Asymmetric maximal ideals in

Author:
Sadahiro Saeki

Journal:
Trans. Amer. Math. Soc. **222** (1976), 241-254

MSC:
Primary 43A10

DOI:
https://doi.org/10.1090/S0002-9947-1976-0415201-7

MathSciNet review:
0415201

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Abstract: Let *G* be a nondiscrete LCA group, the measure algebra of *G*, and the closed ideal of those measures in whose Fourier transforms vanish at infinity. Let and be the spectrum of , the set of all symmetric elements of , and the spectrum of , respectively. In this paper this is shown: Let be a separable subset of . Then there exist a probability measure in and a compact subset *X* of such that for each and each

*G*is -compact and is empty otherwise, and (d) for each the set contains the topological boundary of in the complex plane.

**[1]**Gavin Brown,*𝑀₀(𝐺) has a symmetric maximal ideal off the Šilov boundary*, Proc. London Math. Soc. (3)**27**(1973), 484–504. MR**0324314**, https://doi.org/10.1112/plms/s3-27.3.484**[2]**Gavin Brown and William Moran,*𝐿^{1/2}(𝐺) is the kernel of the asymmetric maximal ideals of 𝑀(𝐺)*, Bull. London Math. Soc.**5**(1973), 179–186. MR**0338686**, https://doi.org/10.1112/blms/5.2.179**[3]**Gavin Brown and William Moran,*𝑀_{𝑂}(𝐺)-boundaries are 𝑀(𝐺)-boundaries*, J. Functional Analysis**18**(1975), 350–368. MR**0361615****[4]**Theodore W. Gamelin,*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR**0410387****[5]**Colin C. Graham,*Measures vanishing off the symmetric maximal ideals of 𝑀(𝐺)*, Proc. Cambridge Philos. Soc.**75**(1974), 51–61. MR**0344800****[6]**Edwin Hewitt and Shizuo Kakutani,*A class of multiplicative linear functionals on the measure algebra of a locally compact Abelian group*, Illinois J. Math.**4**(1960), 553–574. MR**0123198****[7]**Edwin Hewitt and Karl R. Stromberg,*A remark on Fourier-Stieltjes transforms*, An. Acad. Brasil. Ci.**34**(1962), 175–180. MR**0150536****[8]**Keiji Izuchi and Tetsuhiro Shimizu,*Topologies on groups and a certain 𝐿-ideal of measure algebras*, Tôhoku Math. J. (2)**25**(1973), 53–60. MR**0385461**, https://doi.org/10.2748/tmj/1178241414**[9]**L.-È¦. Lindahl and F. Poulsen (eds.),*Thin sets in harmonic analysis*, Marcel Dekker, Inc., New York, 1971. Seminars held at Institute Mittag-Leffler, Univ. Stockholm, Stockholm, 1969–1970; Lecture Notes in Pure and Applied Mathematics, Vol. 2. MR**0393993****[10]**Walter Rudin,*Fourier analysis on groups*, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR**0152834****[11]**Sadahiro Saeki,*Symmetric maximal ideals in 𝑀(𝐺)*, Pacific J. Math.**54**(1974), no. 1, 229–243. MR**0370058****[12]**Khoichi Saka,*A note on subalgebras of a measure algebra vanishing on non-symmetric homomorphisms*, Tôhoku Math. J. (2)**25**(1973), 333–338. MR**0358221**, https://doi.org/10.2748/tmj/1178241333**[13]**Tetsuhiro Shimizu,*𝐿-ideals of measure algebras*, Proc. Japan Acad.**48**(1972), 172–176. MR**0318780****[14]**Joseph L. Taylor,*Measure algebras*, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R. I., 1973. Expository lectures from the CBMS Regional Conference held at the University of Montana, Missoula, Mont., June 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 16. MR**0427949****[15]**J. H. Williamson,*Banach algebra elements with independent powers, and theorems of Wiener-Pitt type*, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 186–197. MR**0198143**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0415201-7

Keywords:
LCA group,
measure algebra,
asymmetric maximal ideal,
-boundary

Article copyright:
© Copyright 1976
American Mathematical Society