Extreme contractions on continuous vector-valued function spaces
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- by Hasan Al-Halees and Richard J. Fleming PDF
- Proc. Amer. Math. Soc. 134 (2006), 2661-2666 Request permission
Abstract:
An old question asks whether extreme contractions on $C(K)$ are necessarily nice; that is, whether the conjugate of such an operator maps extreme points of the dual ball to extreme points. Partial results have been obtained. Determining which operators are extreme seems to be a difficult task, even in the scalar case. Here we consider the case of extreme contractions on $C(K,E)$, where $E$ itself is a Banach space. We show that every extreme contraction $T$ on $C(K,E)$ to itself which maps extreme points to elements of norm one is nice, where $K$ is compact and $E$ is the sequence space $c_{0}$.References
- Hasan Al-Halees and Richard J. Fleming, Extreme point methods and Banach-Stone theorems, J. Aust. Math. Soc. 75 (2003), no.ย 1, 125โ143. MR 1984631, DOI 10.1017/S1446788700003505
- R. M. Blumenthal, Joram Lindenstrauss, and R. R. Phelps, Extreme operators into $C(K)$, Pacific J. Math. 15 (1965), 747โ756. MR 209862, DOI 10.2140/pjm.1965.15.747
- Alan Gendler, Extreme operators in the unit ball of $L(C(X),C(Y))$ over the complex field, Proc. Amer. Math. Soc. 57 (1976), no.ย 1, 85โ88. MR 405173, DOI 10.1090/S0002-9939-1976-0405173-9
- 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. MR 0116199, DOI 10.1007/978-1-4615-7819-2
- Ryszard Grzฤ ลlewicz, Extreme operators on $2$-dimensional $l_{p}$-spaces, Colloq. Math. 44 (1981), no.ย 2, 309โ315 (1982). MR 652589, DOI 10.4064/cm-44-2-309-315
- P. D. Morris and R. R. Phelps, Theorems of Krein-Milman type for certain convex sets of operators, Trans. Amer. Math. Soc. 150 (1970), 183โ200. MR 262804, DOI 10.1090/S0002-9947-1970-0262804-3
- M. Sharir, Characterization and properties of extreme operators into $C(Y)$, Israel J. Math. 12 (1972), 174โ183. MR 317026, DOI 10.1007/BF02764661
Additional Information
- Hasan Al-Halees
- Affiliation: Department of Mathematics, Saginaw Valley State University, University Center, Michigan 48710-0001
- Richard J. Fleming
- Affiliation: Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859
- MR Author ID: 67545
- Received by editor(s): March 15, 2005
- Received by editor(s) in revised form: April 1, 2005
- Published electronically: March 23, 2006
- Communicated by: Joseph A. Ball
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 2661-2666
- MSC (2000): Primary 47B38, 46E40
- DOI: https://doi.org/10.1090/S0002-9939-06-08282-7
- MathSciNet review: 2213745