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Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication but it is not necessarily continuous.

1- Is there any example of a multiplicative character $\gamma :\beta (G)\to \mathbb{T}$ that is not continuos?

2- Is there any concrete example of a multiplicative character $\gamma :\beta (G)\to \mathbb{T}$? I know how to build abstract examples but don't know of any concrete example.

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  • $\begingroup$ Does "multiplicative linear functional" mean "homomorphism"? $\endgroup$ – Nik Weaver Jan 23 '13 at 6:32
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    $\begingroup$ What do you mean by "linear"? Where is the vector space structure? $\endgroup$ – Jochen Wengenroth Jan 23 '13 at 8:08
  • $\begingroup$ Sorry. I should have said "character". $\endgroup$ – nick Jan 23 '13 at 18:37
  • $\begingroup$ Anyway I don't even think $\beta G$ is a semigroup in general. How do you define the product in $\beta {\bf Z}$? $\endgroup$ – Nik Weaver Jan 23 '13 at 19:20
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    $\begingroup$ @ Nik: For any discrete semigroup $(S,.)$, the Stone-Cech compactification $\beta (S)$ of $S$ admits an extension of the operation so that $(\beta S,.)$ is a compact right topological semigroup (meaning right multiplication remains continuous). A good source of information is the book by N.Hindman and D.Struass: books.google.ca/… $\endgroup$ – nick Jan 23 '13 at 20:06

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