An ergodic property of locally compact amenable semigroups
1973; Mathematical Sciences Publishers; Volume: 48; Issue: 2 Linguagem: Inglês
10.2140/pjm.1973.48.615
ISSN1945-5844
Autores Tópico(s)Advanced Topology and Set Theory
ResumoLet M(S) be the Banach algebra of all bounded regular Borel measures on a locally compact semigroup S with variation norm and convolution as multiplication and M 0 (S) the probability measures in M(S).We obtain necessary and sufficient conditions for the semigroup S to have the (ergodic) property that for each veM(S), \u(S)\ = inf {\\v*μ\\: μeM 0 (S)}, an extension of a known result for locally compact groups.1* Notations and terminologies* We shall follow Hewitt and Ross [9] for basic notations and terminologies concerning integration on locally compact spaces.Let S be a locally compact semigroup with jointly continuous multiplication and M(S) the Banach algebra of all bounded regular Borel measures on S with total variation norm and convolution μ*v, μ, v e M(S) as multiplication where fdμ*ι> = \^f{xy)dμ{x)dv{y) = \^f{xy)dv{y)dμ{x) for / e C 0 (S) the space of all continuous functions on S which vanish at infinity.(See for example [1], [6], or [18].)Let M 0 (S) = {μe M(S):μ^ 0 and \\μ\\ = 1} be the set of all probability measures in M(S).Consider the continuous dual M(S)* of M(S).Denote by 1 the element in M(S)* such that l(μ) = \dμ = μ(S), μ e M(S).Clearly |||| 1 2* Convolution of functionals and measures, means* Let F e M(S)*,μeM(S), we define a linear functional l μ F= μ®F on M(S) by μ ® F(v) = F(μ * v), v e M(S).Clearly μ®Fe M(S)*.In fact \\μ®F\\ ^ \\μ\\-\\F\\.Similarly we define F®μ = r μ F. A linear functional ΛfeΛf(S)** is called a mean if M{F) ^ 0 if F ^ 0 and M(l) = 1.Here F ^ 0 means F(μ) ^ 0 for all μ^O in M(S).An equivalent definition is inf {F(μ): μ e M 0 (S)} ^ M(F) £ sup {F(μ): μ e M 0 (S)} for any FeM(S)*.Consequently ||Λf|| = M(l) = 1 for any mean M on M(S)*.It follows that the set of means is weak* compact and convex.Each probability measure μeM 0 (S) is a mean if we put μ(F) = F(μ),Fe
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