## On invariant algebras of continuous functions on balls and Euclidean spaces.(Russian)Zbl 0578.46052

Let $${\mathcal M}$$ denote the group of automorphisms of the unit ball B in $${\mathbb{C}}^ n$$. [We use the notation of [*] $$= ''Function$$ Theory in the Unit Ball of $${\mathbb{C}}^ n$$”, by W. Rudin (1980; Zbl 0495.32001)] A set of functions X on B is said to be $${\mathcal M}$$-invariant if $$f\circ \psi \in X$$ whenever $$f\in X$$ and $$\psi\in {\mathcal M}$$. The closed $${\mathcal M}$$-invariant subalgebras of $${\mathcal C}(\bar B)$$ were classified by Nagel and Rudin, but their work left open the question of a similar classification of $${\mathcal C}(B)$$ (see [*], Ch. 13). This was resolved by W. Rudin, Ann. Inst. Fourier 33, no. 2, 19-41 (1983; Zbl 0487.32012) and independently by the present author, who also considers the $${\mathcal M}$$-invariant subspaces of $$X_ 0$$, the space of invariant harmonic functions on B.
Theorem 1: The only closed $${\mathcal M}$$-invariant subspaces of $$X_ 0$$ are $$X_ 0$$ itself, $$p\ell h = the$$ space of pluriharmonic functions on B, H(B), conj H(B), $${\mathbb{C}}$$ and $$\{$$ $$0\}$$.
Theorem 2: The only closed $${\mathcal M}$$-invariant subalgebras of $${\mathcal C}(B)$$ are $${\mathcal C}(B)$$, H(B), conj H(B), $${\mathbb{C}}$$ and $$\{$$ $$0\}$$. Corollary: Let $$K\subset \subset B$$ be such that $${\mathcal P}(K)\neq {\mathcal C}(K)$$ and let $$f\in {\mathcal C}(B)$$. If $$f\circ \phi |_ K$$ is approximable by polynomials for all $$\phi$$ in $${\mathcal M}$$ then $$f\in H(B)$$. (This answers a question raised in [*] 13.4.6).
Analogues of Theorems 1 and 2 are given with B replaced by $${\mathbb{C}}^ n$$ and $${\mathcal M}$$ replaced by the group of all transformations of the form $$z\to Uz+a$$ where $$a\in {\mathbb{C}}^ n$$ and U is a unitary operator.
Reviewer: J.R.Shackell

### MSC:

 46J30 Subalgebras of commutative topological algebras 46J10 Banach algebras of continuous functions, function algebras 32A38 Algebras of holomorphic functions of several complex variables

### Citations:

Zbl 0495.32001; Zbl 0487.32012
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