On topologically invariant means on a locally compact group

Chou, Ching

doi:10.1090/S0002-9947-1970-0269780-8

On topologically invariant means on a locally compact group
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Trans. Amer. Math. Soc. 151 (1970), 443-456 Request permission

Abstract:

Let $\mathcal {M}$ be the set of all probability measures on $\beta N$. Let G be a locally compact, noncompact, amenable group. Then there is a one-one affine mapping of $\mathcal {M}$ into the set of all left invariant means on ${L^\infty }(G)$. Note that $\mathcal {M}$ is a very big set. If we further assume G to be $\sigma$-compact, then we have a better result: The set $\mathcal {M}$ can be embedded affinely into the set of two-sided topologically invariant means on ${L^\infty }(G)$. We also give a structure theorem for the set of all topologically left invariant means when G is $\sigma$-compact.

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