Invariant means on the continuous bounded functions

Rosenblatt, Joseph

doi:10.1090/S0002-9947-1978-0473714-8

Invariant means on the continuous bounded functions
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Trans. Amer. Math. Soc. 236 (1978), 315-324 Request permission

Abstract:

Let G be a noncompact nondiscrete $\sigma$-compact locally compact metric group. A Baire category argument gives measurable sets $\{ {A_\gamma }:\gamma \in \Gamma \}$ of finite measure with card $(\Gamma ) = c$ which are independent on the open sets. One approximates $\{ {A_\gamma }:\gamma \in \Gamma \}$ by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions $\{ {f_\gamma }:\gamma \in \Gamma \}$ in ${\text {CB}}(G)$ with a certain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on ${\text {CB}}(G)$ there are ${2^c}$ mutually singular elements of LIM each of which is singular to TLIM.

References

Ching Chou, On the size of the set of left invariant means on a semi-group, Proc. Amer. Math. Soc. 23 (1969), 199–205. MR 247444, DOI 10.1090/S0002-9939-1969-0247444-1
Ching Chou, On topologically invariant means on a locally compact group, Trans. Amer. Math. Soc. 151 (1970), 443–456. MR 269780, DOI 10.1090/S0002-9947-1970-0269780-8
Ching Chou, The exact cardinality of the set of invariant means on a group, Proc. Amer. Math. Soc. 55 (1976), no. 1, 103–106. MR 394036, DOI 10.1090/S0002-9939-1976-0394036-3
K. de Leeuw and I. Glicksberg, The decomposition of certain group representations, J. Analyse Math. 15 (1965), 135–192. MR 186755, DOI 10.1007/BF02787692
Edmond Granirer, On the invariant mean on topological semigroups and on topological groups, Pacific J. Math. 15 (1965), 107–140. MR 209388, DOI 10.2140/pjm.1965.15.107
Edmond Granirer, Criteria for compactness and for discreteness of locally compact amenable groups, Proc. Amer. Math. Soc. 40 (1973), 615–624. MR 340962, DOI 10.1090/S0002-9939-1973-0340962-8
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
Teng Sun Liu and Arnoud van Rooij, Invariant means on a locally compact group, Monatsh. Math. 78 (1974), 356–359. MR 358218, DOI 10.1007/BF01294646
Joseph Max Rosenblatt, Invariant means and invariant ideals in $L_{\infty }(G)$ for a locally compact group $G$, J. Functional Analysis 21 (1976), no. 1, 31–51. MR 0397304, DOI 10.1016/0022-1236(76)90027-6
Joseph Max Rosenblatt, Invariant means for the bounded measurable functions on a non-discrete locally compact group, Math. Ann. 220 (1976), no. 3, 219–228. MR 397305, DOI 10.1007/BF01431093
Joseph Max Rosenblatt, The number of extensions of an invariant mean, Compositio Math. 33 (1976), no. 2, 147–159. MR 435729
Ralph A. Raimi, On Banach’s generalized limits, Duke Math. J. 26 (1959), 17–28. MR 117569
Walter Rudin, Invariant means on $L^{\infty }$, Studia Math. 44 (1972), 219–227. MR 304975, DOI 10.4064/sm-44-3-219-227
Walter Rudin, Homomorphisms and translations in $L^{\infty }(G)$, Advances in Math. 16 (1975), 72–90. MR 367555, DOI 10.1016/0001-8708(75)90101-2
Benjamin B. Wells Jr., Homomorphisms and translates of bounded functions, Duke Math. J. 41 (1974), 35–39. MR 336238