1. CJM Online first
 Chen, Yanni; Hadwin, Don; Liu, Zhe; Nordgren, Eric

A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain
The object of this paper is to prove a version of the BeurlingHelsonLowdenslager
invariant subspace theorem for operators on certain Banach spaces
of functions on a multiply connected domain in $\mathbb{C}$. The norms
for these spaces are either the usual Lebesgue and Hardy space
norms or certain continuous gauge norms.
In the Hardy space case the expected corollaries include the
characterization of the cyclic vectors as the outer functions
in this context, a demonstration that the set of analytic multiplication
operators is maximal abelian and reflexive, and a determination
of the closed operators that commute with all analytic multiplication
operators.
Keywords:Beurling theorem, invariant subspace, generalized Hardy space, gauge norm, multiply connected domain, Forelli projection, innerouter factorization, affiliated operator Categories:47L10, 30H10 

2. CJM Online first
 Giannopoulos, Apostolos; Koldobsky, Alexander; Valettas, Petros

Inequalities for the surface area of projections of convex bodies
We provide general inequalities that compare the surface area
$S(K)$ of a convex body $K$ in ${\mathbb R}^n$
to the minimal, average or maximal surface area of its hyperplane
or lower dimensional projections. We discuss the
same questions for all the quermassintegrals of $K$. We examine
separately the dependence of the constants
on the dimension in the case where $K$ is in some of the classical
positions or $K$ is a projection body.
Our results are in the spirit of the hyperplane problem, with
sections replaced by projections and volume by
surface area.
Keywords:surface area, convex body, projection Categories:52A20, 46B05 

3. CJM 2016 (vol 68 pp. 762)
 Colesanti, Andrea; Gómez, Eugenia Saorín; Nicolás, Jesus Yepes

On a Linear Refinement of the PrÃ©kopaLeindler Inequality
If $f,g:\mathbb{R}^n\longrightarrow\mathbb{R}_{\geq0}$ are nonnegative measurable
functions, then the PrÃ©kopaLeindler inequality asserts that
the integral of the Asplund sum (provided that it is measurable)
is greater or equal than the $0$mean of the integrals of $f$
and $g$.
In this paper we prove that under the sole assumption that $f$
and $g$ have
a common projection onto a hyperplane, the PrÃ©kopaLeindler
inequality admits a linear refinement. Moreover, the same inequality
can be obtained when assuming that both projections (not necessarily
equal as functions) have the same integral. An analogous approach
may be also carried out for the socalled BorellBrascampLieb
inequality.
Keywords:PrÃ©kopaLeindler inequality, linearity, Asplund sum, projections, BorellBrascampLieb inequality Categories:52A40, 26D15, 26B25 

4. CJM 2011 (vol 64 pp. 573)
 Nawata, Norio

Fundamental Group of Simple $C^*$algebras with Unique Trace III
We introduce the fundamental group ${\mathcal F}(A)$ of
a simple $\sigma$unital $C^*$algebra $A$ with unique (up to scalar multiple)
densely defined lower semicontinuous trace.
This is a generalization of ``Fundamental Group of Simple
$C^*$algebras with Unique Trace I and II'' by Nawata and Watatani.
Our definition in this paper makes sense for stably projectionless $C^*$algebras.
We show that there exist separable stably projectionless $C^*$algebras such that
their fundamental groups are equal to $\mathbb{R}_+^\times$
by using the classification theorem of Razak and Tsang.
This is a contrast to the unital case in Nawata and Watatani.
This study is motivated by the work of Kishimoto and Kumjian.
Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function Categories:46L05, 46L08, 46L35 

5. CJM 2011 (vol 63 pp. 1161)
 Neuwirth, Stefan; Ricard, Éric

Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group
We inspect the relationship between relative Fourier
multipliers on noncommutative LebesgueOrlicz spaces of a discrete
group $\varGamma$ and relative ToeplitzSchur multipliers on
SchattenvonNeumannOrlicz classes. Four applications are given:
lacunary sets, unconditional Schauder bases for the subspace of a
Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the
norm of the Hilbert transform and the Riesz projection on
SchattenvonNeumann classes with exponent a power of 2, and the norm of
Toeplitz Schur multipliers on SchattenvonNeumann classes with
exponent less than 1.
Keywords:Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projection Categories:47B49, 43A22, 43A46, 46B28 

6. CJM 2007 (vol 59 pp. 638)
 MacDonald, Gordon W.

Distance from Idempotents to Nilpotents
We give bounds on the distance from a nonzero idempotent to the
set of nilpotents in the set of $n\times n$ matrices in terms of
the norm of the idempotent. We construct explicit idempotents and
nilpotents which achieve these distances, and determine exact
distances in some special cases.
Keywords:operator, matrix, nilpotent, idempotent, projection Categories:47A15, 47D03, 15A30 
