We show that any bounded balanced domain of holomorphy is an $L{²}_h$-domain of holomorphy.

We present various characterizations of n-circled domains of holomorphy $G\subset {ℂ}^n$ with respect to some subspaces of ${\mathscr{H}}^{\infty }\left(G\right)$.

To a pair of a Lie group $G$ and an open elliptic convex cone $W$ in its Lie algebra one associates a complex semigroup $S=G\mathrm{Exp}\left(iW\right)$ which permits an action of $G\times G$ by biholomorphic mappings. In the case where $W$ is a vector space $S$ is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain $D\subseteq S$ is Stein is and only if it is of the form $G\mathrm{Exp}\left(D_h\right)$, with $Dh\subseteq iW$ convex, that each holomorphic function on $D$ extends to the smallest biinvariant Stein domain containing $D$,...

Let $M=G/H$ be a real symmetric space and $𝔤=𝔥+𝔮$ the corresponding decomposition of the Lie algebra. To each open $H$-invariant domain $D_{𝔮}\subseteq i𝔮$ consisting of real ad-diagonalizable elements, we associate a complex manifold $\Xi \left(D_{𝔮}\right)$ which is a curved analog of a tube domain with base $D_{𝔮}$, and we have a natural action of $G$ by holomorphic mappings. We show that $\Xi \left(D_{𝔮}\right)$ is a Stein manifold if and only if $D_{𝔮}$ is convex, that the envelope of holomorphy is schlicht and that $G$-invariant plurisubharmonic functions correspond to convex $H$-invariant...