在本論文中，我們討論有關順從的半拓樸半群的作用的固定點性質，以及定義在拓樸群和半群上的富立葉代數。

具體地說，在不假設存在正規結構的前提下，我們探討左順從半拓樸半群在作用於 (對偶的) 巴拿赫空間的弱緊 (弱*緊) 集合時，其變換表現是否具有固定點的性質。
應用這些成果，我們建立如同在文獻 [36] 中所述的，有關可交換的終極非擴張映射族的共同固定點理論。
我們也將某些結果推廣，應用在弗雷歇空間或局部凸空間的架構下的問題。

我們同時研究 (左) 順從半拓樸半群在自反巴拿赫空間的閉凸集合上的作用。 我們所討論的這些作用，將會是滿足一類關於布萊格曼距離的漸近非擴張性映射。
我們將建立這些映射的共同固定點的唯一性，及如何估計這些共同固定點位置的方法。
我們的結果，推廣了在文獻 [46,50] 中所提出的，有關作用在希爾伯特空間的表現的結果。

另一方面，我們也討論了定義在局部緊的拓樸群，以及定義在離散可逆半群上的富立葉代數。
特別地，我們建立了定義在緊群 G 上的富立葉代數 A(G) 的弱* 固定點性質。
對於某類型的離散可逆半群 S, 我們證明了其富立葉代數 A(S) 的所有非零可乘線性泛函的集合，剛巧是由 S 上的點質量所構成。
對於 A(S) 的順從性和特徵順從性，我們也有所討論。

In this thesis, we are interested in the fixed point properties of actions of amenable semitopological semigroups and the Fourier algebra defined on topological groups and semigroups.
More precisely, we investigate the fixed point properties for actions of left amenable semitopological semigroups on weakly/weak* compact convex sets of a (dual) Banach space not necessarily with normal structure. Applying these results, we establish the existence
of a common fixed point for a commutative family of pointwise eventually nonexpansive mappings defined in [36]. Some extensions to the Frechet and locally convex space settings are also discussed.
We also study the actions of a (left) amenable semitopological semigroup on a closed and
convex subset of a reflexive Banach space. The actions we discuss satisfy some asymptotic nonexpansiveness with respect to the Bregman distance. The uniqueness and approximation of the fixed points are established. This extends the results in [46, 50] which are for the
actions on Hilbert spaces.
On the other hand, we study the Fourier algebra defined on a locally compact topological group, or on a discrete inverse semigroup. In particular, we establish a weak* fixed point property for the Fourier algebra A(G) of a compact group G. For some certain discrete inverse semigroups S, we show that the set of all nonzero multiplicative linear functionals on the Fourier algebra A(S) consists of all point mass arising from S. The amenability and the character amenability on A(S) are discussed.

Thesis Validation Letter ....i
Thesis Authorization Letter.... ii
Acknowledgment .....iii
Chinese Abstract .....iv
Abstract .....v
1 Introduction and Preliminaries ..............1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 The amenability of semitopological semigroups . .4
1.2.2 Semigroup actions . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Some geometric properties . . . . . . . . . . . . . . . . . . . . 8
2 Fixed point theorems for amenable semitopological semigroups .......................9
2.1 Some known fixed point theorems for nonexpansive actions . . . . . . . . . 9
2.2 Various nonexpansive actions on Banach spaces . . 11
2.3 Fixed point theorems for pointwise eventually nonexpansive mappings . . . 31
2.4 Semigroup actions in Frechet and locally convex spaces ´ . . . . . . . . . . . 33
3 Bregman nonexpansive type actions of semitopological semigroups .................43
3.1 Bregman distances . . . . . . . . . .. . . . . 43
3.2 Fixed point theorems for Bregman nonexpansive type actions . . . . . . . . 46
3.3 The uniqueness and approximation of fixed points . ...........................55
4 Amenability of Banach algebras and Fourier algebra of inverse semigroups .................61
4.1 Amenability of Banach algebras . . . . . . . . . . . . . . . . 62
4.2 Character amenability of Banach algebras . . . . . . . . 66
4.3 Fourier algebras of locally compact groups and weak* fixed point properties ........................73
4.4 Fourier algebra on inverse semigroups . . . . . . . . . . .81
4.5 A survey of module operator amenability of Fourier algebra on inverse semigroups . . . . . . . . . . . . . . . . . . . . . 89
5 Remarks and open problems .....................98

[1] M. Abtahi and Y. Zhang, A new proof of the amenability of C(X), Bull. Aust. Math. Soc., 81
(2010) 414–417.
[2] D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. (1981), 423–424.
[3] M. Amini, A. Medghalchi and F. Naderi, Pointwise eventually non-expansive action of semitopological semigroups and fixed points, J. Math. Anal. Appl., 437 (2016), 1176–1183.
[4] M. Amini and R. Rezavand, Module operator amenability of the Fourier algebra of an inverse
semigroup, Semigroup Forum 92, 45–70 (2016).
[5] M. Amini and R. Rezavand, Module operator virtual diagonals on the Fourier algebra of an
inverse semigroup. Semigroup Forum 97 (2018), no. 3, 562–570.
[6] A. Aminpour, A. Dianatifar and R. Nasr-Isfahani, Asymptotically nonexpansive actions of
strong amenable semigroups and fixed points, J. Math. Anal. Appl., 461 (2018), 364–377.
[7] L. N. Argabright, Invariant means and fixed points: A sequel to Mitchell’s paper, Trans. Amer.
Math. Soc., (1968), 127–130.
[8] B. A. Barnes, Representations of the l1-algebra of an inverse semigroup, Trans. Amer. Math.
Soc. 218 (1976), 361–396.
[9] L. P. Belluce and W. A. Kirk, Fixed-point theorems for families of contraction mappings, Pacific
Journal of Mathematics, 18, (1966), 213–217.
101[10] J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on Semigroups, John Wiley and Sons,
New York, 1988.
[11] S. Borzdynski and A. Wi ´ snicki, A common fixed point theorem for a commuting family of ´
weak* continuous nonexpansive mappings, Studia Math., 225 (2014), no. 2, 173–181.
[12] F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New YorkHeidelberg, 1973.
[13] D. Butnariu and A. N. Iusem, Totally Convex Functions for Fixed Points Computation and
Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht 2000.
[14] R. D. Bourgin, Geometric aspects of convex sets with the Radon-Nikodym property ´ , SpringerVerlag, Berlin (1983).
[15] W. M. Boyce, Commuting functions with no common fixed point, Trans. Amer. Math. Soc. 137
(1969), 77–92.
[16] H.-Y. Chen, G. P. Geher, C.-N. Liu, L. Moln ´ ar, D. Virosztek and N.-C. Wong, Maps on positive ´
definite operators preserving the quantum χ2 α-divergence, Letters of Math. Physics, 107 (2017),
2267–2290.
[17] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957) 509–544.
[18] M. M. Day, Fixed-point theorems for compact convex sets, Illinois J. Math., 5 (1961), 585–590.
[19] R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math., 13
(1963), 1939–1141.
[20] J. Duncan and I. Namioka, Amenability of inverse semigroups and their semigroup algebras,
Proc. Royal Soc. Edinb., 80 (1978) 309–321.
102[21] M. Enock and J. M. Schwartz, Kac algebras and duality of locally compact groups, SpringerVerlag, Berlin, 1992.
[22] E. G. Effros and Z.-J. Ruan, Operator spaces, London Mathematical Society Monographs. New
Series, 23. The Clarendon Press, Oxford University Press, New York, 2000.
[23] P. Eymard, L’algebre de Fourier d’un groupe localement compact, (French) ` Bull. Soc. Math.
France 92 (1964), 181–236.
[24] B. E. Forrest and V. Runde, Amenability and weak amenability of the Fourier algebra, Math.
Z., 250, 731–744 (2005).
[25] R. Ghamarshoushtar and Y. Zhang, Amenability properties of Banach algebra valued continuous functions, J. Math. Anal. Appl, 15 (2015), 1335–1341.
[26] E. Granirer, On invariant mean on topological semigroups and on topological groups, Pacific J.
Math., 15 (1965), 107–140.
[27] F. Kohsaka and W. Takahashi, Proximal point algorithms with Bregman functions in Banach
spaces, J. Nonlinear Convex Anal., 6:3 (2005), 505–523.
[28] Y.-Y. Huang, J.-C. Jeng, T.-Y. Kuo and C.-C. Hong, Fixed point and weak convergence theorems
for point-dependent λ-hybrid mappings in Banach spaces, Fixed Point Theory and Applications,
2011:105 (2011), 1–15.
[29] R. D. Holmes and A. T.-M. Lau, Asymptotically nonexpansive actions of topological semigroups and fixed points, Bull. London Math. Soc., 3 (1971), 343–347.
[30] R. D. Holmes and P. P. Narayanaswami, On asymptotically nonexpansive semigroups of mappings, Canadian Mathematical Bulletin, 13 (1970), 209–214.
[31] M. Ismail and P. Nyikos, On spaces in which countably compact sets are closed, and hereditary
properties, Topology and its Appl., 11 (1980), 281–292.
103[32] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127 (1972).
[33] B. E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras,
Amer. J. Math., 94 (1972), 685–698.
[34] S. Kakutani, Two fixed point theorems concerning bicompact convex set, Proc. Japan Acad. 14
(1938), 242–245.
[35] W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math.
Monthly, 72 (1965), 1004–1006.
[36] W. A. Kirk and H. K. Xu, Asymptotic pointwise contractions, Nonlinear Anal., 69 (2008),
4706–4712.
[37] E. Kaniuth, A. T. Lau and J. Pym, On φ−amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc., 144 (2008), 85–96.
[38] E. Kaniuth, A. T. Lau and J. Pym, On character amenability of Banach algebras, J. Math. Anal.
Appl. 344 (2008), 942—955.
[39] E. Kaniuth and A. T. Lau, Fourier and Fourier-Stieltjes algebras on locally compact groups,
Mathematical Surveys and Monographs, 231. American Mathematical Society, Providence, RI,
2018.
[40] F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly
nonexpansive-type mappings in Banach spaces, SIAM J. Optim., 19 (2008) 824–835.
[41] T.-Y. Kuo, J.-C. Jeng, Y.-Y. Huang and C.-C. Hong, Fixed point and weak convergence theorems
for (α, β)-hybrid mappings in Banach spaces, Abstr. Appl. Anal., 2012 (2012), Art. ID 789043.
[42] A. T.-M. Lau, Invariant means on almost periodic functions and fixed point properties, Rocky
Mountain J. Math., 3 (1973), 69–76.
104[43] A. T.-M. Lau, The Fourier–Stieltjes algebra of a topological semigroup with involution, Pacific
J. Math. 77 (1978) 165–181.
[44] A. T.-M. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on
locally compact groups and semigroups, Fund. Math., 118 (1983), 161–175.
[45] A. T.-M. Lau and P. Mah, Fixed point properties for Banach algebras associated to locally
compact groups, J. Funct. Anal., 258 (2010), 357–372.
[46] A. T.-M. Lau and W. Takahashi, Invariant means and fixed point properties for nonexpansive
representations of topological semigroups, Topol. Methods Nonlinear Anal., 5 (1995), 39–57.
[47] A. T.-M. Lau and W. Takahashi, Fixed point properties for semigroup of nonexpansive mappings on Frechet spaces, ´ Nonlinear Anal., 11 (2009), 3837–3841.
[48] A. T.-M. Lau and Y. Zhang, Fixed point properties of semigroups of nonexpansive mappings,
J. Funct. Anal., 254 (2008), 2534–2554.
[49] A. T.-M. Lau and Y. Zhang, Fixed point properties for semigroups of nonlinear mappings and
amenability, J. Funct. Anal., 263 (2012), 2949–2977.
[50] A. T.-M. Lau and Y. Zhang, Fixed point properties for semigroups of nonlinear mappings on
unbounded sets, J. Math. Anal. Appl., 433 (2016), 1204–1219.
[51] T. C. Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific
J. Math., 90 (1980), 135-143.
[52] T. C. Lim, Characterization of normal structure, Proc. Amer. Math. Soc. 43 (1974) 313–319.
[53] J. Lindenstrauss, Weakly compact sets — their topological properties and the Banach spaces
they generate, Symposium on Infinite-Dimensional Topology, Ann. of Math. Studies, No. 69,
Princeton Univ. Press, Princeton, N. J., (1972), 235–273.
105[54] M. Megrelishvili, Fragmentability and continuity of semigroup actions, Semigroup Forum, 57
(1998), 101–126.
[55] T. Mitchell, Function algebras, means and fixed points, Trans. Amer. Math. Soc., (1968) 117–
126.
[56] T. Mitchell, Topological semigroups and fixed points, Illinois J. Math., 14 (1970), 630–641.
[57] B. N. Muoi and N.-C. Wong, Bregman nonexpansive type actions of semitopological semigroups, J. Nonlinear Convex Anal., 4 (2021) 871–885.
[58] B. N. Muoi and N.-C. Wong, Fixed point theorems of various nonexpansive actions of semitopological semigroups on weakly/weak* compact convex sets, arXiv:2006.15393v3 [math.FA],
(submitted).
[59] B. N. Muoi and N.-C. Wong, Super asymptotically nonexpansive actions of semitopological
semigroups on Frechet and locally convex spaces, ´ Proceedings of NACA-ICOTA2019, (2021)
53–72.
[60] I. Namioka, Neighborhoods of extreme points, Israel J. Math., 5 (1967), 145–152.
[61] I. Namioka, Right topological groups, distal flows and a fixed point theorem, Math. Syst. Theory, 6 (1972) 193–209.
[62] E. Naraghirad, N.-C. Wong and J.-C. Yao, Applications of Bregman-Opial property to Bregman
nonspreading mappings in Banach spaces, Abstr. Appl. Anal., 2014 (2014), Art. ID 272867.
[63] A. L. T. Paterson, Amenability, volume 29 of Mathematical Surveys and Monographs. American
Mathematical Society, Providence, RI, 1988.
[64] Z.-J. Ruan, The operator amenability of A(G), Amer. J. Math., 117 (1995), no. 6, 1449–1474.
106[65] W. Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics,
No. 12 Interscience Publishers, New York-London 1962.
[66] W. Rudin, Functional analysis, International Series in Pure and Applied Mathematics,
McGraw-Hill, Inc., New York, 1991.
[67] S. Saeidi, F. Golkar and A. M. Forouzanfar, Existence of fixed points for asymptotically nonexpansive type actions of semigroups, J. Fixed Point Theory Appl., 20 (2018), no. 2, Art. 72.
[68] K. Salame, On Lau’s conjecture II, Proc. Amer. Math. Soc., 148 (2020), no. 5, 1999–2008.
[69] H. H. Schaefer, Topological Vector Spaces, Springer-Verlag, 1971.
[70] J. Schauder, Der Fixpunktsatz in Funktionalraumen, Studia Math., 2 (1930), 171–180.
[71] W. Takahashi, Fixed point theorem and nonlinear ergodic theorem for nonexpansive semigroups without convexity, Canad. J. Math., 44 (1992), 880–887.
[72] W. Takahashi, Fixed point theorems for new nonlinear maps in a Hilbert space, J. Nonlinear
Convex Anal., 11 (2010), 79–88.
[73] W. Takahashi, N.-C. Wong and J.-C. Yao, Fixed point theorems and convergence theorems for
generalized nonspreading maps in Banach spaces, J. Fixed Point Theory and Appl., 11 (2012),
159–183.
[74] A. Wisnicki, Around the nonlinear Ryll-Nardzewski theorem, ´ Math. Ann., 377 (2020), 267–
279.
[75] A. Wisnicki, Amenable semigroups and nonexpansive dynamical systems, ´ preprint.
arXiv:1909.09723 [math.FA].