It is proved that the Fréchet algebra $C\left(B\right)$ has exactly three closed subalgebras $Y$ which contain nonconstant functions and which are invariant, in the sense that $f\circ \Psi \in Y$ whenever $f\in Y$ and $\Psi$ is a biholomorphic map of the open unit ball $B$ of ${\mathbf{C}}^n$ onto $B$. One of these consists of the holomorphic functions in $B$, the second consists of those whose complex conjugates are holomorphic, and the third is $C\left(B\right)$.

One proves the density of an ideal of analytic functions into the closure of analytic functions in a $L^p\left(\mu \right)$-space, under some geometric conditions on the support of the measure $\mu$ and the zero variety of the ideal.

For a domain $\Omega \subset {ℂ}^n$ let $H\left(\Omega \right)$ be the holomorphic functions on $\Omega$ and for any $k\in \mathbb{N}$ let $A^k\left(\Omega \right)=H\left(\Omega \right)\cap C^k\left(\overline{\Omega }\right)$. Denote by ${𝒜}_D^k\left(\Omega \right)$ the set of functions $f\Omega \to [0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k\left(\Omega \right)$ such that $\left\{\right|f_j\left|\right\}$ is a nonincreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }\left|f_j\left(z\right)\right|$. By ${𝒜}_I^k\left(\Omega \right)$ denote the set of functions $f\Omega \to (0,\infty )$ with the property that there exists a sequence of functions $f_j\in A^k\left(\Omega \right)$ such that $\left\{\right|f_j\left|\right\}$ is a nondecreasing sequence and such that $f\left(z\right)={lim}_{j\to \infty }\left|f_j\left(z\right)\right|$. Let $k\in \mathbb{N}$ and let ${\Omega }_1$ and ${\Omega }_2$ be bounded $A^k$-domains of holomorphy in ${ℂ}^{m_1}$ and ${ℂ}^{m_2}$ respectively. Let $g_1\in {𝒜}_D^k\left({\Omega }_1\right)$, $g_2\in {𝒜}_I^k\left({\Omega }_1\right)$ and $h\in {𝒜}_D^k\left({\Omega }_2\right)\cap {𝒜}_I^k\left({\Omega }_2\right)$. We prove that the...

We study some properties of the maximal ideal space of the bounded holomorphic functions in several variables. Two examples of bounded balanced domains are introduced, both having non-trivial maximal ideals.