Quantitative bounds for random embeddings of Rk into Lorentz sequence spaces are given, with improved dependence on ε.
给出了Rk随机嵌入到Lorentz序列空间的定量界，改进了对ε的依赖。
{"title":"Random Euclidean embeddings in finite-dimensional Lorentz spaces","authors":"Daniel J. Fresen","doi":"10.4064/sm210612-26-8","DOIUrl":"https://doi.org/10.4064/sm210612-26-8","url":null,"abstract":"Quantitative bounds for random embeddings of Rk into Lorentz sequence spaces are given, with improved dependence on ε.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45251411","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 double iterated outer L spaces, namely the outer L spaces associated with three exponents and defined on sets endowed with a measure and two outer measures. We prove that in the case of finite sets, under certain conditions between the outer measures, the double iterated outer L spaces are isomorphic to Banach spaces uniformly in the cardinality of the set. We achieve this by showing the expected duality properties between them. We also provide counterexamples demonstrating that the uniformity does not hold in any arbitrary setting on finite sets, at least in a certain range of exponents. We prove the isomorphism to Banach spaces and the duality properties between the double iterated outer L spaces also in the upper half 3-space infinite setting described by Uraltsev, going beyond the case of finite sets.
{"title":"Duality for double iterated outer $L^p$ spaces","authors":"Marco Fraccaroli","doi":"10.4064/sm210423-19-6","DOIUrl":"https://doi.org/10.4064/sm210423-19-6","url":null,"abstract":"We study the double iterated outer L spaces, namely the outer L spaces associated with three exponents and defined on sets endowed with a measure and two outer measures. We prove that in the case of finite sets, under certain conditions between the outer measures, the double iterated outer L spaces are isomorphic to Banach spaces uniformly in the cardinality of the set. We achieve this by showing the expected duality properties between them. We also provide counterexamples demonstrating that the uniformity does not hold in any arbitrary setting on finite sets, at least in a certain range of exponents. We prove the isomorphism to Banach spaces and the duality properties between the double iterated outer L spaces also in the upper half 3-space infinite setting described by Uraltsev, going beyond the case of finite sets.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44793856","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}
Heshan Aravinda, Arnaud Marsiglietti, J. Melbourne
. We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for the intrinsic volumes of a convex body, which generalizes and improves a result of Lotz, McCoy, Nourdin, Peccati, and Tropp (2019).
{"title":"Concentration inequalities for ultra log-concave distributions","authors":"Heshan Aravinda, Arnaud Marsiglietti, J. Melbourne","doi":"10.4064/sm210605-2-10","DOIUrl":"https://doi.org/10.4064/sm210605-2-10","url":null,"abstract":". We establish concentration inequalities in the class of ultra log-concave distributions. In particular, we show that ultra log-concave distributions satisfy Poisson concentration bounds. As an application, we derive concentration bounds for the intrinsic volumes of a convex body, which generalizes and improves a result of Lotz, McCoy, Nourdin, Peccati, and Tropp (2019).","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45123568","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}
Consider a finite collection {T1, . . . , TJ} of differential operators with constant coefficients on Tn (n ≥ 2) and the space of smooth functions generated by this collection, namely, the space of functions f such that Tjf ∈ C(T n), 1 ≤ j ≤ J . We prove that if there are at least two linearly independent operators among their senior parts (relative to some mixed pattern of homogeneity), then this space does not have local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of C(S).
{"title":"Absence of local unconditional structure in spaces of smooth functions on the torus of arbitrary dimension","authors":"A. Tselishchev","doi":"10.4064/sm200629-21-12","DOIUrl":"https://doi.org/10.4064/sm200629-21-12","url":null,"abstract":"Consider a finite collection {T1, . . . , TJ} of differential operators with constant coefficients on Tn (n ≥ 2) and the space of smooth functions generated by this collection, namely, the space of functions f such that Tjf ∈ C(T n), 1 ≤ j ≤ J . We prove that if there are at least two linearly independent operators among their senior parts (relative to some mixed pattern of homogeneity), then this space does not have local unconditional structure. This fact generalizes the previously known result that such spaces are not isomorphic to a complemented subspace of C(S).","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41759045","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}
Let $Theta^{(n)}$ be a random vector uniformly distributed on the unit sphere $mathbb S^{n-1}$ in $mathbb R^n$. Consider the projection of the uniform distribution on the cube $[-1,1]^n$ to the line spanned by $Theta^{(n)}$. The projected distribution is the random probability measure $mu_{Theta^{(n)}}$ on $mathbb R$ given by [ mu_{Theta^{(n)}}(A) := frac 1 {2^n} int_{[-1,1]^n} mathbb 1{langle u, Theta^{(n)} rangle in A} du, ] for Borel subets $A$ of $mathbb{R}$. It is well known that, with probability $1$, the sequence of random probability measures $mu_{Theta^{(n)}}$ converges weakly to the centered Gaussian distribution with variance $1/3$. We prove a large deviation principle for the sequence $mu_{Theta^{(n)}}$ on the space of probability measures on $mathbb R$ with speed $n$. The (good) rate function is explicitly given by $I(nu(alpha)) := - frac{1}{2} log ( 1 - |alpha|_2^2)$ whenever $nu(alpha)$ is the law of a random variable of the form begin{align*} sqrt{1 - |alpha|_2^2 } frac{Z}{sqrt 3} + sum_{ k = 1}^infty alpha_k U_k, end{align*} where $Z$ is standard Gaussian independent of $U_1,U_2,ldots$ which are i.i.d. $text{Unif}[-1,1]$, and $alpha_1 geq alpha_2 geq ldots $ is a non-increasing sequence of non-negative reals with $|alpha|_2<1$. We obtain a similar result for random projections of the uniform distribution on the discrete cube ${-1,+1}^n$.
{"title":"Projections of the uniform distribution on the cube: a large deviation perspective","authors":"S. Johnston, Z. Kabluchko, J. Prochno","doi":"10.4064/sm210413-16-9","DOIUrl":"https://doi.org/10.4064/sm210413-16-9","url":null,"abstract":"Let $Theta^{(n)}$ be a random vector uniformly distributed on the unit sphere $mathbb S^{n-1}$ in $mathbb R^n$. Consider the projection of the uniform distribution on the cube $[-1,1]^n$ to the line spanned by $Theta^{(n)}$. The projected distribution is the random probability measure $mu_{Theta^{(n)}}$ on $mathbb R$ given by [ mu_{Theta^{(n)}}(A) := frac 1 {2^n} int_{[-1,1]^n} mathbb 1{langle u, Theta^{(n)} rangle in A} du, ] for Borel subets $A$ of $mathbb{R}$. It is well known that, with probability $1$, the sequence of random probability measures $mu_{Theta^{(n)}}$ converges weakly to the centered Gaussian distribution with variance $1/3$. We prove a large deviation principle for the sequence $mu_{Theta^{(n)}}$ on the space of probability measures on $mathbb R$ with speed $n$. The (good) rate function is explicitly given by $I(nu(alpha)) := - frac{1}{2} log ( 1 - |alpha|_2^2)$ whenever $nu(alpha)$ is the law of a random variable of the form begin{align*} sqrt{1 - |alpha|_2^2 } frac{Z}{sqrt 3} + sum_{ k = 1}^infty alpha_k U_k, end{align*} where $Z$ is standard Gaussian independent of $U_1,U_2,ldots$ which are i.i.d. $text{Unif}[-1,1]$, and $alpha_1 geq alpha_2 geq ldots $ is a non-increasing sequence of non-negative reals with $|alpha|_2<1$. We obtain a similar result for random projections of the uniform distribution on the discrete cube ${-1,+1}^n$.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46000851","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 a previous work, “compact versions” of Rubio de Francia’s weighted extrapolation theorem were proved, which allow one to extrapolate the compactness of an linear operator from just one space to the full range of weighted Lebesgue spaces, where this operator is bounded. In this paper, we extend these results to the setting of weighted Morrey spaces. As applications, we easily obtain new results on the weighted compactness of commutators of Calderón–Zygmund singular integrals, rough singular integrals and Bochner–Riesz multipliers.
在之前的工作中，我们证明了Rubio de Francia的加权外推定理的“紧化版本”，它允许我们将线性算子的紧性从一个空间外推到该算子有界的加权Lebesgue空间的整个范围。在本文中，我们将这些结果推广到加权Morrey空间的设置。作为应用，我们很容易得到Calderón-Zygmund奇异积分、粗糙奇异积分和Bochner-Riesz乘子对易子的加权紧性的新结果。
{"title":"Extrapolation of compactness on weighted Morrey spaces","authors":"S. Lappas","doi":"10.4064/sm210607-20-9","DOIUrl":"https://doi.org/10.4064/sm210607-20-9","url":null,"abstract":"In a previous work, “compact versions” of Rubio de Francia’s weighted extrapolation theorem were proved, which allow one to extrapolate the compactness of an linear operator from just one space to the full range of weighted Lebesgue spaces, where this operator is bounded. In this paper, we extend these results to the setting of weighted Morrey spaces. As applications, we easily obtain new results on the weighted compactness of commutators of Calderón–Zygmund singular integrals, rough singular integrals and Bochner–Riesz multipliers.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45471431","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}
The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a 2-isometry is subnormal. In this paper, we address this problem for cyclic 2-isometries. In view of some recent developments in operator theory on function spaces (see [4, 22]), one may recast CDSP as the problem of subnormality of the Cauchy dual M ′ z of the multiplication operator Mz acting on a de BrangesRovnyak space H(B), where B is a vector-valued rational function. The main result of this paper characterizes the subnormality of M ′ z on H(B) provided B is a vector-valued rational function with simple poles. As an application, we provide affirmative solution to CDSP for the Dirichlettype spaces D(μ) associated with measures μ supported on two antipodal points of the unit circle. 1. Cauchy dual subnormality problem for 2-isometries The Cauchy dual subnormality problem (for short, CDSP) for 2-isometries can be seen as the manifestation of the rich interplay between positive definite and negative definite functions on abelian semigroups. Indeed, CDSP can be considered as the non-commutative variant of the fact from the harmonic analysis on semigroups that the reciprocal of a Bernstein function f : [0,∞) → (0,∞) is completely monotone (see [31, Theorem 3.6]). This fact turns out to be somewhat equivalent to the solution of CDSP for completely hyperexpansive weighted shifts (see [7, Proposition 6] for a generalization). Another early result towards the solution of CDSP asserts that the Cauchy dual of any concave operator is a hyponormal contraction (see [33, Equation (26)]). Later this fact was generalized in [13, Theorem 3.1] by deducing power hyponormality of the Cauchy dual of any concave operator. Around the same time CDSP was settled affirmatively for ∆T -regular 2-isometries in [8, Theorem 3.4] and for 2-isometric operator-valued weighted shifts in [5, Theorems 2.5 and 3.3] (see also [14, Corollary 6.2] for the solution for yet another subclass of 2-isometries). Further, it was shown in [5, Examples 6.6 and 7.10] that there exist 2-isometric weighted shifts on directed trees (that include adjacency operators) whose Cauchy dual is not necessarily subnormal. Recently, a class of cyclic 2-isometric composition operators without subnormal Cauchy dual has been exhibited in [6, Theorem 4.4]. 2000 Mathematics Subject Classification. Primary 47B32, 47B38; Secondary 44A60, 31C25.
{"title":"The Cauchy dual subnormality problem via de Branges–Rovnyak spaces","authors":"S. Chavan, S. Ghara, M. R. Reza","doi":"10.4064/sm210419-9-12","DOIUrl":"https://doi.org/10.4064/sm210419-9-12","url":null,"abstract":"The Cauchy dual subnormality problem (for short, CDSP) asks whether the Cauchy dual of a 2-isometry is subnormal. In this paper, we address this problem for cyclic 2-isometries. In view of some recent developments in operator theory on function spaces (see [4, 22]), one may recast CDSP as the problem of subnormality of the Cauchy dual M ′ z of the multiplication operator Mz acting on a de BrangesRovnyak space H(B), where B is a vector-valued rational function. The main result of this paper characterizes the subnormality of M ′ z on H(B) provided B is a vector-valued rational function with simple poles. As an application, we provide affirmative solution to CDSP for the Dirichlettype spaces D(μ) associated with measures μ supported on two antipodal points of the unit circle. 1. Cauchy dual subnormality problem for 2-isometries The Cauchy dual subnormality problem (for short, CDSP) for 2-isometries can be seen as the manifestation of the rich interplay between positive definite and negative definite functions on abelian semigroups. Indeed, CDSP can be considered as the non-commutative variant of the fact from the harmonic analysis on semigroups that the reciprocal of a Bernstein function f : [0,∞) → (0,∞) is completely monotone (see [31, Theorem 3.6]). This fact turns out to be somewhat equivalent to the solution of CDSP for completely hyperexpansive weighted shifts (see [7, Proposition 6] for a generalization). Another early result towards the solution of CDSP asserts that the Cauchy dual of any concave operator is a hyponormal contraction (see [33, Equation (26)]). Later this fact was generalized in [13, Theorem 3.1] by deducing power hyponormality of the Cauchy dual of any concave operator. Around the same time CDSP was settled affirmatively for ∆T -regular 2-isometries in [8, Theorem 3.4] and for 2-isometric operator-valued weighted shifts in [5, Theorems 2.5 and 3.3] (see also [14, Corollary 6.2] for the solution for yet another subclass of 2-isometries). Further, it was shown in [5, Examples 6.6 and 7.10] that there exist 2-isometric weighted shifts on directed trees (that include adjacency operators) whose Cauchy dual is not necessarily subnormal. Recently, a class of cyclic 2-isometric composition operators without subnormal Cauchy dual has been exhibited in [6, Theorem 4.4]. 2000 Mathematics Subject Classification. Primary 47B32, 47B38; Secondary 44A60, 31C25.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43139767","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}
It is shown that a locally compact second countable group G has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free G-action T = (Tg)g∈G on an infinite σ-finite standard measure space (X, μ) admitting a T -Følner sequence (i.e. a sequence (An)∞n=1 of measured subsets of finite measure such that A1 ⊂ A2 ⊂ · · · , ⋃ ∞ n=1 An = X and limn→∞ supg∈K μ(TgAn△An) μ(An) = 0 for each compact K ⊂ G). A pair of groups H ⊂ G has property (T) if and only if there is a μ-preserving G-action S on X admitting an S-Følner sequence and such that S ↾ H is weakly mixing. These refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint. 0. Introduction Throughout this paper G is a non-compact locally compact second countable group. It has the Haagerup property if there is a weakly continuous unitary representation V of G in a separable Hilbert space H such that limg→∞ V (g) = 0 in the weak operator topology and (∗) for each ǫ > 0 and every compact subset K ⊂ G, there is a unit vector ξ ∈ H such that supg∈G ‖V (g)ξ − ξ‖ < ǫ. Of course, the amenable groups have the Haagerup property. The class of discrete countable Haagerup groups contains the free groups and is closed under free products and wreath products [CoStVa]. For more information about the Haagerup property we refer to [Ch–Va]. There is a purely dynamical description of this property: G is Haagerup if and only if there exists a mixing non-strongly ergodic probability preserving free G-action [Ch–Va, Theorem 2.2.2] (see §1 for the definitions). Recently, an infinite measure preserving counterpart of this result was discovered in [DeJoZu]: Theorem A. G has the Haagerup property if and only if there is a 0-type measure preserving G-action T = (Tg)g∈G on an infinite σ-finite measure space (X,B, μ) admitting a sequence of non-negative unit vectors (ξn) ∞ n=1 in L (X,μ) such that limn→∞ supg∈K〈ξn ◦ Tg, ξn〉 = 1 for each compact K ⊂ G. We recall that T is called of 0-type if limg→∞ μ(TgA ∩ B) = 0 for all subsets A,B ∈ B of finite measure. In this paper we provide a much shorter alternative proof of Theorem A which is grounded on the Moore-Hill concept of restricted infinite products of probability measures [Hi]. We note that the 0-type for infinite measure preserving systems is a natural counterpart of the mixing for probability preserving systems. However unlike mixing, Typeset by AMS-TEX 1 the 0-type is not a “strong” asymptotic property. It implies neither weak mixing nor ergodicity. Moreover, the totally dissipative actions are all of 0-type. In view of that the description in Theorem A does not look sharp from the ergodic theory point of view. Our first main result in this work is the following finer ergodic criterion of the Haagerup property. Theorem B. The following are equivalent. (i) G has the Haagerup property. (ii) There exists a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard meas
{"title":"Haagerup property and Kazhdan pairs\u0000via ergodic infinite measure preserving actions","authors":"A. I. Danilenko","doi":"10.4064/sm210702-27-10","DOIUrl":"https://doi.org/10.4064/sm210702-27-10","url":null,"abstract":"It is shown that a locally compact second countable group G has the Haagerup property if and only if there exists a sharply weak mixing 0-type measure preserving free G-action T = (Tg)g∈G on an infinite σ-finite standard measure space (X, μ) admitting a T -Følner sequence (i.e. a sequence (An)∞n=1 of measured subsets of finite measure such that A1 ⊂ A2 ⊂ · · · , ⋃ ∞ n=1 An = X and limn→∞ supg∈K μ(TgAn△An) μ(An) = 0 for each compact K ⊂ G). A pair of groups H ⊂ G has property (T) if and only if there is a μ-preserving G-action S on X admitting an S-Følner sequence and such that S ↾ H is weakly mixing. These refine some recent results by Delabie-Jolissaint-Zumbrunnen and Jolissaint. 0. Introduction Throughout this paper G is a non-compact locally compact second countable group. It has the Haagerup property if there is a weakly continuous unitary representation V of G in a separable Hilbert space H such that limg→∞ V (g) = 0 in the weak operator topology and (∗) for each ǫ > 0 and every compact subset K ⊂ G, there is a unit vector ξ ∈ H such that supg∈G ‖V (g)ξ − ξ‖ < ǫ. Of course, the amenable groups have the Haagerup property. The class of discrete countable Haagerup groups contains the free groups and is closed under free products and wreath products [CoStVa]. For more information about the Haagerup property we refer to [Ch–Va]. There is a purely dynamical description of this property: G is Haagerup if and only if there exists a mixing non-strongly ergodic probability preserving free G-action [Ch–Va, Theorem 2.2.2] (see §1 for the definitions). Recently, an infinite measure preserving counterpart of this result was discovered in [DeJoZu]: Theorem A. G has the Haagerup property if and only if there is a 0-type measure preserving G-action T = (Tg)g∈G on an infinite σ-finite measure space (X,B, μ) admitting a sequence of non-negative unit vectors (ξn) ∞ n=1 in L (X,μ) such that limn→∞ supg∈K〈ξn ◦ Tg, ξn〉 = 1 for each compact K ⊂ G. We recall that T is called of 0-type if limg→∞ μ(TgA ∩ B) = 0 for all subsets A,B ∈ B of finite measure. In this paper we provide a much shorter alternative proof of Theorem A which is grounded on the Moore-Hill concept of restricted infinite products of probability measures [Hi]. We note that the 0-type for infinite measure preserving systems is a natural counterpart of the mixing for probability preserving systems. However unlike mixing, Typeset by AMS-TEX 1 the 0-type is not a “strong” asymptotic property. It implies neither weak mixing nor ergodicity. Moreover, the totally dissipative actions are all of 0-type. In view of that the description in Theorem A does not look sharp from the ergodic theory point of view. Our first main result in this work is the following finer ergodic criterion of the Haagerup property. Theorem B. The following are equivalent. (i) G has the Haagerup property. (ii) There exists a sharply weak mixing (conservative) 0-type measure preserving free G-action T on an infinite σ-finite standard meas","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42317351","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 vector-valued spectrum Mu,∞(Bl2 , Bl2 ) which is the set of nonzero algebra homomorphisms from Au(Bl2 ) (the algebra of uniformly continuous holomorphic functions on Bl2) to H∞(Bl2 ) (the algebra of bounded holomorphic functions on Bl2 ). This set is naturally projected onto the closed unit ball of H∞(Bl2 , l2) giving rise to an associated fibering. Extending the classical notion of cluster sets introduced by I. J. Schark (1961) to the vector-valued spectrum we define vector-valued cluster sets. The aim of the article is to look at the relationship between fibers and cluster sets obtaining results regarding the existence of analytic balls into these sets.
{"title":"A look into homomorphisms between uniform algebras\u0000over a Hilbert space","authors":"Verónica Dimant, J. Singer","doi":"10.4064/sm210219-10-6","DOIUrl":"https://doi.org/10.4064/sm210219-10-6","url":null,"abstract":"We study the vector-valued spectrum Mu,∞(Bl2 , Bl2 ) which is the set of nonzero algebra homomorphisms from Au(Bl2 ) (the algebra of uniformly continuous holomorphic functions on Bl2) to H∞(Bl2 ) (the algebra of bounded holomorphic functions on Bl2 ). This set is naturally projected onto the closed unit ball of H∞(Bl2 , l2) giving rise to an associated fibering. Extending the classical notion of cluster sets introduced by I. J. Schark (1961) to the vector-valued spectrum we define vector-valued cluster sets. The aim of the article is to look at the relationship between fibers and cluster sets obtaining results regarding the existence of analytic balls into these sets.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45406466","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 $pgeq 2$, the $p$-adic ring $C^*$-algebra $mathcal{Q}_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $chi_kin{rm End}(mathcal{Q}_p)$ by setting $chi_k(U):=U^k$ and $chi_k(S_p):=S_p$. We then compute the entropy of $chi_k$, which turns out to be $log |k|$. Finally, for selected values of $k$ we also compute the Watatani index of $chi_k$ showing that the entropy is the natural logarithm of the index.
{"title":"On the entropy and index of the winding endomorphisms of $p$-adic ring $C^*$-algebras","authors":"Valeriano Aiello, S. Rossi","doi":"10.4064/sm201125-9-2","DOIUrl":"https://doi.org/10.4064/sm201125-9-2","url":null,"abstract":"For $pgeq 2$, the $p$-adic ring $C^*$-algebra $mathcal{Q}_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $chi_kin{rm End}(mathcal{Q}_p)$ by setting $chi_k(U):=U^k$ and $chi_k(S_p):=S_p$. We then compute the entropy of $chi_k$, which turns out to be $log |k|$. Finally, for selected values of $k$ we also compute the Watatani index of $chi_k$ showing that the entropy is the natural logarithm of the index.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42789738","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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