The powers of a maximal ideal in a Banach algebra and analytic structure

Author:
T. T. Read

Journal:
Trans. Amer. Math. Soc. **161** (1971), 235-248

MSC:
Primary 46J20

MathSciNet review:
0435853

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Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions are given for the existence of an analytic variety at an element of the spectrum of a commutative Banach algebra with identity. An associated graded algebra first considered by S. J. Sidney is used to determine the dimension of the analytic variety in terms of the closed powers of the maximal ideal which is the kernel of .

**[1]**Andrew Browder,*Point derivations and analytic structure in the spectrum of a Banach algebra*, J. Functional Analysis**7**(1971), 156–164. MR**0273407****[2]**Philip C. Curtis Jr. and Alessandro Figà-Talamanca,*Factorization theorems for Banach algebras*, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 169–185. MR**0203500****[3]**Bernard R. Gelbaum,*Tensor products of Banach algebras*, Canad. J. Math.**11**(1959), 297–310. MR**0104162****[4]**Andrew M. Gleason,*Finitely generated ideals in Banach algebras*, J. Math. Mech.**13**(1964), 125–132. MR**0159241****[5]**Seymour Goldberg,*Unbounded linear operators: Theory and applications*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0200692****[6]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[7]**Marshall Hall Jr.,*Combinatorial theory*, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1967. MR**0224481****[8]**Kenneth Hoffman,*Bounded analytic functions and Gleason parts*, Ann. of Math. (2)**86**(1967), 74–111. MR**0215102****[9]**Lars Hörmander,*An introduction to complex analysis in several variables*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR**0203075****[10]**S. J. Sidney,*Properties of the sequence of closed powers of a maximal ideal in a sup-norm algebra*, Trans. Amer. Math. Soc.**131**(1968), 128–148. MR**0222651**, 10.1090/S0002-9947-1968-0222651-6**[11]**Oscar Zariski and Pierre Samuel,*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR**0120249**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0435853-0

Keywords:
Cmmutative Banach algebra with identity,
powers of a maximal ideal,
analytic structure,
dimension of an analytic variety,
graded algebra

Article copyright:
© Copyright 1971
American Mathematical Society