26. CMB 2016 (vol 60 pp. 77)
 Christ, Michael; Rieffel, Marc A.

Nilpotent Group C*algebras as Compact Quantum Metric Spaces
Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$
denote the
operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$.
Following Connes,
$M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced
group C*algebra $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the
state space of
$C_r^*(G)$. We show that
for any length function satisfying a strong form of polynomial
growth on a discrete group,
the topology from this metric
coincides with the
weak$*$ topology (a key property for the
definition of a ``compact quantum metric
space''). In particular, this holds for all wordlength functions
on finitely generated nilpotentbyfinite groups.
Keywords:group C*algebra, Dirac operator, quantum metric space, discrete nilpotent group, polynomial growth Categories:46L87, 20F65, 22D15, 53C23, 58B34 

27. CMB 2016 (vol 59 pp. 769)
 GarcíaPacheco, Francisco Javier; Hill, Justin R.

Geometric Characterizations of Hilbert Spaces
We study some geometric properties related to the set $\Pi_X:=
\{
(x,x^*
)\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^*
(x
)=1
\}$ obtaining two characterizations of Hilbert spaces
in the category of Banach spaces. We also compute the distance
of a generic element $
(h,k
)\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space.
Keywords:Hilbert space, extreme point, smooth, $\mathsf{L}^2$summands Categories:46B20, 46C05 

28. CMB 2016 (vol 59 pp. 606)
 Mihăilescu, Mihai; Moroşanu, Gheorghe

Eigenvalues of $ \Delta_p \Delta_q $ Under Neumann Boundary Condition
The
eigenvalue problem $\Delta_p u\Delta_q u=\lambdau^{q2}u$
with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to
the
corresponding homogeneous Neumann boundary condition is
investigated on a bounded open set with smooth boundary from
$\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads
us to a complete description of the set of eigenvalues as being
a
precise interval $(\lambda_1, +\infty )$ plus an isolated point
$\lambda =0$. This comprehensive result is strongly related to
our
framework which is complementary to the wellknown case $p=q\neq
2$ for which a full description of the set of eigenvalues is
still
unavailable.
Keywords:eigenvalue problem, Sobolev space, Nehari manifold, variational methods Categories:35J60, 35J92, 46E30, 49R05 

29. CMB 2016 (vol 59 pp. 878)
 Wang, Jianfei

The Carleson Measure Problem Between Analytic Morrey Spaces
The purpose of this paper is to characterize positive measure
$\mu$ on the unit disk such that the analytic
Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly
embedded to the tent space
$\mathcal{T}_{q,1\frac{q}{p}(1\eta)}^{\infty}(\mu)$ for the
case $1\leq q\leq p\lt \infty$
respectively. As an application, these results are used to
establish the boundedness and compactness of integral operators
and multipliers between analytic Morrey spaces.
Keywords:Morrey space, Carleson measure problem, boundedness, compactness Categories:30H35, 28A12, 47B38, 46E15 

30. CMB 2016 (vol 59 pp. 320)
 Ino, Shoji

Perturbations of Von Neumann Subalgebras with Finite Index
In this paper, we study uniform perturbations of von Neumann
subalgebras of a von Neumann algebra.
Let $M$ and $N$ be von Neumann subalgebras of a von Neumann algebra
with finite probabilistic index in the sense of PimsnerPopa.
If $M$ and $N$ are sufficiently close,
then $M$ and $N$ are unitarily equivalent.
The implementing unitary can be chosen as being close to the
identity.
Keywords:von Neumann algebras, perturbations Categories:46L10, 46L37 

31. CMB 2015 (vol 59 pp. 435)
 Yao, Hongliang

On Extensions of Stably Finite C*algebras (II)
For any $C^*$algebra $A$ with an approximate
unit of projections, there is a smallest ideal $I$ of $A$ such
that the quotient $A/I$ is stably finite.
In this paper, a sufficient and necessary condition is obtained
for an ideal of a $C^*$algebra with real rank zero is this smallest
ideal by $K$theory.
Keywords:extension, stably finite C*algebra, index map Categories:46L05, 46L80 

32. CMB 2015 (vol 59 pp. 104)
 He, Ziyi; Yang, Dachun; Yuan, Wen

LittlewoodPaley Characterizations of SecondOrder Sobolev Spaces via Averages on Balls
In this paper, the authors characterize secondorder 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 LittlewoodPaley $g_\lambda^\ast$function in
terms of ball means.
Keywords:Sobolev space, ball means, Lusinarea function, $g_\lambda^*$function Categories:46E35, 42B25, 42B20, 42B35 

33. CMB 2015 (vol 58 pp. 846)
 Sundar, S.

A Computation with the ConnesThom Isomorphism
Let $A \in M_{n}(\mathbb{R})$ be an invertible matrix. Consider
the semidirect 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
ConnesThom map if $\det(A)\gt 0$ and anticommutes if $\det(A)\lt 0$.
As an application, we recompute the $K$groups of the CuntzLi
algebra associated to an integer dilation matrix.
Keywords:Ktheory, ConnesThom isomorphism, CuntzLi algebras Categories:46L80, 58B34 

34. 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
tailbound for a hypergeometric random variable.
Keywords:Khintchine inequality, Kahane inequality, Rademacher random variables, hypergeometric distribution. Categories:46B06, 60E15, 52A23, 46B09 

35. 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 

36. 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 CuntzKrieger
uniqueness theorem for graph algebras by means of branching systems.
Keywords:C*algebra, graph algebra, Leavitt path algebra, branching system, representation Categories:46L05, 37A55 

37. CMB 2015 (vol 58 pp. 757)
38. 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 

39. 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 semifinite $\mathrm{C}^*$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 Murrayvon Neumann semigroup
and the space of densely defined lower semicontinuous traces.
Finally, we prove that these criteria are satisfied by notnecessarilyunital
approximately subhomogeneous algebras of slow dimension growth.
Combined with results of the firstnamed 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 

40. CMB 2015 (vol 58 pp. 350)
 MerinoCruz, 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 

41. 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, KKtheory, Etheory, circle actions Categories:46L55, 19K35 

42. 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
firstorder Sobolev spaces. Moreover, two key pointwise
inequalities for the partial derivatives of the multisublinear
maximal functions are established. As an application, the
quasicontinuity on the multisublinear maximal function is also
obtained.
Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuity Categories:42B25, 46E35 

43. 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 oneparameter groups of surjective isometries Categories:46E15, 47B15, 47B38 

44. CMB 2014 (vol 58 pp. 432)
 Yang, Dachun; Yang, Sibei

Secondorder Riesz Transforms and Maximal Inequalities Associated with Magnetic SchrÃ¶dinger Operators
Let $A:=(\nablai\vec{a})\cdot(\nablai\vec{a})+V$ 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
secondorder Riesz transforms $VA^{1}$ and
$(\nablai\vec{a})^2A^{1}$ are bounded from the
MusielakOrliczHardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$,
to the MusielakOrlicz 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:MusielakOrliczHardy space, magnetic SchrÃ¶dinger operator, atom, secondorder Riesz transform, maximal inequality Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30 

45. CMB 2014 (vol 58 pp. 51)
 De Nitties, Giuseppe; SchulzBaldes, 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 

46. 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 DruryArveson Hilbert module is a prototype) with the property that any
two nested unitarily equivalent submodules are indeed equal.
Keywords:boundary representations, subnormal, joint pisometry Categories:47A13, 46E22 

47. CMB 2014 (vol 58 pp. 150)
 Ostrovskii, Mikhail I.

Connections Between Metric Characterizations of Superreflexivity and the RadonNikodÃ½ Property for Dual Banach Spaces
Johnson and Schechtman (2009)
characterized superreflexivity in terms of finite diamond graphs.
The present author characterized the RadonNikodÃ½m 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, RadonNikodÃ½m property, superreflexivity Categories:46B85, 46B07, 46B22 

48. 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 quasicomplementary, noncomplementary subspaces of
$\ell_\infty$ that are isometric to $\ell_\infty$.
Keywords:$L_1$, Tauberian operator, $\ell_\infty$ Categories:46E30, 46B08, 47A53 

49. 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 

50. CMB 2014 (vol 58 pp. 3)