Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Asymmetric maximal ideals in $ M(G)$


Author: Sadahiro Saeki
Journal: Trans. Amer. Math. Soc. 222 (1976), 241-254
MSC: Primary 43A10
MathSciNet review: 0415201
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let G be a nondiscrete LCA group, $ M(G)$ the measure algebra of G, and $ {M_0}(G)$ the closed ideal of those measures in $ M(G)$ whose Fourier transforms vanish at infinity. Let $ {\Delta _G},{\Sigma _G}$ and $ {\Delta _0}$ be the spectrum of $ M(G)$, the set of all symmetric elements of $ {\Delta _G}$, and the spectrum of $ {M_0}(G)$, respectively. In this paper this is shown: Let $ \Phi $ be a separable subset of $ M(G)$. Then there exist a probability measure $ \tau $ in $ {M_0}(G)$ and a compact subset X of $ {\Delta _0}\backslash {\Sigma _G}$ such that for each $ \vert c\vert \leqslant 1$ and each

$\displaystyle \nu \in \Phi \;{\text{Card}}\;\{ f \in X:\hat \tau (f) = c\;{\text{and}}\;\vert\hat \nu (f)\vert = r(\nu )\} \geqslant {2^{\text{c}}}.$

Here $ r(\nu ) = \sup \{ \vert\hat \nu (f)\vert:f \in {\Delta _G}\backslash \hat G\} $. As immediate consequences of this result, we have (a) every boundary for $ {M_0}(G)$ is a boundary for $ M(G)$ (a result due to Brown and Moran), (b) $ {\Delta _G}\backslash {\Sigma _G}$ is dense in $ {\Delta _G}\backslash \hat G$, (c) the set of all peak points for $ M(G)$ is $ \hat G$ if G is $ \sigma $-compact and is empty otherwise, and (d) for each $ \mu \in M(G)$ the set $ \hat \mu ({\Delta _0}\backslash {\Sigma _G})$ contains the topological boundary of $ \hat \mu ({\Delta _G}\backslash \hat G)$ in the complex plane.

References [Enhancements On Off] (What's this?)

  • [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
  • [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
  • [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
  • [9] L.-Ȧ. 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
  • [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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 43A10

Retrieve articles in all journals with MSC: 43A10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0415201-7
Keywords: LCA group, measure algebra, asymmetric maximal ideal, $ {M_0}$-boundary
Article copyright: © Copyright 1976 American Mathematical Society