Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group
HTML articles powered by AMS MathViewer
- by Edmond E. Granirer
- Trans. Amer. Math. Soc. 189 (1974), 371-382
- DOI: https://doi.org/10.1090/S0002-9947-1974-0336241-0
- PDF | Request permission
Abstract:
We define for any locally compact group G, the space of bounded uniformly continuous functionals on Ĝ, $UCB(\hat G)$, in the context of P. Eymard [Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208] (for notations see next section). For $u \in A(G)$ let ${u^ \bot } = \{ \phi \in VN(G);\phi [A(G)u] = 0\}$. Theorem. If for some norm separable subspace $X \subset VN(G)$ and some positive definite $0 \ne u \in A(G),UCB(\hat G) \subset$ norm closure $[W(\hat G) + X + {u^ \bot }]$ then G is discrete. If G is discrete then $UCB(\hat G) \subset AP(\hat G) \subset W(\hat G)$.References
- Charles A. Akemann, The dual space of an operator algebra, Trans. Amer. Math. Soc. 126 (1967), 286–302. MR 206732, DOI 10.1090/S0002-9947-1967-0206732-8
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Charles F. Dunkl and Donald E. Ramirez, Existence and nonuniqueness of invariant means on ${\scr L}^{\infty }(\hat G)$, Proc. Amer. Math. Soc. 32 (1972), 525–530. MR 296609, DOI 10.1090/S0002-9939-1972-0296609-1
- Charles F. Dunkl and Donald E. Ramirez, Weakly almost periodic functionals carried by hypercosets, Trans. Amer. Math. Soc. 164 (1972), 427–434. MR 291370, DOI 10.1090/S0002-9947-1972-0291370-3
- Charles F. Dunkl and Donald E. Ramirez, Topics in harmonic analysis, The Appleton-Century Mathematics Series, Appleton-Century-Crofts [Meredith Corporation], New York, 1971. MR 0454515
- Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236 (French). MR 228628, DOI 10.24033/bsmf.1607
- Edmond E. Granirer, Exposed points of convex sets and weak sequential convergence, Memoirs of the American Mathematical Society, No. 123, American Mathematical Society, Providence, R.I., 1972. Applications to invariant means, to existence of invariant measures for a semigroup of Markov operators etc. . MR 0365090
- 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
- Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
- Horst Leptin, Sur l’algèbre de Fourier d’un groupe localement compact, C. R. Acad. Sci. Paris Sér. A-B 266 (1968), A1180–A1182 (French). MR 239002
- Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
- Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto-London, 1969. MR 0251549
- R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
- Shôichirô Sakai, On topological properties of $W^*$-algebras, Proc. Japan Acad. 33 (1957), 439–444. MR 98992
- Charles F. Dunkl and Donald E. Ramirez, Existence and nonuniqueness of invariant means on ${\scr L}^{\infty }(\hat G)$, Proc. Amer. Math. Soc. 32 (1972), 525–530. MR 296609, DOI 10.1090/S0002-9939-1972-0296609-1
- P. F. Renaud, Invariant means on a class of von Neumann algebras, Trans. Amer. Math. Soc. 170 (1972), 285–291. MR 304553, DOI 10.1090/S0002-9947-1972-0304553-0
- Isaac Namioka, Partially ordered linear topological spaces, Mem. Amer. Math. Soc. 24 (1957), 50. MR 94681
Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 189 (1974), 371-382
- MSC: Primary 43A60
- DOI: https://doi.org/10.1090/S0002-9947-1974-0336241-0
- MathSciNet review: 0336241