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K be defined by fr(X I , x2,t) = tx I + (1 t - )x 2 . Then the image Qr of fr is closed in E w and convex. Moreover, Qr % K: For otherwise, each extreme point z of K would be of the form z = tx I + ( 1 where x i ~ Ki, would and, t)x 2, i = 1,2, and t ~ [ r , ~ . But that imply that each extreme point of K was in K 1 therefore, Finally, thst K = K1, a contradiction.

Ii) The following functions are continuous: Proof: (i), I. ( g , s ) ~ E s : G x S --~S; 2. (s,g)~gs : S x G --, S; 3. (g,s,h)e~gsh : G x S x G - - ~ S; 4. (g,s) "1~Ssg -I : G x S --~ S (inversion i n G By Ellis' Theorem (ii) (I), and is continuous, Lemma: e&E(T), (ii) (2). Since g ~ g - 1 : G ( ( g , h ) , s ) ~ , g s h -I is a group action. 1) we immediately get --, G : (G x G) x S - - 9 S From this we get Let T be a semitopological (li)~)an8 (ii)(4). semigroup. If then Te, eT, eTe are all retracts of T, and are therefore closed.

G,s) "1~Ssg -I : G x S --~ S (inversion i n G By Ellis' Theorem (ii) (I), and is continuous, Lemma: e&E(T), (ii) (2). Since g ~ g - 1 : G ( ( g , h ) , s ) ~ , g s h -I is a group action. 1) we immediately get --, G : (G x G) x S - - 9 S From this we get Let T be a semitopological (li)~)an8 (ii)(4). semigroup. If then Te, eT, eTe are all retracts of T, and are therefore closed. Proof: Clear. 3 If S is a right group, then S is a topological group, and S is isomorphic topological semigroups) semi- (in the category of to G x Y, where G is a locally compact topological group, and Y is a right zero is (topological) semlgroup.

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Compact Semitopological Semigroups and Weakly Almost Periodic Functions by J. F. Berglund, K. H. Hofmann


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