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Search: All articles in the CMB digital archive with keyword UBH

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1. CMB 2017 (vol 60 pp. 705)

Benelkourchi, Slimane
Envelope Approach to Degenerate Complex Monge-Ampère Equations on Compact Kähler Manifolds
We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.

Keywords:degenerate complex Monge-Ampère equation, compact Kähler manifold, big cohomology, plurisubharmonic function
Categories:32W20, 32Q25, 32U05

2. CMB 2017 (vol 60 pp. 736)

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 non--constant holomorphic functions.

Keywords:complex homogeneous manifold, plurisubharmonic exhaustion function, holomorphic reduction, Stein manifold, Remmert reduction, Hirschowitz annihilator
Categories:32M10, 32U10, 32A10, 32Q28

3. 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 semi-finite $\mathrm{C}^*$-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 Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first-named 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

4. CMB 2011 (vol 55 pp. 242)

Cegrell, Urban
Convergence in Capacity
In this note we study the convergence of sequences of Monge-Ampère measures $\{(dd^cu_s)^n\}$, where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.

Keywords:complex Monge-Ampère operator, convergence in capacity, plurisubharmonic function
Categories:32U20, 31C15

5. 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 quasi-Banach 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 quasi-Banach space.

Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic function
Categories:46B20, 52A07, 60G44

6. CMB 2003 (vol 46 pp. 373)

Laugesen, Richard S.; Pritsker, Igor E.
Potential Theory of the Farthest-Point Distance Function
We study the farthest-point 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

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