Nonseparability of quotient spaces of function algebras on topological semigroups
Author:
Heneri A. M. Dzinotyiweyi
Journal:
Trans. Amer. Math. Soc. 272 (1982), 223235
MSC:
Primary 43A60; Secondary 22A20
MathSciNet review:
656487
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Abstract: Let be a topological semigroup, the space of all bounded realvalued continuous functions on . We define the subspace of consisting of all weakly uniformly continuous functions and the space of all weakly almost periodic functions in . Among other results, for a large class of topological semigroups , for which noncompact locally compact topological groups are a very special case, we prove that the quotient spaces and, for nondiscrete , are nonseparable. (The actual setting of these results is more general.) For locally compact topological groups, parts of our results answer affirmatively certain questions raised earlier by Ching Chou and E. E. Granirer.
 [1]
Anne
C. Baker and J.
W. Baker, Algebras of measures on a locally compact semigroup.
III, J. London Math. Soc. (2) 4 (1972),
685–695. MR 0306806
(46 #5928)
 [2]
R.
B. Burckel, Weakly almost periodic functions on semigroups,
Gordon and Breach Science Publishers, New York, 1970. MR 0263963
(41 #8562)
 [3]
Ching
Chou, Weakly almost periodic functions and
almost convergent functions on a group, Trans.
Amer. Math. Soc. 206 (1975), 175–200. MR 0394062
(52 #14868), http://dx.doi.org/10.1090/S00029947197503940628
 [4]
H.
A. M. Dzinotyiweyi, Algebras of measures on 𝐶distinguished
topological semigroups, Math. Proc. Cambridge Philos. Soc.
84 (1978), no. 2, 323–336. MR 0493158
(58 #12189)
 [5]
Heneri
A. M. Dzinotyiweyi, Weakly almost periodic functions and the
irregularity of multiplication in semigroup algebras, Math. Scand.
46 (1980), no. 1, 157–172. MR 585239
(82d:43006a)
 [6]
Heneri
A. M. Dzinotyiweyi, Continuity of semigroup actions on normed
linear spaces, Quart. J. Math. Oxford Ser. (2) 31
(1980), no. 124, 415–421. MR 596977
(82a:22001), http://dx.doi.org/10.1093/qmath/31.4.415
 [7]
Edmond
E. Granirer, Exposed points of convex sets and weak sequential
convergence, American Mathematical Society, Providence, R.I., 1972.
Applications to invariant means, to existence of invariant measures for a
semigroup of Markov operators etc. . ; Memoirs of the American Mathematical
Society, No. 123. MR 0365090
(51 #1343)
 [8]
Edmond
E. Granirer, Weakly almost periodic and uniformly
continuous functionals on the Fourier algebra of any locally compact
group, Trans. Amer. Math. Soc. 189 (1974), 371–382. MR 0336241
(49 #1017), http://dx.doi.org/10.1090/S00029947197403362410
 [9]
A.
Grothendieck, Critères de compacité dans les espaces
fonctionnels généraux, Amer. J. Math.
74 (1952), 168–186 (French). MR 0047313
(13,857e)
 [10]
E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, SpringerVerlag, Berlin and New York, 1963.
 [11]
Gerald
L. Itzkowitz, Continuous measures, Baire category, and uniform
continuity in topological groups, Pacific J. Math. 54
(1974), no. 2, 115–125. MR 0372114
(51 #8331)
 [12]
J.
M. Kister, Uniform continuity and compactness in
topological groups, Proc. Amer. Math. Soc.
13 (1962), 37–40.
MR
0133392 (24 #A3226), http://dx.doi.org/10.1090/S00029939196201333928
 [13]
Theodore
Mitchell, Topological semigroups and fixed points, Illinois J.
Math. 14 (1970), 630–641. MR 0270356
(42 #5245)
 [14]
Gérard
L. G. Sleijpen, Locally compact semigroups and continuous
translations of measures, Proc. London Math. Soc. (3)
37 (1978), no. 1, 75–97. MR 0499939
(58 #17682a)
 [15]
Gérard
L. G. Sleijpen, Emaciated sets and measures with continuous
translations, Proc. London Math. Soc. (3) 37 (1978),
no. 1, 98–119. MR 0499940
(58 #17682b)
 [1]
 A. C. Baker and J. W. Baker, Algebras of measures on a locally compact topological semigroup. III, J. London Math. Soc. 4 (1972), 685695. MR 0306806 (46:5928)
 [2]
 R. B. Burckel, Weakly almost periodic functions and semigroups, Gordon and Breach, New York, 1970. MR 0263963 (41:8562)
 [3]
 Ching Chou, Weakly almost periodic functions and almost convergent functions on a group, Trans. Amer. Math. Soc. 206 (1975), 175200. MR 0394062 (52:14868)
 [4]
 H. A. M. Dzinotyiweyi, Algebras of measures on distinguished topological semigroups, Proc. Cambridge Philos. Soc. 84 (1979), 323336. MR 0493158 (58:12189)
 [5]
 , Weakly almost periodic functions and the irregularity of multiplication in semigroup algebras, Math. Scand. 46 (1980), 157172. MR 585239 (82d:43006a)
 [6]
 , Continuity of semigroup actions on normed linear spaces, Quart. J. Math. Oxford Ser. (2) 31 (1980), 415421. MR 596977 (82a:22001)
 [7]
 E. E. Granirer, Exposed points of convex sets and weak sequential convergence, Mem. Amer. Math. Soc. No. 123 (1972). MR 0365090 (51:1343)
 [8]
 , Weakly almost periodic and uniformly continuous functional on the Fourier algebra of any locally compact group, Trans. Amer. Math. Soc. 189 (1974), 371382. MR 0336241 (49:1017)
 [9]
 A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168186. MR 0047313 (13:857e)
 [10]
 E. Hewitt and K. A. Ross, Abstract harmonic analysis. I, SpringerVerlag, Berlin and New York, 1963.
 [11]
 G. L. Itzkowitz, Continuous measures, Baire category, and uniform continuity in topological groups, Pacific J. Math. 54 (1974), 115125. MR 0372114 (51:8331)
 [12]
 J. M. Kister, Uniform continuity and compactness in topological groups, Proc. Amer. Math. Soc. 13 (1962), 3740. MR 0133392 (24:A3226)
 [13]
 T. Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630641. MR 0270356 (42:5245)
 [14]
 G. L. G. Sleijpen, Locally compact semigroups and continuous translations of measures, Proc. London Math. Soc. 37 (1978), 7597. MR 0499939 (58:17682a)
 [15]
 , Emaciated sets and measures with continuous translations, Proc. London Math. Soc. 37 (1978), 98119. MR 0499940 (58:17682b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206564876
PII:
S 00029947(1982)06564876
Keywords:
Topological semigroup,
uniformly continuous functions,
weakly almost periodic functions
Article copyright:
© Copyright 1982 American Mathematical Society
