## Banach algebras on semigroups and on their compactifications.(English)Zbl 1192.43001

Mem. Am. Math. Soc. 966, v, 165 p. (2010).
Let $$S$$ be a (discrete) semigroup. In the memoir under review the authors study the structure of the semigroup Banach algebra $$l^1(S)$$ and its second dual $$l^1(S)''$$. The algebra $$l^1(S)$$ is taken with the convolution product $$*$$ and the second dual $$l^1(S)''$$ is taken with respect to the first and second Arens products, $$\square$$ and $$\lozenge$$; these second dual algebras are identified with the Banach algebras $$M(\beta S, \square)$$ and $$M(\beta S, \lozenge)$$, which are, hrespectively, the right and left topological semigroups of measures defined on $$\beta S$$, the Stone-Čech compactification of $$S$$.
The authors determine exactly when a semigroup algebra $$l^1(S)$$ is amenable, so answering an open question, and discuss its amenability constant. It is proved that $$S$$ is finite whenever $$M(\beta S)$$ is amenable and it is discussed when $$M(\beta S)$$ is weakly amenable. The authors show that the second dual of $$L^1(G)$$, for $$G$$ a locally compact group, is weakly amenable if and only if $$G$$ is finite. Also, they discuss left-invariant means on $$S$$ as elements of the space $$M(\beta S)$$ and the radical of the algebras $$l^1(\beta S)$$ and $$M(\beta S)$$. The memoir concludes with a list of selected open problems.

### MSC:

 43A07 Means on groups, semigroups, etc.; amenable groups 43-02 Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis 43A20 $$L^1$$-algebras on groups, semigroups, etc. 43A10 Measure algebras on groups, semigroups, etc. 46J10 Banach algebras of continuous functions, function algebras 46J45 Radical Banach algebras
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### References:

 [1] P. D. Adams, Algebraic topics in the Stone-Čech compactifications of semigroups, Thesis, University of Hull, 2001. [2] Richard Arens, Operations induced in function classes, Monatsh. Math. 55 (1951), 1-19. · Zbl 0042.35601 [3] Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839-848. · Zbl 0044.32601 [4] J. W. Baker and Ali Rejali, On the Arens regularity of weighted convolution algebras, J. London Math. Soc. (2) 40 (1989), no. 3, 535-546. · Zbl 0705.43003 [5] M. Lashkarizadeh Bami, The topological centers of $$LUC(S)^*$$ and $$M_a(S)^{**}$$ of certain foundation semigroups, Glasg. Math. J. 42 (2000), no. 3, 335-343. · Zbl 0973.43005 [6] B. A.; Barnes and J. Duncan, The Banach algebra $$l^{1}(S)$$, J. Funct. Anal. 18 (1975), 96-113. · Zbl 0299.46047 [7] Mathias Beiglböck, Vitaly Bergelson, Neil Hindman, and Dona Strauss, Multiplicative structures in additively large sets, J. Combin. Theory Ser. A 113 (2006), no. 7, 1219-1242. · Zbl 1105.05071 [8] Christian Berg, Jens Peter Reus Christensen, and Paul Ressel, Positive definite functions on abelian semigroups, Math. Ann. 223 (1976), no. 3, 253-274. · Zbl 0331.43010 [9] Christian Berg, Jens Peter Reus Christensen, and Paul Ressel, Harmonic analysis on semigroups, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. Theory of positive definite and related functions. · Zbl 0619.43001 [10] John F. Berglund, Hugo D. Junghenn, and Paul Milnes, Analysis on semigroups, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1989. Function spaces, compactifications, representations; A Wiley-Interscience Publication. · Zbl 0727.22001 [11] T. D. Blackmore, Weak amenability of discrete semigroup algebras, Semigroup Forum 55 (1997), no. 2, 196-205. · Zbl 0878.43006 [12] David P. Blecher, Are operator algebras Banach algebras?, Banach algebras and their applications, Contemp. Math., vol. 363, Amer. Math. Soc., Providence, RI, 2004, pp. 53-58. · Zbl 1078.46040 [13] Shea D. Burns, The existence of disjoint smallest ideals in the two natural products on $$\beta S$$, Semigroup Forum 63 (2001), no. 2, 191-201. · Zbl 0989.22008 [14] R. J. Butcher, Thesis, University of Sheffield, 1975. [15] Timothy J. Carlson, Neil Hindman, Jillian McLeod, and Dona Strauss, Almost disjoint large subsets of semigroups, Topology Appl. 155 (2008), no. 5, 433-444. · Zbl 1261.54026 [16] Paul Civin and Bertram Yood, The second conjugate space of a Banach algebra as an algebra, Pacific J. Math. 11 (1961), 847-870. · Zbl 0119.10903 [17] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. · Zbl 0111.03403 [18] W. W. Comfort and S. Negrepontis, The theory of ultrafilters, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 211. · Zbl 0298.02004 [19] H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. · Zbl 0981.46043 [20] H. G. Dales and H. V. Dedania, Weighted convolution algebras on subsemigroups of the real line, Dissertationes Math. (Rozprawy Mat.) 459 (2009), 60. · Zbl 1177.46037 [21] H. G. Dales and A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), no. 836, vi+191. · Zbl 1075.43003 [22] H. G. Dales and W. H. Woodin, An introduction to independence for analysts, London Mathematical Society Lecture Note Series, vol. 115, Cambridge University Press, Cambridge, 1987. · Zbl 0629.03030 [23] H. G. Dales, F. Ghahramani, and A. Ya. Helemskii, The amenability of measure algebras, J. London Math. Soc. (2) 66 (2002), no. 1, 213-226. · Zbl 1015.43002 [24] H. G. Dales, A. T.-M. Lau, and D. Strauss, Second duals of measure algebras, Dissertationes Math., submitted. [25] H. G. Dales, Niels Jakob Laustsen, and Charles J. Read, A properly infinite Banach $$\ast$$-algebra with a non-zero, bounded trace, Studia Math. 155 (2003), no. 2, 107-129. · Zbl 1025.46019 [26] H. G. Dales, R. J. Loy, and Y. Zhang, Approximate amenability for Banach sequence algebras, Studia Math. 177 (2006), no. 1, 81-96. · Zbl 1117.46030 [27] H. G. Dales, A. Rodríguez-Palacios, and M. V. Velasco, The second transpose of a derivation, J. London Math. Soc. (2) 64 (2001), no. 3, 707-721. · Zbl 1023.46051 [28] Matthew Daws, Dual Banach algebras: representations and injectivity, Studia Math. 178 (2007), no. 3, 231-275. · Zbl 1115.46038 [29] Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. · Zbl 0078.29402 [30] M. Despić and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bull. 37 (1994), no. 2, 165-167. · Zbl 0813.43001 [31] J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 309-325. · Zbl 0427.46028 [32] J. Duncan and I. Namioka, Amenability of inverse semigroups and their semigroup algebras, Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), no. 3-4, 309-321. · Zbl 0393.22004 [33] J. Duncan and A. L. T. Paterson, Amenability for discrete convolution semigroup algebras, Math. Scand. 66 (1990), no. 1, 141-146. · Zbl 0748.46027 [34] Charles F. Dunkl and Donald E. Ramirez, Weakly almost periodic functionals on the Fourier algebra, Trans. Amer. Math. Soc. 185 (1973), 501-514. · Zbl 0271.43009 [35] Charles F. Dunkl and Donald E. Ramirez, Representations of commutative semitopological semigroups, Lecture Notes in Mathematics, Vol. 435, Springer-Verlag, Berlin-New York, 1975. · Zbl 0302.22001 [36] M. Eshaghi Gordji and M. Filali, Weak amenability of the second dual of a Banach algebra, Studia Math. 182 (2007), no. 3, 205-213. · Zbl 1135.46027 [37] G. H. Esslamzadeh, Banach algebra structure and amenability of a class of matrix algebras with applications, J. Funct. Anal. 161 (1999), no. 2, 364-383. · Zbl 0927.46027 [38] G. H. Esslamzadeh, Duals and topological center of a class of matrix algebras with applications, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3493-3503. · Zbl 0968.43003 [39] Pierre Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181-236 (French). · Zbl 0169.46403 [40] Lonnie Fairchild, Extreme invariant means without minimal support, Trans. Amer. Math. Soc. 172 (1972), 83-93. · Zbl 0227.43002 [41] Stefano Ferri and Matthias Neufang, On the topological centre of the algebra $${\mathrm LUC}(\scr G)^\ast$$ for general topological groups, J. Funct. Anal. 244 (2007), no. 1, 154-171. · Zbl 1145.43001 [42] S. Ferri and D. Strauss, Ideals, idempotents and right cancelable elements in the uniform compactification, Semigroup Forum 63 (2001), no. 3, 449-456. · Zbl 1010.22003 [43] S. Ferri and D. Strauss, A note on the $$\scr {WAP}$$-compactification and the $$\scr {LUC}$$-compactification of a topological group, Semigroup Forum 69 (2004), no. 1, 87-101. · Zbl 1057.22002 [44] M. Filali, Finite-dimensional right ideals in some algebras associated with a locally compact group, Proc. Amer. Math. Soc. 127 (1999), no. 6, 1729-1734. · Zbl 0918.43002 [45] Mahmoud Filali and Pekka Salmi, Slowly oscillating functions in semigroup compactifications and convolution algebras, J. Funct. Anal. 250 (2007), no. 1, 144-166. · Zbl 1125.43002 [46] Mahmoud Filali and Ajit Iqbal Singh, Recent developments on Arens regularity and ideal structure of the second dual of a group algebra and some related topological algebras, General topological algebras (Tartu, 1999) Math. Stud. (Tartu), vol. 1, Est. Math. Soc., Tartu, 2001, pp. 95-124. · Zbl 1008.22001 [47] Brian Forrest, Weak amenability and the second dual of the Fourier algebra, Proc. Amer. Math. Soc. 125 (1997), no. 8, 2373-2378. · Zbl 0883.46029 [48] Brian E. Forrest and Volker Runde, Amenability and weak amenability of the Fourier algebra, Math. Z. 250 (2005), no. 4, 731-744. · Zbl 1080.22002 [49] F. Ghahramani and J. Laali, Amenability and topological centres of the second duals of Banach algebras, Bull. Austral. Math. Soc. 65 (2002), no. 2, 191-197. · Zbl 1029.46116 [50] Fereidoun Ghahramani and Anthony To Ming Lau, Multipliers and ideals in second conjugate algebras related to locally compact groups, J. Funct. Anal. 132 (1995), no. 1, 170-191. · Zbl 0832.22007 [51] F. Ghahramani and R. J. Loy, Generalized notions of amenability, J. Funct. Anal. 208 (2004), no. 1, 229-260. · Zbl 1045.46029 [52] F. Ghahramani and J. P. McClure, Module homomorphisms of the dual modules of convolution Banach algebras, Canad. Math. Bull. 35 (1992), no. 2, 180-185. · Zbl 0789.43001 [53] F. Ghahramani, A. T. Lau, and V. Losert, Isometric isomorphisms between Banach algebras related to locally compact groups, Trans. Amer. Math. Soc. 321 (1990), no. 1, 273-283. · Zbl 0711.43002 [54] F. Ghahramani, R. J. Loy, and G. A. Willis, Amenability and weak amenability of second conjugate Banach algebras, Proc. Amer. Math. Soc. 124 (1996), no. 5, 1489-1497. · Zbl 0851.46035 [55] F. Ghahramani, R. J. Loy, and Y. Zhang, Generalized notions of amenability. II, J. Funct. Anal. 254 (2008), no. 7, 1776-1810. · Zbl 1146.46023 [56] Mahya Ghandehari, Hamed Hatami, and Nico Spronk, Amenability constants for semilattice algebras, Semigroup Forum 79 (2009), no. 2, 279-297. · Zbl 1176.43001 [57] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. · Zbl 0093.30001 [58] Irving Glicksberg, Uniform boundedness for groups, Canad. J. Math. 14 (1962), 269-276. · Zbl 0109.02001 [59] Frédéric Gourdeau, Amenability and the second dual of a Banach algebra, Studia Math. 125 (1997), no. 1, 75-81. · Zbl 0907.46040 [60] E. Granirer, On amenable semigroups with a finite-dimensional set of invariant means. I, Illinois J. Math. 7 (1963), 32-48. · Zbl 0113.09801 [61] E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 177-197. · Zbl 0136.27202 [62] 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. · Zbl 0258.46001 [63] Edmond E. Granirer, The radical of $$L^{\infty }(G)^{\ast }$$, Proc. Amer. Math. Soc. 41 (1973), 321-324. · Zbl 0278.43009 [64] Michael Grosser, Bidualräume und Vervollständigungen von Banachmoduln, Lecture Notes in Mathematics, vol. 717, Springer, Berlin, 1979 (German). · Zbl 0412.46005 [65] Michael Grosser and Viktor Losert, The norm-strict bidual of a Banach algebra and the dual of $$C_{u}(G)$$, Manuscripta Math. 45 (1984), no. 2, 127-146. · Zbl 0527.46037 [66] Niels Groenbaek, A characterization of weakly amenable Banach algebras, Studia Math. 94 (1989), no. 2, 149-162. · Zbl 0704.46030 [67] Niels Grønbæk, Amenability of weighted discrete convolution algebras on cancellative semigroups, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), no. 3-4, 351-360. · Zbl 0678.46038 [68] Niels Grønbæk, Amenability of discrete convolution algebras, the commutative case, Pacific J. Math. 143 (1990), no. 2, 243-249. · Zbl 0662.43002 [69] A. Grothendieck, Critères de compacité dans les espaces fonctionnels généraux, Amer. J. Math. 74 (1952), 168-186 (French). · Zbl 0046.11702 [70] U. Haagerup, All nuclear $$C^{\ast }$$-algebras are amenable, Invent. Math. 74 (1983), no. 2, 305-319. · Zbl 0529.46041 [71] S. Hartman and C. Ryll-Nardzewski, Almost periodic extensions of functions. II, Colloq. Math. 15 (1966), 79-86. · Zbl 0145.32102 [72] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. · Zbl 0416.43001 [73] Edwin Hewitt and Herbert S. Zuckerman, The $$l_1$$-algebra of a commutative semigroup, Trans. Amer. Math. Soc. 83 (1956), 70-97. · Zbl 0072.12701 [74] Neil Hindman, Minimal ideals and cancellation in $$\beta {\mathbf N}$$, Semigroup Forum 25 (1982), no. 3-4, 291-310. · Zbl 0499.22002 [75] Neil Hindman, The ideal structure of the space of $$\kappa$$-uniform ultrafilters on a discrete semigroup, Rocky Mountain J. Math. 16 (1986), no. 4, 685-701. · Zbl 0624.22001 [76] Neil Hindman and John Pym, Free groups and semigroups in $$\beta {\mathbf N}$$, Semigroup Forum 30 (1984), no. 2, 177-193. · Zbl 0536.22005 [77] Neil Hindman and Dona Strauss, Prime properties of the smallest ideal of $$\beta \mathbf N$$, Semigroup Forum 52 (1996), no. 3, 357-364. · Zbl 0867.22002 [78] Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, de Gruyter Expositions in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 1998. Theory and applications. · Zbl 0918.22001 [79] Neil Hindman and Dona Strauss, Characterization of simplicity and cancellativity in $$\beta S$$, Semigroup Forum 75 (2007), no. 1, 70-76. · Zbl 1132.22003 [80] M. Hochster, Subsemigroups of amenable groups, Proc. Amer. Math. Soc. 21 (1969), 363-364. · Zbl 0174.30801 [81] John M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, vol. 12, The Clarendon Press, Oxford University Press, New York, 1995. Oxford Science Publications. · Zbl 0835.20077 [82] Nilgün Işık, John Pym, and Ali Ülger, The second dual of the group algebra of a compact group, J. London Math. Soc. (2) 35 (1987), no. 1, 135-148. · Zbl 0585.43001 [83] Barry Edward Johnson, Cohomology in Banach algebras, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. · Zbl 0256.18014 [84] B. E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972), 685-698. · Zbl 0246.46040 [85] B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), no. 3, 281-284. · Zbl 0757.43002 [86] B. E. Johnson, Non-amenability of the Fourier algebra of a compact group, J. London Math. Soc. (2) 50 (1994), no. 2, 361-374. · Zbl 0829.43004 [87] Maria Klawe, On the dimension of left invariant means and left thick subsets, Trans. Amer. Math. Soc. 231 (1977), no. 2, 507-518. · Zbl 0371.43005 [88] M. Koçak and D. Strauss, Near ultrafilters and compactifications, Semigroup Forum 55 (1997), no. 1, 94-109. · Zbl 0878.22001 [89] Anthony To-ming Lau, Topological semigroups with invariant means in the convex hull of multiplicative means, Trans. Amer. Math. Soc. 148 (1970), 69-84. · Zbl 0201.02901 [90] Anthony To Ming Lau, Operators which commute with convolutions on subspaces of $$L_{\infty }(G)$$, Colloq. Math. 39 (1978), no. 2, 351-359. · Zbl 0411.47025 [91] Anthony To Ming Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups, Fund. Math. 118 (1983), no. 3, 161-175. · Zbl 0545.46051 [92] Anthony To Ming Lau, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 2, 273-283. · Zbl 0591.43003 [93] Karl Heinrich Hofmann, Jimmie D. Lawson, and John S. Pym , The analytical and topological theory of semigroups, de Gruyter Expositions in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin, 1990. Trends and developments. · Zbl 0702.00013 [94] A. T. Lau, Fourier and Fourier-Stieltjes algebras of a locally compact group and amenability, Topological vector spaces, algebras and related areas (Hamilton, ON, 1994), Pitman Res. Notes Math. Ser., vol. 316, Longman Sci. Tech., Harlow, 1994, pp. 79-92. · Zbl 0856.43002 [95] Anthony To Ming Lau and Viktor Losert, On the second conjugate algebra of $$L_1(G)$$ of a locally compact group, J. London Math. Soc. (2) 37 (1988), no. 3, 464-470. · Zbl 0608.43002 [96] A. T.-M. Lau and R. J. Loy, Weak amenability of Banach algebras on locally compact groups, J. Funct. Anal. 145 (1997), no. 1, 175-204. · Zbl 0890.46036 [97] Anthony To Ming Lau and Alan L. T. Paterson, The exact cardinality of the set of topological left invariant means on an amenable locally compact group, Proc. Amer. Math. Soc. 98 (1986), no. 1, 75-80. · Zbl 0595.43003 [98] Anthony To Ming Lau and John Pym, The topological centre of a compactification of a locally compact group, Math. Z. 219 (1995), no. 4, 567-579. · Zbl 0828.22008 [99] Anthony To Ming Lau and Ali Ülger, Topological centers of certain dual algebras, Trans. Amer. Math. Soc. 348 (1996), no. 3, 1191-1212. · Zbl 0859.43001 [100] A. T.-M. Lau, R. J. Loy, and G. A. Willis, Amenability of Banach and $$C^*$$-algebras on locally compact groups, Studia Math. 119 (1996), no. 2, 161-178. · Zbl 0858.46038 [101] A. T. Lau, A. R. Medghalchi, and J. S. Pym, On the spectrum of $$L^\infty (G)$$, J. London Math. Soc. (2) 48 (1993), no. 1, 152-166. · Zbl 0788.43005 [102] E. S. Ljapin, Semigroups, 3rd ed., American Mathematical Society, Providence, R.I., 1974. Translated from the 1960 Russian original by A. A. Brown, J. M. Danskin, D. Foley, S. H. Gould, E. Hewitt, S. A. Walker and J. A. Zilber; Translations of Mathematical Monographs, Vol. 3. [103] V. Losert, Weakly compact multipliers on group algebras, J. Funct. Anal. 213 (2004), no. 2, 466-472. · Zbl 1069.43001 [104] Nicholas Macri, The continuity of Arens’ product on the Stone-Čech compactification of semigroups, Trans. Amer. Math. Soc. 191 (1974), 185-193. · Zbl 0277.22001 [105] Paul Milnes, Uniformity and uniformly continuous functions for locally compact groups, Proc. Amer. Math. Soc. 109 (1990), no. 2, 567-570. · Zbl 0697.22005 [106] Theodore Mitchell, Fixed points and multiplicative left invariant means, Trans. Amer. Math. Soc. 122 (1966), 195-202. · Zbl 0146.12101 [107] Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630-641. · Zbl 0219.22003 [108] W. D. Munn, On semigroup algebras, Proc. Cambridge Philos. Soc. 51 (1955), 1-15. · Zbl 0064.02001 [109] I. Namioka, On certain actions of semi-groups on $$L$$-spaces, Studia Math. 29 (1967), 63-77. · Zbl 0232.22009 [110] Matthias Neufang, A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis, Arch. Math. (Basel) 82 (2004), no. 2, 164-171. · Zbl 1052.22004 [111] Matthias Neufang, On a conjecture by Ghahramani-Lau and related problems concerning topological centres, J. Funct. Anal. 224 (2005), no. 1, 217-229. · Zbl 1085.43002 [112] Matthias Neufang, On the topological centre problem for weighted convolution algebras and semigroup compactifications, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1831-1839. · Zbl 1149.22007 [113] Theodore W. Palmer, Banach algebras and the general theory of $$^*$$-algebras. Vol. I, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. · Zbl 0809.46052 [114] Theodore W. Palmer, Banach algebras and the general theory of $$*$$-algebras. Vol. 2, Encyclopedia of Mathematics and its Applications, vol. 79, Cambridge University Press, Cambridge, 2001. $$*$$-algebras. · Zbl 0983.46040 [115] D. J. Parsons, The centre of the second dual of a commutative semigroup algebra, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 71-92. · Zbl 0536.43002 [116] Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. · Zbl 0648.43001 [117] I. V. Protasov, Topological center of a semigroup of free ultrafilters, Mat. Zametki 63 (1998), no. 3, 437-441 (Russian, with Russian summary); English transl., Math. Notes 63 (1998), no. 3-4, 384-387. · Zbl 0915.22002 [118] I. V. Protasov and J. S. Pym, Continuity of multiplication in the largest compactification of a locally compact group, Bull. London Math. Soc. 33 (2001), no. 3, 279-282. · Zbl 1024.22002 [119] John S. Pym, The convolution of functionals on spaces of bounded functions, Proc. London Math. Soc. (3) 15 (1965), 84-104. · Zbl 0135.35503 [120] John Pym, Semigroup structure in Stone-Čech compactifications, J. London Math. Soc. (2) 36 (1987), no. 3, 421-428. · Zbl 0599.22007 [121] D. Rees, On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387-400. · JFM 66.1207.01 [122] D. Rees, Note on semi-groups, Proc. Cambridge Philos. Soc. 37 (1941), 434-435. · Zbl 0063.06456 [123] Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. · Zbl 0142.01701 [124] Volker Runde, Amenability for dual Banach algebras, Studia Math. 148 (2001), no. 1, 47-66. · Zbl 1003.46028 [125] Volker Runde, Banach space properties forcing a reflexive, amenable Banach algebra to be trivial, Arch. Math. (Basel) 77 (2001), no. 3, 265-272. · Zbl 1012.46054 [126] Volker Runde, Lectures on amenability, Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin, 2002. · Zbl 0999.46022 [127] Volker Runde, The amenability constant of the Fourier algebra, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1473-1481 (electronic). · Zbl 1117.46031 [128] Volker Runde, Cohen-Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca-Herz algebras, J. Math. Anal. Appl. 329 (2007), no. 1, 736-751. · Zbl 1124.43001 [129] Volker Runde and Nico Spronk, Operator amenability of Fourier-Stieltjes algebras, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 3, 675-686. · Zbl 1052.43003 [130] Volker Runde and Nico Spronk, Operator amenability of Fourier-Stieltjes algebras. II, Bull. Lond. Math. Soc. 39 (2007), no. 2, 194-202. · Zbl 1130.43002 [131] Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. · Zbl 0495.03035 [132] Ross Stokke, Approximate diagonals and Følner conditions for amenable group and semigroup algebras, Studia Math. 164 (2004), no. 2, 139-159. · Zbl 1056.22002 [133] Eric K. van Douwen, The maximal totally bounded group topology on $$G$$ and the biggest minimal $$G$$-space, for abelian groups $$G$$, Topology Appl. 34 (1990), no. 1, 69-91. · Zbl 0696.22003 [134] Carroll Wilde and Klaus Witz, Invariant means and the Stone-Čech compactification, Pacific J. Math. 21 (1967), 577-586. · Zbl 0154.39301 [135] Edward L. Wimmers, The Shelah $$P$$-point independence theorem, Israel J. Math. 43 (1982), no. 1, 28-48. · Zbl 0511.03022 [136] Klaus G. Witz, Applications of a compactification for bounded operator semigroups, Illinois J. Math. 8 (1964), 685-696. · Zbl 0228.47027 [137] Zhuocheng Yang, On the set of invariant means, J. London Math. Soc. (2) 37 (1988), no. 2, 317-330. · Zbl 0644.43001 [138] N. J. Young, Separate continuity and multilinear operations, Proc. London Math. Soc. (3) 26 (1973), 289-319. · Zbl 0247.46027 [139] N. J. Young, Semigroup algebras having regular multiplication, Studia Math. 47 (1973), 191-196. · Zbl 0266.43003 [140] N. J. Young, The irregularity of multiplication in group algebras, Quart J. Math. Oxford Ser. (2) 24 (1973), 59-62. · Zbl 0252.43009 [141] Yong Zhang, Weak amenability of a class of Banach algebras, Canad. Math. Bull. 44 (2001), no. 4, 504-508. · Zbl 1156.46306
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