# Compact Semitopological Semigroups and Weakly Almost by J. F. Berglund, K. H. Hofmann PDF

By J. F. Berglund, K. H. Hofmann

ISBN-10: 3540039139

ISBN-13: 9783540039136

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Extra info for Compact Semitopological Semigroups and Weakly Almost Periodic Functions

Example text

Suppose (I) and mlnlmallty. that e s and Then G = eSe is a group; zero semigroup; Further, IJ = I by I. S, Now IS = (IJ)S = I(JS) c__ IJ = I. 9 Proposition: simple. I c_ j = IJ _~ I for i ~ with Y = E(eS), that eS is righ X = E(Se), a right zero a left semigroup. o- : Y x X --@ G given by o-(y, x) = yx, the function ~e : (X,G,Y)~--* is an inJective S with morphism : s Re(X, g, y) = xgy of semizroups. --9 The function - 48 - defined by ~' ~e(S) = (s(ese) "I , ese, (ese) -I s) (inversion in G = eSe~ is a left inverse of ~e.

Is a semitopological in the topology A(K) has a minimal of K K. ideal M(A(K)); in particular, M(A(K)) = ~ o L ~ A(K) : there is a k ~ K w i t h ~ or is a minimal = ~k~ left ideal. 12 Let K be a compact convex subset of a locally topological vector space E. subsemigroup Suppose T' ~ convex A(K) such that T' is equicontinuous is a on K. Let T be the closure of T' in K K. (i) T is compact in K K and equicontinuous (ii) The uniform and polntwise topologies on K on T are the same. (iii) T ~ A(K). (iv) T is a topological Proof: semigroup.

13 locally establish of the present Proposition: proposition topological semigroup tions that, on K, T is topologically dense in T for all point in K. 6 and infra). prove tT is loss of ~en tT = T for all t a T. T = E(T) where of affine E. of a Then T has a fixed we may assome then we may assume space subset in the pointwlse t s T). case we have independently convex vector be an eguicontinuous Suppose to a chapter. Let K be a compact convex of K. 18 infra, fk = ek = k for all f a E(T). So we may assume group.