Let Ω be a Greenian domain in ℝ^{$d$},$d$≥2, or—more generally—let Ω be a connected $\mathcal{P}$ -Brelot space satisfying axiom D, and let$u$be a numerical function on Ω, $u\not\equiv\infty$ , which is locally bounded from below. A short proof yields the following result: The function$u$is the infimum of its superharmonic majorants if and only if each set {$x$:$u$($x$)>$t$},$t$∈ℝ, differs from an analytic set only by a polar set and $\int u\,d\mu_{x}^{V}\le u(x)$ , whenever$V$is a relatively compact open set in Ω and$x$∈$V$.

We construct Fatou–Bieberbach domains in $\mathbb{C}^{n}$ for$n$>1 which contain a given compact set$K$and at the same time avoid a totally real affine subspace$L$of dimension <$n$, provided that$K$∪$L$is polynomially convex. By using this result, we show that the domain $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ for 1≤$k$<$n$enjoys the basic Oka property with approximation for maps from any Stein manifold of dimension <$n$.

A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which von Neumann algebras are clean as rings? In this paper, we characterize strongly clean von Neumann algebras and prove that all finite von Neumann algebras and all separable infinite factors are clean.

The circular cone is not self-dual under the standard inner product and includes second-order cone as a special case. In this paper, we focus on the monotonicity of and circular cone monotonicity of <i>f</i>. Their relationship is discussed as well. Our results show that the angle <i></i> plays a different role in these two concepts.

For a wide class of 2-dimensional periodic elliptic operators, we show that the global extrema of all spectral band functions are isolated.