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1. CMB Online first

He, Ziyi; Yang, Dachun; Yuan, Wen
Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls
In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means.

Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-function
Categories:46E35, 42B25, 42B20, 42B35

2. CMB Online first

Sundar, S.
A Computation with the Connes-Thom Isomorphism
Let $A \in M_{n}(\mathbb{R})$ be an invertible matrix. Consider the semi-direct product $\mathbb{R}^{n} \rtimes \mathbb{Z}$ where the action of $\mathbb{Z}$ on $\mathbb{R}^{n}$ is induced by the left multiplication by $A$. Let $(\alpha,\tau)$ be a strongly continuous action of $\mathbb{R}^{n} \rtimes \mathbb{Z}$ on a $C^{*}$-algebra $B$ where $\alpha$ is a strongly continuous action of $\mathbb{R}^{n}$ and $\tau$ is an automorphism. The map $\tau$ induces a map $\widetilde{\tau}$ on $B \rtimes_{\alpha} \mathbb{R}^{n}$. We show that, at the $K$-theory level, $\tau$ commutes with the Connes-Thom map if $\det(A)\gt 0$ and anticommutes if $\det(A)\lt 0$. As an application, we recompute the $K$-groups of the Cuntz-Li algebra associated to an integer dilation matrix.

Keywords:K-theory, Connes-Thom isomorphism, Cuntz-Li algebras
Categories:46L80, 58B34

3. CMB Online first

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

4. CMB Online first

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

5. CMB Online first

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, representation
Categories:46L05, 37A55

6. CMB Online first

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, Calderón reproducing formula, Besov and Triebel-Lizorkin spaces, embedding
Categories:42B25, 46F05, 46E35

7. 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, hyperplane
Categories:46B45, 46B04

8. 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 growth
Categories:46L35, 46L05, 46L80, 47L40, 46L85

9. 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 ideal
Categories:46J15, 46J20, 30H50

10. 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 actions
Categories:46L55, 19K35

11. CMB Online first

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, quasicontinuity
Categories:42B25, 46E35

12. 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 isometries
Categories:46E15, 47B15, 47B38

13. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators
Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic Schrödinger 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 Hölder 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 Schrödinger operator, atom, second-order Riesz transform, maximal inequality
Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

14. 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 indices
Categories:19K56, 46L80

15. 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-isometry
Categories:47A13, 46E22

16. CMB 2014 (vol 58 pp. 150)

Ostrovskii, Mikhail I.
Connections Between Metric Characterizations of Superreflexivity and the Radon-Nikodý Property for Dual Banach Spaces
Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-Nikodým 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-Nikodým property, superreflexivity
Categories:46B85, 46B07, 46B22

17. 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

18. 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 length
Category:46L05

19. 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 algebras
Categories:46H20, 43A20

20. CMB 2014 (vol 58 pp. 207)

Moslehian, Mohammad Sal; Zamani, Ali
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 $C^*$-module
Categories:47A30, 46L05, 46L08, 47B47, 15A60

21. 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 functions
Categories:54C30, 46E25

22. 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 decay
Categories:46L54, 20G42, 46L65

23. 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 space
Categories:47H09, 46B20, 47H10, 47E10

24. 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 tightness
Categories:46A03, 54D50, 54A25

25. 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, stability
Categories:46F99, 39B82
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