×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. M. F. O’Connell, ?Real parts of uniform algebras,? Pacific J. Math.,46, 235-247 (1973). · Zbl 0264.46051
[2] W. P. Nowinger, ?Real parts of uniform algebras on the circle,? Pacific J. Math.,57, No. 1, 259-264 (1975). · Zbl 0301.46039
[3] J. Hamelin, Uniform Algebras [Russian translation], Mir, Moscow (1973).
[4] M. I. Zaslavskaya, ?On some representations of uniform algebras on the unit circle and polytorus,? Uch. Zametki ECU (1989). · Zbl 0882.32007
[5] U. Rudin, Theory of Functions on a Disc [Russian translation], Mir, Moscow (1973). · Zbl 0262.40002
[6] U. Rudin, Theory of Functions on the Unit Ball in C [Russian translation], Mir, Moscow (1984). · Zbl 0597.32001
[7] B. T. Batikyan, ?Logarithms of moduli of invertible elements in a Banach algebra,? Mat. Zametki,23, No. 3, 373-376 (1978).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.