10.4064/sm210421-13-10","url":null,"abstract":"We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a Z-random walk in different frameworks: probabilistic (when the r.f. is i.i.d. or a moving average of i.i.d. random variables) and algebraic (when the r.f. is generated by commuting automorphisms of a torus or by commuting hyperbolic flows on homogeneous spaces).","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44845455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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