We investigate some aspects of maximum modulus sets in the boundary of a strictly pseudoconvex domain $D$ of dimension $N$. If $\Sigma \subset bD$ is a smooth manifold of dimension $N$ and a maximum modulus set, then it admits a unique foliation by compact interpolation manifolds. There is a semiglobal converse in the real analytic case. Two functions in $A^2\left(D\right)$ with the same smooth $N$-dimensional maximum modulus set are analytically related and are polynomially related if a certain homology class in $H_1(D,\mathbf{R})$ vanishes or if $\bar{D}\subset {\mathbf{C}}^N$ is polynomially...

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 $C\left(B\right)$.