Vol. 125, No. 2, 1986

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Exposed points of left invariant means

Zhuocheng Yang

Vol. 125 (1986), No. 2, 487–494

If S is a left amenable semigroup, let ML(S) be the set of left invariant means on m(S), the space of bounded real-valued functions on S. We prove in this paper that a left invariant mean on m(S) is an exposed point of ML(S) if and only if it is the arithmetic average on a minimal finite left ideal of S. In particular, ML(S) has no exposed point when S is an infinite group. We also prove that if ML(S) has an exposed point, then it is the w-closed convex hull of all its exposed points. This gives another proof of the Granirer-Klawe theorem on the dimension of ML(S).

Mathematical Subject Classification 2000
Primary: 43A07
Secondary: 22A20
Received: 20 September 1985
Published: 1 December 1986
Zhuocheng Yang