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**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.

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 |