The Šilov boundary of 𝑀₀(𝐺)

Moran, William

doi:10.1090/S0002-9947-1973-0318779-4

The Šilov boundary of $M_{0}(G)$
HTML articles powered by AMS MathViewer

by William Moran PDF

Trans. Amer. Math. Soc. 179 (1973), 455-464 Request permission

Abstract:

Let G be a locally compact abelian group and let ${M_0}(G)$ be the convolution algebra consisting of those Radon measures on G whose Fourier-Stieltjes transforms vanish at infinity. It is shown that the Šilov boundary of ${M_0}(G)$ is a proper subset of the maximal ideal space of ${M_0}(G)$. The measures constructed to prove this theorem are also used to obtain a stronger result for the full measure algebra $M(G)$.

References

W. J. Bailey, G. Brown, and W. Moran, Spectra of independent power measures, Proc. Cambridge Philos. Soc. 72 (1972), 27–35. MR 296608, DOI 10.1017/s0305004100050921
N. K. Bari, Trigonometricheskie ryady, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). With the editorial collaboration of P. L. Ul′janov. MR 0126115
J. R. Blum and Bernard Epstein, On the Fourier transforms of an interesting class of measures, Israel J. Math. 10 (1971), 302–305. MR 298341, DOI 10.1007/BF02771647
G. Brown and W. Moran, A dichotomy of infinite convolutions of discrete measures, Proc. Cambridge Philos. Soc. 73 (1973), 307–316. MR 312153, DOI 10.1017/s0305004100076878
Charles F. Dunkl and Donald E. Ramirez, Homomorphisms on groups and induced maps on certain algebras of measures, Trans. Amer. Math. Soc. 160 (1971), 475–485. MR 283129, DOI 10.1090/S0002-9947-1971-0283129-7
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496, DOI 10.1007/978-1-4419-8638-2
B. E. Johnson, The Šilov boundary of $M(G)$, Trans. Amer. Math. Soc. 134 (1968), 289–296. MR 230048, DOI 10.1090/S0002-9947-1968-0230048-8
B. E. Johnson, Symmetric maximal ideals in $M(G)$, Proc. Amer. Math. Soc. 18 (1967), 1040–1044. MR 218839, DOI 10.1090/S0002-9939-1967-0218839-5
Yu. A. Šreĭder, The structure of maximal ideals in rings of measures with convolution, Mat. Sbornik N.S. 27(69) (1950), 297–318 (Russian). MR 0038567
Joseph L. Taylor, The structure of convolution measure algebras, Trans. Amer. Math. Soc. 119 (1965), 150–166. MR 185465, DOI 10.1090/S0002-9947-1965-0185465-9
Joseph L. Taylor, Inverses, logarithms, and idempotents in $M(G)$, Rocky Mountain J. Math. 2 (1972), no. 2, 183–206. MR 296610, DOI 10.1216/RMJ-1972-2-2-183