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26. CMB 2015 (vol 59 pp. 204)

Spektor, Susanna
 Restricted Khinchine Inequality We prove a Khintchine type inequality under the assumption that the sum of Rademacher random variables equals zero. We also show a new tail-bound for a hypergeometric random variable. Keywords:Khintchine inequality, Kahane inequality, Rademacher random variables, hypergeometric distribution.Categories:46B06, 60E15, 52A23, 46B09

27. CMB 2015 (vol 59 pp. 3)

Alfuraidan, Monther Rashed
 The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph We study the existence of fixed points for contraction multivalued mappings in modular metric spaces endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as a generalization of Nadler's and Edelstein's fixed point theorems to modular metric spaces endowed with a graph. Keywords:fixed point theory, modular metric spaces, multivalued contraction mapping, connected digraph.Categories:47H09, 46B20, 47H10, 47E10

28. CMB 2015 (vol 59 pp. 95)

Gonçalves, Daniel; Li, Hui; Royer, Danilo
 Faithful Representations of Graph Algebras via Branching Systems We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a sufficient condition under which the representation induced from a branching system of a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We finish the paper by providing a proof of the converse of the Cuntz-Krieger uniqueness theorem for graph algebras by means of branching systems. Keywords:C*-algebra, graph algebra, Leavitt path algebra, branching system, representationCategories:46L05, 37A55

29. CMB 2015 (vol 58 pp. 757)

Han, Yanchang
 Embedding Theorem for Inhomogeneous Besov and Triebel-Lizorkin Spaces on RD-spaces In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure. Keywords:spaces of homogeneous type, test function space, distributions, CalderÃ³n reproducing formula, Besov and Triebel-Lizorkin spaces, embeddingCategories:42B25, 46F05, 46E35

30. CMB 2015 (vol 58 pp. 459)

Casini, Emanuele; Miglierina, Enrico; Piasecki, Lukasz
 Hyperplanes in the Space of Convergent Sequences and Preduals of $\ell_1$ The main aim of the present paper is to investigate various structural properties of hyperplanes of $c$, the Banach space of the convergent sequences. In particular, we give an explicit formula for the projection constants and we prove that an hyperplane of $c$ is isometric to the whole space if and only if it is $1$-complemented. Moreover, we obtain the classification of those hyperplanes for which their duals are isometric to $\ell_{1}$ and we give a complete description of the preduals of $\ell_{1}$ under the assumption that the standard basis of $\ell_{1}$ is weak$^{*}$-convergent. Keywords:space of convergent sequences, projection, $\ell_1$-predual, hyperplaneCategories:46B45, 46B04

31. CMB 2015 (vol 58 pp. 402)

Tikuisis, Aaron Peter; Toms, Andrew
 On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple $\mathrm{C}^*$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first-named author, this shows that slow dimension growth coincides with $\mathcal Z$-stability, for approximately subhomogeneous algebras. Keywords:nuclear C*-algebras, Cuntz semigroup, dimension functions, stably projectionless C*-algebras, approximately subhomogeneous C*-algebras, slow dimension growthCategories:46L35, 46L05, 46L80, 47L40, 46L85

32. CMB 2015 (vol 58 pp. 350)

Merino-Cruz, Héctor; Wawrzynczyk, Antoni
 On Closed Ideals in a Certain Class of Algebras of Holomorphic Functions We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of functions which are holomorphic on the unit disc $\mathbb{D}$, continuous up to the boundary and of the class $C^{(n)}$ at all points where the function $G$ does not vanish. Here, $G$ refers to a function of the disc algebra without zeros on $\mathbb{D}$. Then we proved that all closed ideals in $\mathfrak{A}_G^n$ with at most countable hull are standard. In the present paper, on the assumption that $G$ is an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the closed ideals $I$ with hull containing $h(G)$ are standard. Keywords:Banach algebra, disc algebra, holomorphic spaces, standard idealCategories:46J15, 46J20, 30H50

33. CMB 2015 (vol 58 pp. 374)

Szabó, Gábor
 A Short Note on the Continuous Rokhlin Property and the Universal Coefficient Theorem in $E$-Theory Let $G$ be a metrizable compact group, $A$ a separable $\mathrm{C}^*$-algebra and $\alpha\colon G\to\operatorname{Aut}(A)$ a strongly continuous action. Provided that $\alpha$ satisfies the continuous Rokhlin property, we show that the property of satisfying the UCT in $E$-theory passes from $A$ to the crossed product $\mathrm{C}^*$-algebra $A\rtimes_\alpha G$ and the fixed point algebra $A^\alpha$. This extends a similar result by Gardella for $KK$-theory in the case of unital $\mathrm{C}^*$-algebras, but with a shorter and less technical proof. For circle actions on separable, unital $\mathrm{C}^*$-algebras with the continuous Rokhlin property, we establish a connection between the $E$-theory equivalence class of $A$ and that of its fixed point algebra $A^\alpha$. Keywords:Rokhlin property, UCT, KK-theory, E-theory, circle actionsCategories:46L55, 19K35

34. CMB 2015 (vol 58 pp. 808)

Liu, Feng; Wu, Huoxiong
 On the Regularity of the Multisublinear Maximal Functions This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained. Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuityCategories:42B25, 46E35

35. CMB 2015 (vol 58 pp. 241)

Botelho, Fernanda
 Isometries and Hermitian Operators on Zygmund Spaces In this paper we characterize the isometries of subspaces of the little Zygmund space. We show that the isometries of these spaces are surjective and represented as integral operators. We also show that all hermitian operators on these settings are bounded. Keywords:Zygmund spaces, the little Zygmund space, Hermitian operators, surjective linear isometries, generators of one-parameter groups of surjective isometriesCategories:46E15, 47B15, 47B38

36. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
 Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic SchrÃ¶dinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse HÃ¶lder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained. Keywords:Musielak-Orlicz-Hardy space, magnetic SchrÃ¶dinger operator, atom, second-order Riesz transform, maximal inequalityCategories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

37. CMB 2014 (vol 58 pp. 51)

De Nitties, Giuseppe; Schulz-Baldes, Hermann
 Spectral Flows of Dilations of Fredholm Operators Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow which is shown to be equal to the index of the operator. This result is interpreted in terms of the $K$-theory of an associated mapping cone. It is then extended to connect $\mathbb{Z}_2$ indices of odd symmetric Fredholm operators to a $\mathbb{Z}_2$-valued spectral flow. Keywords:spectral flow, Fredholm operators, Z2 indicesCategories:19K56, 46L80

38. CMB 2014 (vol 58 pp. 276)

Johnson, William; Nasseri, Amir Bahman; Schechtman, Gideon; Tkocz, Tomasz
 Injective Tauberian Operators on $L_1$ and Operators with Dense Range on $\ell_\infty$ There exist injective Tauberian operators on $L_1(0,1)$ that have dense, nonclosed range. This gives injective, nonsurjective operators on $\ell_\infty$ that have dense range. Consequently, there are two quasi-complementary, noncomplementary subspaces of $\ell_\infty$ that are isometric to $\ell_\infty$. Keywords:$L_1$, Tauberian operator, $\ell_\infty$Categories:46E30, 46B08, 47A53

39. CMB 2014 (vol 58 pp. 9)

Chavan, Sameer
 Irreducible Tuples Without the Boundary Property We examine spectral behavior of irreducible tuples which do not admit boundary property. In particular, we prove under some mild assumption that the spectral radius of such an $m$-tuple $(T_1, \dots, T_m)$ must be the operator norm of $T^*_1T_1 + \cdots + T^*_mT_m$. We use this simple observation to ensure boundary property for an irreducible, essentially normal joint $q$-isometry provided it is not a joint isometry. We further exhibit a family of reproducing Hilbert $\mathbb{C}[z_1, \dots, z_m]$-modules (of which the Drury-Arveson Hilbert module is a prototype) with the property that any two nested unitarily equivalent submodules are indeed equal. Keywords:boundary representations, subnormal, joint p-isometryCategories:47A13, 46E22

40. CMB 2014 (vol 58 pp. 150)

Ostrovskii, Mikhail I.
 Connections Between Metric Characterizations of Superreflexivity and the Radon-NikodÃ½ Property for Dual Banach Spaces Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-NikodÃ½m property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample. Keywords:Banach space, diamond graph, finite representability, metric characterization, Radon-NikodÃ½m property, superreflexivityCategories:46B85, 46B07, 46B22

41. CMB 2014 (vol 57 pp. 853)

Pan, Qingfei; Wang, Kun
 On the Bound of the $\mathrm{C}^*$ Exponential Length Let $X$ be a compact Hausdorff space. In this paper, we give an example to show that there is $u\in \mathrm{C}(X)\otimes \mathrm{M}_n$ with $\det (u(x))=1$ for all $x\in X$ and $u\sim_h 1$ such that the $\mathrm{C}^*$ exponential length of $u$ (denoted by $cel(u)$) can not be controlled by $\pi$. Moreover, in simple inductive limit $\mathrm{C}^*$-algebras, similar examples also exist. Keywords:exponential lengthCategory:46L05

42. CMB 2014 (vol 58 pp. 207)

 Exact and Approximate Operator Parallelism Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}(\mathscr{H})$ of bounded linear operators acting on a Hilbert space $\mathscr{H}$. Among other things, we investigate the relationship between approximate parallelism and norm of inner derivations on $\mathbb{B}(\mathscr{H})$. We also characterize the parallel elements of a $C^*$-algebra by using states. Finally we utilize the linking algebra to give some equivalence assertions regarding parallel elements in a Hilbert $C^*$-module. Keywords:$C^*$-algebra, approximate parallelism, operator parallelism, Hilbert $C^*$-moduleCategories:47A30, 46L05, 46L08, 47B47, 15A60

43. CMB 2014 (vol 58 pp. 3)

Alaghmandan, Mahmood
 Approximate Amenability of Segal Algebras II We prove that every proper Segal algebra of a SIN group is not approximately amenable. Keywords:Segal algebras, approximate amenability, SIN groups, commutative Banach algebrasCategories:46H20, 43A20

44. CMB 2014 (vol 58 pp. 7)

Boulabiar, Karim
 Characters on $C(X)$ The precise condition on a completely regular space $X$ for every character on $C(X)$ to be an evaluation at some point in $X$ is that $X$ be realcompact. Usually, this classical result is obtained relying heavily on involved (and even nonconstructive) extension arguments. This note provides a direct proof that is accessible to a large audience. Keywords:characters, realcompact, evaluation, real-valued continuous functionsCategories:54C30, 46E25

45. CMB 2014 (vol 58 pp. 297)

Khamsi, M. A.
 Approximate Fixed Point Sequences of Nonlinear Semigroup in Metric Spaces In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings $\{T_t\}_{t \geq 0}$, i.e., a family such that $T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where the domain is a metric space $(M,d)$. In particular we prove that under suitable conditions, the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings. Keywords:approximate fixed point, fixed point, hyperbolic metric space, Ishikawa iterations, nonexpansive mapping, semigroup of mappings, uniformly convex hyperbolic spaceCategories:47H09, 46B20, 47H10, 47E10

46. CMB 2014 (vol 57 pp. 708)

Brannan, Michael
 Strong Asymptotic Freeness for Free Orthogonal Quantum Groups It is known that the normalized standard generators of the free orthogonal quantum group $O_N^+$ converge in distribution to a free semicircular system as $N \to \infty$. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator norm of any non-commutative polynomial in the normalized standard generators of $O_N^+$ converges as $N \to \infty$ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well known $L^2$-$L^\infty$ norm equivalence for non-commutative polynomials in free semicircular systems. Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decayCategories:46L54, 20G42, 46L65

47. CMB 2014 (vol 57 pp. 803)

Gabriyelyan, S. S.
 Free Locally Convex Spaces and the $k$-space Property Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. Then $L(X)$ is a $k$-space if and only if $X$ is a countable discrete space. We prove also that $L(D)$ has uncountable tightness for every uncountable discrete space $D$. Keywords:free locally convex space, $k$-space, countable tightnessCategories:46A03, 54D50, 54A25

48. CMB 2014 (vol 58 pp. 30)

Chung, Jaeyoung
 On an Exponential Functional Inequality and its Distributional Version Let $G$ be a group and $\mathbb K=\mathbb C$ or $\mathbb R$. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions $f\colon G\to \mathbb K$ satisfying the inequality $\Bigl|f \Bigl(\sum_{k=1}^n x_k \Bigr)-\prod_{k=1}^n f(x_k) \Bigr|\le \phi(x_2, \dots, x_n),\quad \forall\, x_1, \dots, x_n\in G,$ where $\phi\colon G^{n-1}\to [0, \infty)$. Also, as a distributional version of the above inequality we consider the stability of the functional equation \begin{equation*} u\circ S - \overbrace{u\otimes \cdots \otimes u}^{n-\text {times}}=0, \end{equation*} where $u$ is a Schwartz distribution or Gelfand hyperfunction, $\circ$ and $\otimes$ are the pullback and tensor product of distributions, respectively, and $S(x_1, \dots, x_n)=x_1+ \dots +x_n$. Keywords:distribution, exponential functional equation, Gelfand hyperfunction, stabilityCategories:46F99, 39B82

49. CMB 2014 (vol 57 pp. 780)

Erzakova, Nina A.
 Measures of Noncompactness in Regular Spaces Previous results by the author on the connection between three of measures of non-compactness obtained for $L_p$, are extended to regular spaces of measurable functions. An example of advantage in some cases one of them in comparison with another is given. Geometric characteristics of regular spaces are determined. New theorems for $(k,\beta)$-boundedness of partially additive operators are proved. Keywords:measure of non-compactness, condensing map, partially additive operator, regular space, ideal spaceCategories:47H08, 46E30, 47H99, 47G10

50. CMB 2014 (vol 58 pp. 71)

Ghenciu, Ioana
 Limited Sets and Bibasic Sequences Bibasic sequences are used to study relative weak compactness and relative norm compactness of limited sets. Keywords:limited sets, $L$-sets, bibasic sequences, the Dunford-Pettis propertyCategories:46B20, 46B28, 28B05
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