1. CMB Online first
 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 

2. 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 

3. CMB 2012 (vol 57 pp. 37)
 Dashti, Mahshid; NasrIsfahani, 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 

4. 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 FrÃ©chet
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 FrÃ©chet differentiability, representation Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50 

5. CMB 2011 (vol 54 pp. 654)
 Forrest, Brian E.; Runde, Volker

Norm One Idempotent $cb$Multipliers with Applications to the Fourier Algebra in the $cb$Multiplier Norm
For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely
bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We
characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm
one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we
describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize
those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$amenable in the sense of B. E. Johnson. (We can even slightly
relax the norm bounds.)
Keywords:amenability, bounded approximate identity, $cb$multiplier norm, Fourier algebra, norm one idempotent Categories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25 

6. CMB 2011 (vol 54 pp. 680)
 JiménezVargas, A.; VillegasVallecillos, Moisés

$2$Local Isometries on Spaces of Lipschitz Functions
Let $(X,d)$ be a metric space, and let $\mathop{\textrm{Lip}}(X)$ denote the Banach
space of all scalarvalued bounded Lipschitz functions $f$ on $X$
endowed with one of the natural norms
$
\ f\ =\max \{\ f\ _\infty ,L(f)\}$ or $\f\ =\
f\ _\infty +L(f),
$
where $L(f)$ is the
Lipschitz constant of $f.$ It is said that the isometry
group of $\mathop{\textrm{Lip}}(X)$ is canonical if every
surjective linear isometry of
$\mathop{\textrm{Lip}}(X) $ is induced by a surjective isometry of $X$.
In this paper
we prove that if $X$ is bounded separable and the isometry group of
$\mathop{\textrm{Lip}}(X)$ is canonical, then every $2$local isometry
of $\mathop{\textrm{Lip}}(X)$ is
a surjective linear isometry. Furthermore, we give a complete
description of all $2$local isometries of $\mathop{\textrm{Lip}}(X)$ when $X$ is
bounded.
Keywords:isometry, local isometry, Lipschitz function Categories:46B04, 46J10, 46E15 

7. CMB 2007 (vol 50 pp. 172)
8. CMB 2007 (vol 50 pp. 3)
 Basener, Richard F.

Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra
In this paper we introduce a nested family of spaces of continuous functions defined
on the spectrum of a uniform algebra. The smallest space in the family is the
uniform algebra itself. In the ``finite dimensional'' case, from some point on the
spaces will be the space of all continuous complexvalued functions on the
spectrum. These spaces are defined in terms of solutions to the nonlinear
CauchyRiemann equations as introduced by the author in 1976, so they are not
generally linear spaces of functions. However, these spaces do shed light on the
higher dimensional properties of a uniform algebra. In particular, these spaces are
directly related to the generalized Shilov boundary of the uniform algebra (as
defined by the author and, independently, by Sibony in the early 1970s).
Categories:32A99, 46J10 

9. CMB 2003 (vol 46 pp. 632)
 Runde, Volker

The Operator Amenability of Uniform Algebras
We prove a quantized version of a theorem by M.~V.~She\u{\i}nberg:
A uniform algebra equipped with its canonical, {\it i.e.}, minimal,
operator space structure is operator amenable if and only if it is
a commutative $C^\ast$algebra.
Keywords:uniform algebras, amenable Banach algebras, operator amenability, minimal, operator space Categories:46H20, 46H25, 46J10, 46J40, 47L25 
