By J. F. Berglund, K. H. Hofmann
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Extra info for Compact Semitopological Semigroups and Weakly Almost Periodic Functions
Ideal. eI = ele ~- eSe ~ I . 8 (hence, then Se is left simple Consequently, eSe 4T- ideal Thus in S and I = Se. I a right ideal in J. , J is simple; If I is minimal ideal Proof: in S IJ ~_ I; (ii) If J is minimal, (iii) ideal in J, then I is a right in S. (I) (IJ)S = l(JS) c ij. (li) ~ (I) and mlnlmality, (111) For each I E l , IJ = I. 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.
T. Proof: sK = tsK c. tK = stK ~ s K . 1 . 1 9 S u p p o s e S has a c o m p l e t e l y s i m p l e m i n i m a l ideal M(S). ~ S, t h e n the f o l l o w i n g are e q u i v a l e n t : (a) se ~. eS; (b) se = ese; (c) sR c R, w h e r e R is the m i n i m a l (d) sf & fS for all f 6 E(R) If, in a d d i t i o n , equivalent S = T, then right ideal eS = E(M(S))F~R. the a b o v e conditions are to (e) s e K <~eK. Proof: (a)==>(b): so ese = e(et) (b)==>(c): By (a), t h e r e is a t & S w i t h se=et; = e2t = et = se.
Hence, is continuous we have - ~3 - REFERENCES I . : El~ments de Math~matlque. XVIII. Premiere partie. Les structures'fondam~ntales de l'analyse. Livre V. Espaces vectorlelles topologique. Chap. III-V. Actualit~s Sci. et Ind. 1229, Paris: Hermann & Cie. 1955. f / . / . : Elements de ~athematique. XXV. s partle. Les structures fondamentales de l'analyseo Livre VI. Int~gratlon. Chap. V~ Actuallt~s Sci. et Ind. 1281. Paris: Hermann & Cie. 1 959. : Applications of almost periodic compactifications.