1. CMB Online first
 Abrahamsen, Trond A.; Hájek, Petr; Nygaard, Olav; Troyanski, Stanimir L.

Strongly extreme points and approximation properties
We show that if $x$ is a strongly extreme point of a bounded closed
convex subset of a Banach space and the identity has a geometrically
and topologically good enough local approximation at $x$, then $x$
is already a denting point. It turns out that such an approximation
of the identity exists at any strongly extreme point of the unit
ball of a Banach space with the unconditional compact approximation
property. We also prove that every Banach space with a Schauder
basis can be equivalently renormed to satisfy the sufficient
conditions mentioned.
Keywords:denting point, strongly extreme point, unconditional compact approximation property Categories:46B20, 46B04 

2. CMB 2017 (vol 60 pp. 855)
3. CMB Online first
 Bénéteau, Catherine Anne; Fleeman, Matthew C.; Khavinson, Dmitry S.; Seco, Daniel; Sola, Alan

Remarks on inner functions and optimal approximants
We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to $1/f$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modified to produce inner functions.
Keywords:inner function, reproducing Kernel Hilbert Space, operatortheoretic function theory Categories:46E22, 30J05 

4. CMB 2017 (vol 60 pp. 673)
 Abtahi, Fatemeh; Azizi, Mohsen; Rejali, Ali

Character Amenability of the Intersection of Lipschitz Algebras
Let $(X,d)$ be a metric space and $J\subseteq [0,\infty)$ be
nonempty. We study the structure of the arbitrary intersections
of
Lipschitz algebras, and define a special Banach subalgebra of
$\bigcap_{\gamma\in J}\operatorname{Lip}_\gamma X$, denoted by
$\operatorname{ILip}_J X$. Mainly,
we investigate $C$character amenability of $\operatorname{ILip}_J X$, in
particular Lipschitz algebras. We address a gap in the proof
of a
recent result in this field. Then we remove this gap, and obtain
a
necessary and sufficient condition for $C$character amenability
of $\operatorname{ILip}_J X$, specially Lipschitz algebras, under an additional
assumption.
Keywords:amenability, character amenability, Lipschitz algebra, metric space Categories:46H05, 46J10, 11J83 

5. CMB Online first
6. CMB Online first
 Figiel, Tadeusz; Johnson, William

Quotients of Essentially Euclidean Spaces
A precise quantitative version of the following qualitative statement
is proved: If a finite dimensional normed space contains approximately
Euclidean subspaces of all proportional dimensions, then every
proportional dimensional quotient space has the same property.
Keywords:essentially euclidean space Categories:46B20, 46B07, 46B99 

7. CMB 2017 (vol 60 pp. 402)
 Shravan Kumar, N.

Invariant Means on a Class of von Neumann Algebras Related to Ultraspherical Hypergroups II
Let $K$ be an ultraspherical hypergroup associated to a locally
compact group $G$ and a spherical projector $\pi$ and let $VN(K)$
denote the dual of the Fourier algebra $A(K)$ corresponding to
$K.$ In this note, we show that the set of invariant means on
$VN(K)$ is singleton if and only if $K$ is discrete. Here $K$
need not be second countable. We also study invariant means on
the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$
in the $cb$multiplier norm. Finally, we consider generalized
translations and generalized invariant means.
Keywords:ultraspherical hypergroup, Fourier algebra, FourierStieltjes algebra, invariant mean, generalized translation, generalized invariant mean Categories:43A62, 46J10, 43A30, 20N20 

8. CMB Online first
 Józiak, Paweł

Remarks on Hopf images and quantum permutation groups $S_n^+$
Motivated by a question of A. Skalski and P.M. SoÅtan (2016)
about inner faithfulness of the S. Curran's map of extending
a quantum increasing sequence to a quantum permutation, we revisit
the results and techniques of T. Banica and J. Bichon (2009)
and study some grouptheoretic properties of the quantum permutation
group on $4$ points. This enables us not only to answer the aforementioned
question in positive in case $n=4, k=2$, but also to classify
the automorphisms of $S_4^+$, describe all the embeddings $O_{1}(2)\subset
S_4^+$ and show that all the copies of $O_{1}(2)$ inside $S_4^+$
are conjugate. We then use these results to show that the converse
to the criterion we applied to answer the aforementioned question
is not valid.
Keywords:Hopf image, quantum permutation group, compact quantum group Categories:20G42, 81R50, 46L89, 16W35 

9. CMB Online first
 Haralampidou, Marina; Oudadess, Mohamed; Palacios, Lourdes; Signoret, Carlos

A characterization of $C^{\ast}$normed algebras via positive functionals
We give a characterization of $C^{\ast}$normed algebras, among
certain involutive normed ones. This is done through the existence
of enough specific positive functionals. The same question is
also
examined in some non normed (topological) algebras.
Keywords:$C^{\ast}$normed algebra, $C^*$algebra, (pre)locally $C^*$algebra, pre$C^*$bornological algebra, positive functional, locally uniformly $A$convex algebra, perfect locally $m$convex algebra, $C^*$(resp. $^*$) subnormable algebra Categories:46H05, 46K05 

10. CMB 2017 (vol 60 pp. 449)
 Alaghmandan, Mahmood; Crann, Jason

Character Density in Central Subalgebras of Compact Quantum Groups
We investigate quantum group generalizations
of various density results from Fourier analysis on compact groups.
In particular, we establish the density of characters in the
space of fixed points of the conjugation action on $L^2(\mathbb{G})$, and
use this result to show the weak* density and norm density of
characters in $ZL^\infty(\mathbb{G})$ and $ZC(\mathbb{G})$, respectively. As a corollary,
we partially answer an open question of Woronowicz.
At the level of $L^1(\mathbb{G})$, we show that the center
$\mathcal{Z}(L^1(\mathbb{G}))$
is precisely the closed linear span of the quantum characters
for a large class of compact quantum groups, including arbitrary
compact Kac algebras. In the latter setting, we show, in addition,
that $\mathcal{Z}(L^1(\mathbb{G}))$ is a completely complemented
$\mathcal{Z}(L^1(\mathbb{G}))$submodule
of $L^1(\mathbb{G})$.
Keywords:compact quantum group, irreducible character Categories:43A20, 43A40, 46J40 

11. CMB 2017 (vol 60 pp. 791)
 Jiang, Chunlan

Reduction to Dimension Two of Local Spectrum for $AH$ Algebra with Ideal Property
A $C^{*}$algebra $A$ has the ideal property if any ideal
$I$ of $A$ is generated as a closed two sided ideal by the projections
inside the ideal. Suppose that the limit $C^{*}$algebra $A$
of inductive limit of direct sums of matrix algebras over spaces
with uniformly bounded dimension has ideal property. In this
paper we will prove that $A$ can be written as an inductive limit
of certain very special subhomogeneous algebras, namely, direct
sum of dimension drop interval algebras and matrix algebras over
2dimensional spaces with torsion $H^{2}$ groups.
Keywords:AH algebra, reduction, local spectrum, ideal property Category:46L35 

12. CMB 2017 (vol 60 pp. 690)
 Bao, Guanlong; Göğüş, Nıhat Gökhan; Pouliasis, Stamatis

$\mathcal{Q}_p$ Spaces and Dirichlet Type Spaces
In this paper, we show that the MÃ¶bius invariant
function space $\mathcal {Q}_p$ can be generated by variant
Dirichlet type spaces
$\mathcal{D}_{\mu, p}$ induced by finite positive Borel measures
$\mu$ on the open unit disk. A criterion for the equality between
the space $\mathcal{D}_{\mu, p}$ and the usual Dirichlet type
space $\mathcal {D}_p$ is given. We obtain a sufficient condition
to construct different $\mathcal{D}_{\mu, p}$ spaces
and we provide examples.
We establish decomposition theorems for $\mathcal{D}_{\mu,
p}$ spaces, and prove that the nonHilbert space $\mathcal
{Q}_p$ is equal to the intersection of Hilbert spaces $\mathcal{D}_{\mu,
p}$. As an application of the relation between $\mathcal {Q}_p$
and $\mathcal{D}_{\mu, p}$ spaces, we also obtain that there
exist different $\mathcal{D}_{\mu, p}$ spaces; this is a trick
to prove the existence without constructing examples.
Keywords:$\mathcal {Q}_p$ space, Dirichlet type space, MÃ¶bius invariant function space Categories:30H25, 31C25, 46E15 

13. CMB 2017 (vol 60 pp. 816)
 Moslehian, Mohammad Sal; Zamani, Ali

Characterizations of Operator BirkhoffJames Orthogonality
In this paper, we obtain some characterizations of the (strong)
BirkhoffJames orthogonality for elements of Hilbert $C^*$modules
and certain elements of $\mathbb{B}(\mathscr{H})$.
Moreover, we obtain a kind of Pythagorean relation for bounded
linear operators.
In addition, for $T\in \mathbb{B}(\mathscr{H})$ we prove that if the
norm attaining
set $\mathbb{M}_T$ is a unit sphere of some finite dimensional
subspace $\mathscr{H}_0$ of $\mathscr{H}$ and $\T\_{{{\mathscr{H}}_0}^\perp}
\lt \T\$, then for every $S\in\mathbb{B}(\mathscr{H})$, $T$ is the strong
BirkhoffJames orthogonal to $S$ if and only if there exists
a unit vector $\xi\in {\mathscr{H}}_0$ such that $\T\\xi =
T\xi$ and $S^*T\xi = 0$.
Finally, we introduce a new type of approximate orthogonality
and investigate this notion in the setting of inner product $C^*$modules.
Keywords:Hilbert $C^*$module, BirkhoffJames orthogonality, strong BirkhoffJames orthogonality, approximate orthogonality Categories:46L05, 46L08, 46B20 

14. CMB 2016 (vol 60 pp. 655)
 Zhuo, Ciqiang; Sickel, Winfried; Yang, Dachun; Yuan, Wen

Characterizations of BesovType and TriebelLizorkinType Spaces via Averages on Balls
Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article,
the authors establish
equivalent characterizations
of Besovtype spaces, TriebelLizorkintype
spaces and BesovMorrey spaces via the sequence
$\{fB_{\ell,2^{k}}f\}_{k}$ consisting of the difference between
$f$ and
the ball average $B_{\ell,2^{k}}f$. These results give a way
to introduce Besovtype spaces,
TriebelLizorkintype spaces and BesovMorrey spaces with any
smoothness order
on metric measure spaces. As special cases, the authors obtain
a new characterization of MorreySobolev spaces
and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent
interest.
Keywords:Besov space, TriebelLizorkin space, ball average, CalderÃ³n reproducing formula Categories:42B25, 46E35, 42B35 

15. CMB 2016 (vol 60 pp. 350)
 Ma, Yumei

Isometry on Linear $n$Gquasi Normed Spaces
This paper generalizes the Aleksandrov problem: the MazurUlam
theorem on $n$Gquasi normed spaces. It proves that a one$n$distance
preserving mapping is an $n$isometry if and only if it has the
zero$n$Gquasi preserving property, and two kinds of $n$isometries
on $n$Gquasi normed space are equivalent; we generalize the
Benz theorem to nnormed spaces with no restrictions on the dimension
of spaces.
Keywords:$n$Gquasi norm, MazurUlam theorem, Aleksandrov problem, $n$isometry, $n$0distance Categories:46B20, 46B04, 51K05 

16. CMB 2016 (vol 60 pp. 217)
 Wang, Yuanyi

Condition $C'_{\wedge}$ of Operator Spaces
In this paper, we study condition $C'_{\wedge}$ which is a
projective tensor product analogue of condition $C'$. We show
that
the finitedimensional OLLP operator spaces have condition
$C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property.
Keywords:operator space, local theory, tensor product Category:46L07 

17. CMB 2016 (vol 60 pp. 104)
18. CMB 2016 (vol 60 pp. 122)
 Ghanei, Mohammad Reza; NasrIsfahani, Rasoul; Nemati, Mehdi

A Homological Property and Arens Regularity of Locally Compact Quantum Groups
We characterize two important notions of amenability and compactness
of
a locally compact quantum group ${\mathbb G}$ in terms of certain
homological
properties. For this, we show that ${\mathbb G}$ is character
amenable if and only if it is both amenable and coamenable.
We finally apply our results to
Arens regularity problems of the quantum group algebra
$L^1({\mathbb G})$; in particular, we improve an interesting result
by Hu, Neufang and Ruan.
Keywords:amenability, Arens regularity, coamenability, locally compact quantum group, homological property Categories:46L89, 43A07, 46H20, 46M10, 58B32 

19. CMB 2016 (vol 60 pp. 173)
 Oubbi, Lahbib

On Ulam Stability of a Functional Equation in Banach Modules
Let $X$ and $Y$ be Banach spaces and $f : X \to Y$ an odd mapping.
For any rational number $r \ne 2$, C. Baak, D. H.
Boo, and Th. M. Rassias have proved the HyersUlam stability
of the following functional equation:
\begin{align*}
r f
\left(\frac{\sum_{j=1}^d x_j}{r}
\right)
& + \sum_{\substack{i(j) \in \{0,1\}
\\ \sum_{j=1}^d i(j)=\ell}} r f
\left(
\frac{\sum_{j=1}^d (1)^{i(j)}x_j}{r}
\right)
= (C^\ell_{d1}  C^{\ell 1}_{d1} + 1) \sum_{j=1}^d
f(x_j)
\end{align*}
where $d$ and $\ell$ are positive integers so that $1 \lt \ell
\lt \frac{d}{2}$, and $C^p_q := \frac{q!}{(qp)!p!}$,
$p, q \in \mathbb{N}$ with $p \le q$.
In this note we solve this equation for arbitrary nonzero scalar
$r$ and show that it is actually HyersUlam stable.
We thus extend and generalize Baak et al.'s result.
Different questions concerning the *homomorphisms and the
multipliers between C*algebras are also
considered.
Keywords:linear functional equation, HyersUlam stability, Banach modules, C*algebra homomorphisms. Categories:39A30, 39B10, 39A06, 46Hxx 

20. CMB 2016 (vol 60 pp. 586)
 Liu, Feng; Wu, Huoxiong

Endpoint Regularity of Multisublinear Fractional Maximal Functions
In this paper we investigate
the endpoint regularity properties of the multisublinear
fractional maximal operators, which include the multisublinear
HardyLittlewood maximal operator. We obtain some new bounds
for the derivative of the onedimensional multisublinear
fractional maximal operators acting on vectorvalued function
$\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$functions.
Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variation Categories:42B25, 46E35 

21. 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 

22. 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 

23. 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 

24. 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 

25. 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 
