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51. 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

52. 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

53. 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

54. 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

55. 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

56. 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

57. 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

58. 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

59. 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

60. CMB 2014 (vol 58 pp. 110)

Kamalov, F.
 Property T and Amenable Transformation Group $C^*$-algebras It is well known that a discrete group which is both amenable and has Kazhdan's Property T must be finite. In this note we generalize the above statement to the case of transformation groups. We show that if $G$ is a discrete amenable group acting on a compact Hausdorff space $X$, then the transformation group $C^*$-algebra $C^*(X, G)$ has Property T if and only if both $X$ and $G$ are finite. Our approach does not rely on the use of tracial states on $C^*(X, G)$. Keywords:Property T, $C^*$-algebras, transformation group, amenableCategories:46L55, 46L05

61. CMB 2014 (vol 57 pp. 810)

Godefroy, G.
 Uniqueness of Preduals in Spaces of Operators We show that if $E$ is a separable reflexive space, and $L$ is a weak-star closed linear subspace of $L(E)$ such that $L\cap K(E)$ is weak-star dense in $L$, then $L$ has a unique isometric predual. The proof relies on basic topological arguments. Categories:46B20, 46B04

62. CMB 2013 (vol 57 pp. 640)

 Equilateral Sets and a SchÃ¼tte Theorem for the $4$-norm A well-known theorem of SchÃ¼tte (1963) gives a sharp lower bound for the ratio of the maximum and minimum distances between $n+2$ points in $n$-dimensional Euclidean space. In this note we adapt BÃ¡rÃ¡ny's elegant proof (1994) of this theorem to the space $\ell_4^n$. This gives a new proof that the largest cardinality of an equilateral set in $\ell_4^n$ is $n+1$, and gives a constructive bound for an interval $(4-\varepsilon_n,4+\varepsilon_n)$ of values of $p$ close to $4$ for which it is known that the largest cardinality of an equilateral set in $\ell_p^n$ is $n+1$. Categories:46B20, 52A21, 52C17

63. CMB 2013 (vol 57 pp. 463)

Bownik, Marcin; Jasper, John
 Constructive Proof of Carpenter's Theorem We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem. Keywords:diagonals of projections, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theoryCategories:42C15, 47B15, 46C05

64. CMB 2013 (vol 57 pp. 598)

Lu, Yufeng; Yang, Dachun; Yuan, Wen
 Interpolation of Morrey Spaces on Metric Measure Spaces In this article, via the classical complex interpolation method and some interpolation methods traced to Gagliardo, the authors obtain an interpolation theorem for Morrey spaces on quasi-metric measure spaces, which generalizes some known results on ${\mathbb R}^n$. Keywords:complex interpolation, Morrey space, Gagliardo interpolation, CalderÃ³n product, quasi-metric measure spaceCategories:46B70, 46E30

65. CMB 2013 (vol 57 pp. 364)

Li, Lei; Wang, Ya-Shu
 How Lipschitz Functions Characterize the Underlying Metric Spaces Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that both $X,Y$ are realcompact, or both $E,F$ are realcompact. The zero set of a vector-valued function $f$ is denoted by $z(f)$. A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions if $z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),$ or $z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,$ respectively. Every zero-set containment preserver, and every nonvanishing function preserver when $\dim E =\dim F\lt +\infty$, is a weighted composition operator $(Tf)(y)=J_y(f(\tau(y)))$. We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism. Keywords:(generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating mapsCategories:46E40, 54D60, 46E15

66. CMB 2013 (vol 57 pp. 546)

 Compact Operators in Regular LCQ Groups We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains non-zero compact operators on $\mathcal{L}^{2}(\mathbb{G})$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where $\mathcal{L}^{1}(\mathbb{G})$ has the Radon--Nikodym property. Keywords:locally compact quantum groups, regularity, compact operatorsCategory:46L89

67. CMB 2012 (vol 57 pp. 424)

Sołtan, Piotr M.; Viselter, Ami
 A Note on Amenability of Locally Compact Quantum Groups In this short note we introduce a notion called quantum injectivity'' of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups. Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivityCategories:20G42, 22D25, 46L89

68. CMB 2012 (vol 57 pp. 90)

Lazar, Aldo J.
 Compact Subsets of the Glimm Space of a $C^*$-algebra If $A$ is a $\sigma$-unital $C^*$-algebra and $a$ is a strictly positive element of $A$ then for every compact subset $K$ of the complete regularization $\mathrm{Glimm}(A)$ of $\mathrm{Prim}(A)$ there exists $\alpha \gt 0$ such that $K\subset \{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq \alpha\}$. This extends a result of J. Dauns to all $\sigma$-unital $C^*$-algebras. However, there are a $C^*$-algebra $A$ and a compact subset of $\mathrm{Glimm}(A)$ that is not contained in any set of the form $\{G\in \mathrm{Glimm}(A) \mid \Vert a + G\Vert \geq \alpha\}$, $a\in A$ and $\alpha \gt 0$. Keywords:primitive ideal space, complete regularizationCategory:46L05

69. CMB 2012 (vol 57 pp. 42)

 Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls e prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball. Keywords:point finite coverings, slices, polyhedral spaces, Hilbert spacesCategories:46B20, 46C05, 52C17

70. CMB 2012 (vol 57 pp. 166)

Öztop, Serap; Spronk, Nico
 On Minimal and Maximal $p$-operator Space Structures We show that for $p$-operator spaces, there are natural notions of minimal and maximal structures. These are useful for dealing with tensor products. Keywords:$p$-operator space, min space, max spaceCategories:46L07, 47L25, 46G10

71. CMB 2012 (vol 57 pp. 3)

Adamczak, Radosław; Latała, Rafał; Litvak, Alexander E.; Oleszkiewicz, Krzysztof; Pajor, Alain; Tomczak-Jaegermann, Nicole
 A Short Proof of Paouris' Inequality We give a short proof of a result of G.~Paouris on the tail behaviour of the Euclidean norm $|X|$ of an isotropic log-concave random vector $X\in\mathbb{R}^n,$ stating that for every $t\geq 1$, $\mathbb{P} \big( |X|\geq ct\sqrt n\big)\leq \exp(-t\sqrt n).$ More precisely we show that for any log-concave random vector $X$ and any $p\geq 1$, $(\mathbb{E}|X|^p)^{1/p}\sim \mathbb{E} |X|+\sup_{z\in S^{n-1}}(\mathbb{E} |\langle z,X\rangle|^p)^{1/p}.$ Keywords:log-concave random vectors, deviation inequalitiesCategories:46B06, 46B09, 52A23

72. CMB 2012 (vol 57 pp. 37)

Dashti, Mahshid; Nasr-Isfahani, Rasoul; Renani, Sima Soltani
 Character Amenability of Lipschitz Algebras Let ${\mathcal X}$ be a locally compact metric space and let ${\mathcal A}$ be any of the Lipschitz algebras ${\operatorname{Lip}_{\alpha}{\mathcal X}}$, ${\operatorname{lip}_{\alpha}{\mathcal X}}$ or ${\operatorname{lip}_{\alpha}^0{\mathcal X}}$. In this paper, we show, as a consequence of rather more general results on Banach algebras, that ${\mathcal A}$ is $C$-character amenable if and only if ${\mathcal X}$ is uniformly discrete. Keywords:character amenable, character contractible, Lipschitz algebras, spectrumCategories:43A07, 46H05, 46J10

73. CMB 2012 (vol 56 pp. 551)

Handelman, David
 Real Dimension Groups Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like, i.e., it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over the a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In the Appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs. Keywords:dimension group, simplicial vector space, direct limit, Riesz interpolationCategories:46A40, 06F20, 13J25, 19K14

74. CMB 2012 (vol 56 pp. 870)

Wei, Changguo
 Note on Kasparov Product of $C^*$-algebra Extensions Using the Dadarlat isomorphism, we give a characterization for the Kasparov product of $C^*$-algebra extensions. A certain relation between $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ is also considered when $B$ is not stable and it is proved that $KK(A, \mathcal q(B))$ and $KK(A, \mathcal q(\mathcal k B))$ are not isomorphic in general. Keywords:extension, Kasparov product, $KK$-groupCategory:46L80

75. CMB 2012 (vol 56 pp. 503)

Bu, Qingying
 Weak Sequential Completeness of $\mathcal K(X,Y)$ For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are weakly sequentially complete and every weakly compact operator from $X$ to $Y$ is compact then the space of all compact operators from $X$ to $Y$ is weakly sequentially complete. The converse is also true if, in addition, either $X^\ast$ or $Y$ has the bounded compact approximation property. Keywords:weak sequential completeness, reflexivity, compact operator spaceCategories:46B25, 46B28
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