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Real parts of functional algebras. (English. Russian original) Zbl 0763.46047
Math. Notes 50, No. 3, 898-902 (1991); translation from Mat. Zametki 50, No. 3, 20-26 (1991).
Let $$n\geq 2$$, $$T_ r^ n$$ be the torus with vector-radius $$r$$ in $$\mathbb{C}^ n$$ and $$S^ n$$ be the unit sphere in $$\mathbb{C}^ n$$. Let $$A_ 1$$ be the algebra of such $$\mathbb{C}$$-valued continuous functions on $$T_ r^ n$$ which are analytic in the open ball (with vector-radius $$r$$) of $$\mathbb{C}^ n$$, $$A_ 2$$ be the algebra of such $$\mathbb{C}$$-valued continuous functions on $$S^ n$$ which are analytic in the open unit ball of $$\mathbb{C}^ n$$ and $$B_ 1(B_ 2)$$ be an algebra of $$\mathbb{C}$$-valued functions on $$T_ r^ n$$ (respectively on $$S^ n$$) which contains constants and separates the points of $$T_ r^ n$$ (respectively of $$S^ n$$). For each algebra $$A$$ of $$\mathbb{C}$$-valued functions on a subset of $$\mathbb{C}^ n$$ let $$\text{Re }A=\{\text{Re }f$$: $$f\in A\}$$ and $$\bar A=\{\bar f$$: $$f\in A\}$$ where $$\text{Re }f$$ denotes the real part of $$f$$ and $$\bar f$$ denotes the conjugate function of $$f$$. It is proved that either $$B_ k=A_ k$$ or $$B_ k=\bar A_ k$$ if $$\text{Re }B_ k=\text{Re }A_ k$$ with $$k=1,2$$.
Reviewer: M.Abel (Tartu)
##### MSC:
 46J10 Banach algebras of continuous functions, function algebras
##### Keywords:
real parts of functional algebras
Full Text:
##### References:
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