1. CMB Online first
 Gilligan, Bruce

Levi's problem for pseudoconvex homogeneous manifolds
Suppose $G$ is a connected complex Lie group and $H$ is a closed
complex subgroup.
Then there exists a closed complex subgroup $J$ of $G$ containing
$H$ such that
the fibration $\pi:G/H \to G/J$ is the holomorphic reduction
of $G/H$, i.e., $G/J$ is holomorphically
separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$.
In this paper we prove that if $G/H$ is pseudoconvex, i.e.,
if
$G/H$ admits a continuous plurisubharmonic exhaustion function,
then $G/J$ is Stein and $J/H$ has no nonconstant holomorphic
functions.
Keywords:complex homogeneous manifold, plurisubharmonic exhaustion function, holomorphic reduction, Stein manifold, Remmert reduction, Hirschowitz annihilator Categories:32M10, 32U10, 32A10, 32Q28 

2. 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 

3. CMB 2011 (vol 55 pp. 242)
 Cegrell, Urban

Convergence in Capacity
In this note we study the convergence of sequences of MongeAmpÃ¨re measures $\{(dd^cu_s)^n\}$,
where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.
Keywords:complex MongeAmpÃ¨re operator, convergence in capacity, plurisubharmonic function Categories:32U20, 31C15 

4. CMB 2004 (vol 47 pp. 481)
 Bekjan, Turdebek N.

A New Characterization of Hardy Martingale Cotype Space
We give a new characterization of Hardy martingale cotype
property of complex quasiBanach space by using the existence of a
kind of plurisubharmonic functions. We also characterize the best
constants of Hardy martingale inequalities with values
in the complex quasiBanach space.
Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic function Categories:46B20, 52A07, 60G44 

5. CMB 2003 (vol 46 pp. 373)
 Laugesen, Richard S.; Pritsker, Igor E.

Potential Theory of the FarthestPoint Distance Function
We study the farthestpoint distance function, which measures the
distance from $z \in \mathbb{C}$ to the farthest point or points of
a given compact set $E$ in the plane.
The logarithm of this distance is subharmonic as a function of $z$,
and equals the logarithmic potential of a unique probability measure
with unbounded support. This measure $\sigma_E$ has many interesting
properties that reflect the topology and geometry of the compact set
$E$. We prove $\sigma_E(E) \leq \frac12$ for polygons inscribed in a
circle, with equality if and only if $E$ is a regular $n$gon for some
odd $n$. Also we show $\sigma_E(E) = \frac12$ for smooth convex sets of
constant width. We conjecture $\sigma_E(E) \leq \frac12$ for all~$E$.
Keywords:distance function, farthest points, subharmonic function, representing measure, convex bodies of constant width Categories:31A05, 52A10, 52A40 
