zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Fixed point set of semigroups of non-expansive mappings and amenability. (English) Zbl 1142.47035
The author studies the fixed point set of a strongly continuous nonexpansive semigroup $\{T(t)\}_{t\in S}$ acting in a Banach space. He investigates the cases when $S$ is a semi-topological semigroup for which $CB(S)$ (the space of bounded continuous functions) is $n$-extremely amenable, and when $S$ is an additive sub-semigroup of a locally convex vector space. Some applications to harmonic analysis are also provided.

47H20Semigroups of nonlinear operators
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
Full Text: DOI
[1] Dulst, D. V.: Equivalent norms and the fixed point property for nonexpansive mappings, J. London math. Soc. 25, 139-144 (1982) · Zbl 0453.46017 · doi:10.1112/jlms/s2-25.1.139
[2] Folland, G. B.: A course in abstract harmonic analysis, Stud. adv. Math. (1995) · Zbl 0857.43001
[3] Gossez, J. P.; Dozo, E. Lami: Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40, 565-573 (1972) · Zbl 0223.47025
[4] Granirer, E.: Extremely amenable semigroups, Math. scand. 17, 177-197 (1965) · Zbl 0136.27202
[5] Granirer, E.: Functional analytic properties of extremely amenable semigroups, Trans. amer. Math. soc. 137, 53-75 (1969) · Zbl 0202.40703 · doi:10.2307/1994787
[6] Granirer, E.; Lau, A. T.: Invariant means on locally compact groups, Illinois J. Math. 15, 249-257 (1971) · Zbl 0212.14904
[7] Gromov, M.; Milman, V. D.: A topological application of the isoperimetric inequality, Amer. J. Math. 105, 843-854 (1983) · Zbl 0522.53039 · doi:10.2307/2374298
[8] Hirano, N.; Kido, K.; Takahashi, W.: Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces, Nonlinear anal. 12, 1269-1281 (1988) · Zbl 0679.47031 · doi:10.1016/0362-546X(88)90059-4
[9] J.I. Kang, Fixed point of non-expansive mappings associated to invariant means in a Banach space, Nonlinear Anal., in press · Zbl 1223.47062 · doi:10.1016/j.na.2007.03.018
[10] Krein, M.; Milman, D.: On extreme points of regular convex sets, Studia math. 9, 133-138 (1940) · Zbl 0063.03360
[11] Lau, A. T.: Extremely amenable algebras, Pacific J. Math. 33, 329-336 (1970) · Zbl 0179.46301
[12] Lau, A. T.: Functional analytic properties of topological semigroups and n-extreme amenability, Trans. amer. Math. soc. 152, 431-439 (1970) · Zbl 0216.17002 · doi:10.2307/1995580
[13] Lau, A. T.: Topological semigroups with invariant means in the convex hull of multiplicative means, Trans. amer. Math. soc. 148, 69-84 (1970) · Zbl 0201.02901 · doi:10.2307/1995038
[14] Lau, A. T.: Action of topological semigroups, invariant means, and fixed points, Studia math. 43, 139-156 (1972) · Zbl 0232.43001
[15] Lau, A. T.: Invariant means on almost periodic functions and fixed point properties, Rocky mountain J. Math. 3, 69-75 (1973) · Zbl 0279.43002 · doi:10.1216/RMJ-1973-3-1-69
[16] A.T. Lau, M. Leinert, Fixed point property and the Fourier algebra of a locally compact group, Trans. Amer. Math. Soc., in press · Zbl 1153.43008 · doi:10.1090/S0002-9947-08-04622-9
[17] Lau, A. T.; Mah, P. F.: Normal structure in dual Banach spaces associated with a locally compact group, Trans. amer. Math. soc. 310, 341-353 (1988) · Zbl 0706.43003 · doi:10.2307/2001126
[18] Lau, A. T.; Mah, P. F.; Ülger, A.: Fixed point property and normal structure for Banach spaces associated to locally compact groups, Proc. amer. Math. soc. 125, 2021-2027 (1997) · Zbl 0868.43001 · doi:10.1090/S0002-9939-97-03773-8
[19] Lau, A. T.; Miyake, H.; Takahashi, W.: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces, Nonlinear anal. 67, 1211-1225 (2007) · Zbl 1123.47048 · doi:10.1016/j.na.2006.07.008
[20] Mitchell, T.: Fixed points and multiplicative left invariant means, Trans. amer. Math. soc. 122, 195-202 (1966) · Zbl 0146.12101 · doi:10.2307/1994510
[21] Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. amer. Math. soc. 73, 591-597 (1967) · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[22] Rodé, G.: An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. math. Anal. appl. 85, 172-178 (1982) · Zbl 0485.47041 · doi:10.1016/0022-247X(82)90032-4
[23] Suzuki, T.: The set of common fixed points of a one-parameter continuous semigroup of mappings is $F(T(1))\cap F(T(2))$, Proc. amer. Math. soc. 134, 673-681 (2005) · Zbl 1119.47057 · doi:10.1090/S0002-9939-05-08361-9
[24] Takahashi, W.: A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. amer. Math. soc. 81, 253-256 (1981) · Zbl 0456.47054 · doi:10.2307/2044205