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51. 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, Calderón product, quasi-metric measure space
Categories:46B70, 46E30

52. 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 maps
Categories:46E40, 54D60, 46E15

53. CMB 2013 (vol 57 pp. 546)

Kalantar, Mehrdad
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 operators

54. 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 injectivity
Categories:20G42, 22D25, 46L89

55. 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 regularization

56. CMB 2012 (vol 57 pp. 42)

Fonf, Vladimir P.; Zanco, Clemente
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 spaces
Categories:46B20, 46C05, 52C17

57. 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 space
Categories:46L07, 47L25, 46G10

58. 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 inequalities
Categories:46B06, 46B09, 52A23

59. 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, spectrum
Categories:43A07, 46H05, 46J10

60. 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 interpolation
Categories:46A40, 06F20, 13J25, 19K14

61. 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 space
Categories:46B25, 46B28

62. 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, $KK$-group

63. CMB 2012 (vol 56 pp. 534)

Filali, M.; Monfared, M. Sangani
A Cohomological Property of $\pi$-invariant Elements
Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$ be a continuous representation of $A$ on a separable Hilbert space $H$ with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of $\pi$ with respect to an orthonormal basis and suppose that for each $1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m} \|\pi_{ij}\|_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$ left $\pi$-invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all $a\in A$. In this paper we prove a link between the existence of left $\pi$-invariant elements and the vanishing of certain Hochschild cohomology groups of $A$. Our results extend an earlier result by Lau on $F$-algebras and recent results of Kaniuth-Lau-Pym and the second named author in the special case that $\pi \colon A \longrightarrow \mathbf C$ is a non-zero character on $A$.

Keywords:Banach algebras, $\pi$-invariance, derivations, representations
Categories:46H15, 46H25, 13N15

64. CMB 2012 (vol 56 pp. 630)

Sundar, S.
Inverse Semigroups and Sheu's Groupoid for the Odd Dimensional Quantum Spheres
In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid $C^{*}$-algebras. We show that the $C^{*}$-algebra $C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the inverse semigroup $T$ by Exel is exactly the same as the groupoid considered by Sheu.

Keywords:inverse semigroups, groupoids, odd dimensional quantum spheres
Categories:46L99, 20M18

65. CMB 2011 (vol 56 pp. 337)

Fan, Qingzhai
Certain Properties of $K_0$-monoids Preserved by Tracial Approximation
We show that the following $K_0$-monoid properties of $C^*$-algebras in the class $\Omega$ are inherited by simple unital $C^*$-algebras in the class $TA\Omega$: (1) weak comparability, (2) strictly unperforated, (3) strictly cancellative.

Keywords:$C^*$-algebra, tracial approximation, $K_0$-monoid
Categories:46L05, 46L80, 46L35

66. CMB 2011 (vol 55 pp. 783)

Motallebi, M. R.; Saiflu, H.
Products and Direct Sums in Locally Convex Cones
In this paper we define lower, upper, and symmetric completeness and discuss closure of the sets in product and direct sums. In particular, we introduce suitable bases for these topologies, which leads us to investigate completeness of the direct sum and its components. Some results obtained about $X$-topologies and polars of the neighborhoods.

Keywords:product and direct sum, duality, locally convex cone
Categories:20K25, 46A30, 46A20

67. CMB 2011 (vol 56 pp. 400)

Prunaru, Bebe
A Factorization Theorem for Multiplier Algebras of Reproducing Kernel Hilbert Spaces
Let $(X,\mathcal B,\mu)$ be a $\sigma$-finite measure space and let $H\subset L^2(X,\mu)$ be a separable reproducing kernel Hilbert space on $X$. We show that the multiplier algebra of $H$ has property $(A_1(1))$.

Keywords:reproducing kernel Hilbert space, Berezin transform, dual algebra
Categories:46E22, 47B32, 47L45

68. CMB 2011 (vol 56 pp. 272)

Cheng, Lixin; Luo, Zhenghua; Zhou, Yu
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate
In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$: a closed bounded convex set $K\subset X$ is super weakly compact if and only if there exists a $w^*$ lower semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly Fréchet differentiable on each bounded set of $X^*$. Then we present a representation theorem for the dual of the semigroup $\textrm{swcc}(X)$ consisting of all the nonempty super weakly compact convex sets of the space $X$.

Keywords:super weakly compact set, dual of normed semigroup, uniform Fréchet differentiability, representation
Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

69. CMB 2011 (vol 56 pp. 388)

Mursaleen, M.
Application of Measure of Noncompactness to Infinite Systems of Differential Equations
In this paper we determine the Hausdorff measure of noncompactness on the sequence space $n(\phi)$ of W. L. C. Sargent. Further we apply the technique of measures of noncompactness to the theory of infinite systems of differential equations in the Banach sequence spaces $n(\phi)$ and $m(\phi)$. Our aim is to present some existence results for infinite systems of differential equations formulated with the help of measures of noncompactness.

Keywords:sequence spaces, BK spaces, measure of noncompactness, infinite system of differential equations
Categories:46B15, 46B45, 46B50, 34A34, 34G20

70. CMB 2011 (vol 56 pp. 434)

Wnuk, Witold
Some Remarks on the Algebraic Sum of Ideals and Riesz Subspaces
Following ideas used by Drewnowski and Wilansky we prove that if $I$ is an infinite dimensional and infinite codimensional closed ideal in a complete metrizable locally solid Riesz space and $I$ does not contain any order copy of $\mathbb R^{\mathbb N}$ then there exists a closed, separable, discrete Riesz subspace $G$ such that the topology induced on $G$ is Lebesgue, $I \cap G = \{0\}$, and $I + G$ is not closed.

Keywords:locally solid Riesz space, Riesz subspace, ideal, minimal topological vector space, Lebesgue property
Categories:46A40, 46B42, 46B45

71. CMB 2011 (vol 56 pp. 65)

Ghenciu, Ioana
The Uncomplemented Subspace $\mathbf K(X,Y) $
A vector measure result is used to study the complementation of the space $K(X,Y)$ of compact operators in the spaces $W(X,Y)$ of weakly compact operators, $CC(X,Y)$ of completely continuous operators, and $U(X,Y)$ of unconditionally converging operators. Results of Kalton and Emmanuele concerning the complementation of $K(X,Y)$ in $L(X,Y)$ and in $W(X,Y)$ are generalized. The containment of $c_0$ and $\ell_\infty$ in spaces of operators is also studied.

Keywords:compact operators, weakly compact operators, uncomplemented subspaces of operators
Categories:46B20, 46B28

72. CMB 2011 (vol 56 pp. 136)

Munteanu, Radu-Bogdan
On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type
Product type equivalence relations are hyperfinite measured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

Keywords:property A, hyperfinite equivalence relation, non-product type
Categories:37A20, 37A35, 46L10

73. CMB 2011 (vol 55 pp. 821)

Perez-Garcia, C.; Schikhof, W. H.
New Examples of Non-Archimedean Banach Spaces and Applications
The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm, without non-zero vectors that are $\|\cdot\|$-orthogonal and such that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$ into a valued field.

Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases
Categories:46S10, 12J25

74. CMB 2011 (vol 55 pp. 673)

Aizenbud, Avraham; Gourevitch, Dmitry
Multiplicity Free Jacquet Modules
Let $F$ be a non-Archimedean local field or a finite field. Let $n$ be a natural number and $k$ be $1$ or $2$. Consider $G:=\operatorname{GL}_{n+k}(F)$ and let $M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup. Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup. Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$. In this paper we prove that $J$ is a multiplicity free functor, i.e., $\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$, for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$. We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.

Keywords:multiplicity one, Gelfand pair, invariant distribution, finite group
Categories:20G05, 20C30, 20C33, 46F10, 47A67

75. CMB 2011 (vol 55 pp. 697)

Borwein, Jonathan M.; Vanderwerff, Jon
Constructions of Uniformly Convex Functions
We give precise conditions under which the composition of a norm with a convex function yields a uniformly convex function on a Banach space. Various applications are given to functions of power type. The results are dualized to study uniform smoothness and several examples are provided.

Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, norm
Categories:52A41, 46G05, 46N10, 49J50, 90C25
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