### A Generalization of a Theorem of Frostman.

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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 $C\left(B\right)$.

A simple theorem is proved which states a sufficient condition for the sum ot two closed subspaces of a Banach space to be closed. This leads to several analogues of Sarason’s theorem which states that ${H}^{\infty}+C$ is a closed subalgebra of ${L}^{\infty}$. In these analogues, the unit circle is replaces by other groups, and the unit disc is replaced by polydiscs or by balls in spaces of several complex variables. Sums of closed ideals in Banach algebras are also studied.

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