Amenability and Hahn-Banach extension property for set-valued mappings
Keywords
Amenability, Hahn--Banach extension, invariant mean, semigroup, set-valued mappingAbstract
Amenability is an important notion in harmonic analysis on groups and semigroups, and their associated Banach algebras. In this paper, we present some characterization of a semitopological semigroup $S$ on the existence of a left invariant mean on $\text{\rm LUC}(S)$, $\text{\rm AP}(S)$ and $\text{\rm WAP}(S)$ in terms of Hahn-Banach extension theorem, which extend the first author's early results in 1970s. Moreover, we refine and extend the well known Day's result and Mitchell's results on fixed point properties for set-valued mappings. As an application, we give an application of our result to a class of Banach algebras related to amenability of groups and semigroups.References
L.N. Argabright, Invariant means and fixed point: a sequel to Mitchell’s paper’, Trans. Amer. Math. Soc. 130 (1968), 127–130.
E. Bédos and L. Tuset, Amenability and co-amenability for locally compact quantum groups, Internat. J. Math. 14 (2003), 865–884.
J.F. Berglund, H.D. Junghenn and P. Milnes, Analysis on Semigroups: Function Spaces, Compactifications, Representations, Wiley–Interscience, New York, 1989.
R. Cross, Multivalued Linear Operators, Marcel Dekker, 1998.
H.G. Dales, A.T.-M. Lau and D. Strauss, Second duals of measure algebras, Dissertationes Math. 481 (2012), 1–121.
M.M. Day, Means for the bounded functions and ergodicity of the bounded representations of semi-groups, Trans. Amer. Math. Soc. 69 (1950), 276–291.
M.M. Day, Amenable semigroup, Illinois J. Math. 1 (1957), 509–544.
M.M. Day, Fixed-point theorems for compact convex set, Illinois J. Math. 5 (1961), 585–590.
M.M. Day, Semigroups and Amenability, Semigroups (K.W. Folley, ed.), Academic Press, 1969, 1–53.
K. Deleeuw and I. Glicksberg, Application of almost periodic functions, Acta Math. 105 (1961), 63–97.
K. Fan, Extension of invariant linear functionals, Proc. Amer. Math. Soc. 66 (1977), 23–29.
F.P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Math. Studies, vol. 16, Van Nostrand Rheinhold, Princeton, N.J., 1969.
E. Hewitt and K.A. Ross, Abstract Harmonic Analysis: Volume I. Structure of Topological Groups, Integration Theory, Group Representations, Springer–Verlag, 1963.
R.D. Holmes and A.T.-M. Lau, Non-expansive actions of topological semigroups and fixed points, J. London Math. Soc. 5 (1972), 330–336.
B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972), 96 pp.
E. Kaniuth and A.T.-M. Lau, Fourier and Fourier–Stieltjes Algebras on Locally Compact Groups, Math. Surveys and Monographs, Amer. Math. Soc., 2018, 306 pp.
A.T.-M. Lau, Invariant means on almost periodic functions and fixed point propertie, Rocky Mountain J. Math. 3 (1973), 69–76.
A.T.-M. Lau, Amenability and invariant subspaces, J. Australian Math. Soc. 2 (1974), 200–204.
A.T.-M. Lau, Invariant means on almost periodic functions and equicontinuous actions, Proc. Amer. Math. Soc. 49 (1975), 379–382.
A.T.-M. Lau, Extension of invariant linear functionals: a sequel to Fan’s paper, Proc. Amer. Math. Soc. 2 (1977), 259–262.
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.
A.T.-M. Lau and J. Ludwig, Fourier–Stieltjes algebra of a topological group, Adv. Math. 229 (2012), 2000–2023.
A.T.-M. Lau and L. Yao, Hahn–Banach extension and optimization related to fixed point properties and amenability, J. Nonlinear Variational Anal. 1 (2017), 127–143.
A.T.-M. Lau and L. Yao, Common fixed point properties for a family of set-valued mappings, J. Math. Anal. Appl. 459 (2018), 203–216.
A.T.-M. Lau and Y. Zhang, Finite-dimensional invariant subspace property and amenability for a class of Banach algebras, Trans. Amer. Math. Soc. 368 (2016), 3755–3775.
T. Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630–641.
A.L.T. Paterson, Amenability, Math. Surveys Monogr., vol. 29, Amer. Math. Soc., 1988.
J.W. Peng, H.W.J. Lee, W.D. Rong and X.M. Yang, Hahn–Banach theorems and subgradients of set-valued maps, Math. Methods Oper. Res. 61 (2005), 281–297.
J. Peng, H.W.J. Lee, W. Rong and X. Yang, A generalization of Hahn–Banach extension theorem, J. Math. Anal. Appl. 302 (2005), 441–449.
J.-P. Pier, Amenable Banach Algebras, Pitman Research Notes in Mathematics Series, vol. 172, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1988.
C.R. Rao, Invariant means on spaces of continuous or measurable functions, Trans. Amer. Math. Soc. 114 (1965), 187–196.
Z.-J. Ruan, Amenability of Hopf von Neumann algebras and Kac algebras, J. Funct. Anal. 139 (1996), 466–499.
R. Rudin, Functional Analysis, Second Edition, McGraw–Hill, 1991.
R.J. Silverman, Means on semigroups and the Hahn–Banach extension property, Trans. Amer. Math. Soc. 83 91986), 222–237.
M. Skantharajah, Amenable hypergroups, Illinois J. Math. 36 (1992), 15–46.
D. Voiculescu, Amenability and Kac algebras, Algèbres d’opérateurs et leurs applications en physique mathématique (Proc. Colloq., Marseille, 1977), Colloq. Internat. CNRS, vol. 274, 1979, 451–457.
B. Willson, Invariant nets for amenable groups and hypergroups, Ph.D. thesis, University of Alberta, 2011.
B. Willson, Configurations and invariant nets for amenable hypergroups and related algebras, Trans. Amer. Math. Soc. 366 (2014), 5087–5112.
J.C.S. Wong, Topological invariant means on locally compact groups and fixed points, Proc. Amer. Math. Soc. 27 (1971), 572–578.
L. Yao, On Monotone Linear Relations and the Sum Problem in Banach Spaces, Ph.D. thesis, University of British Columbia (Okanagan), 2011, http://hdl.handle.net/2429/39970.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0
