# Continuity of multiplicative character

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.

-
Does "multiplicative linear functional" mean "homomorphism"? – Nik Weaver Jan 23 '13 at 6:32
What do you mean by "linear"? Where is the vector space structure? – Jochen Wengenroth Jan 23 '13 at 8:08
Sorry. I should have said "character". – nick Jan 23 '13 at 18:37
Anyway I don't even think $\beta G$ is a semigroup in general. How do you define the product in $\beta {\bf Z}$? – Nik Weaver Jan 23 '13 at 19:20
@ 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/… – nick Jan 23 '13 at 20:06