On the failure of close-to-normal structure type conditions and pathological Kannan maps

Smyth, Michael

doi:10.1090/S0002-9939-96-03466-1

On the failure of close-to-normal structure type conditions and pathological Kannan maps
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by Michael A. Smyth PDF

Proc. Amer. Math. Soc. 124 (1996), 3063-3069 Request permission

Abstract:

We consider the failure of close-to-normal structure type conditions and show that a Banach space can be renormed to fail close-to-weak normal structure exactly when it contains a norm inseparable weakly compact subset. Included is an example of a particularly pathological fixed point free Kannan map.

References

D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. (2) 25 (1982), no. 1, 139–144. MR 645871, DOI 10.1112/jlms/s2-25.1.139
Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104. MR 552464, DOI 10.1515/crll.1980.313.72
Anthony To Ming Lau and Peter F. Mah, Quasinormal structures for certain spaces of operators on a Hilbert space, Pacific J. Math. 121 (1986), no. 1, 109–118. MR 815037, DOI 10.2140/pjm.1986.121.109
Chris Lennard, ${\scr C}_1$ is uniformly Kadec-Klee, Proc. Amer. Math. Soc. 109 (1990), no. 1, 71–77. MR 943795, DOI 10.1090/S0002-9939-1990-0943795-4
Delfina Roux and Clemente Zanco, Kannan maps in normed spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 65 (1978), no. 6, 252–258 (1979) (English, with Italian summary). MR 586609
Brailey Sims, “Ultra”-techniques in Banach space theory, Queen’s Papers in Pure and Applied Mathematics, vol. 60, Queen’s University, Kingston, ON, 1982. MR 778727
Ivan Singer, Bases in Banach spaces. II, Editura Academiei Republicii Socialiste România, Bucharest; Springer-Verlag, Berlin-New York, 1981. MR 610799, DOI 10.1007/978-3-642-67844-8
M. Smyth, Remarks on the weak star fixed point property in the dual of $C(\Omega )$, J. Math. Anal. Appl. 195 (1995), 294–306.
Kok Keong Tan, A note on asymptotic normal structure and close-to-normal structure, Canad. Math. Bull. 25 (1982), no. 3, 339–343. MR 668951, DOI 10.4153/CMB-1982-047-3
Chi Song Wong, Close-to-normal structure and its applications, J. Functional Analysis 16 (1974), 353–358. MR 0348452, DOI 10.1016/0022-1236(74)90054-8
Chi Song Wong, On Kannan maps, Proc. Amer. Math. Soc. 47 (1975), 105–111. MR 358468, DOI 10.1090/S0002-9939-1975-0358468-0