zbMATH — the first resource for mathematics

Complex extremal structure in spaces of continuous functions. (English) Zbl 0920.46028
Let \(T\) be a completely regular space, \(X\) be a complex normed space and \(C(T,X)\) be the space of all continuous and bounded \(X\)-valued functions on \(T\) equipped with the uniform norm. It is shown that if the dimension of \(X\) is finite then every function in the unit ball of \(C(T,X)\) is expressible as a convex combination of three unitary functions if and only if \(\dim T< \dim X_R\) (here \(\dim T\) denotes the covering dimension of \(T\) and \(X_R\) denotes \(X\) considered as a real normed space). If \(X\) is an infinite-dimensional space the same kind decomposition is possible without restrictions for \(T\). Moreover, it is proved that the identity mapping on the unit ball of an infinite-dimensional complex normed space \(X\) can be expressed as the average of three retractions of the unit ball of \(X\) onto the unit sphere of \(X\). The case, when \(X\) is a complex strictly convex normed space, is considered separately.
Reviewer: Mati Abel (Tartu)

46E40 Spaces of vector- and operator-valued functions
46E15 Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI
[1] Aron, R.M.; Lohman, R.H., A geometric function determined by extreme points of the unit ball of a normed space, Pacific J. math., 127, 209-231, (1987) · Zbl 0662.46020
[2] Bogachev, V.I.; Mena-Jurado, J.F.; Navarro-Pascual, J.C., Extreme points in spaces of continuous functions, Proc. amer. math. soc., 123, 1061-1067, (1061-1067) · Zbl 0832.46030
[3] Brown, L.G.; Pedersen, G.K., Approximation and convex decomposition by extremals in aC, Math. scand., (1997) · Zbl 0898.46051
[4] Dugundji, J., An extension of Tietze’s theorem, Pacific J. math., 1, 353-367, (1951) · Zbl 0043.38105
[5] Engelking, R., General topology, (1989), Heldermann Berlin · Zbl 0684.54001
[6] Istratescu, V.I., Strict convexity and complex strict convexity, Lecture notes in pure and applied mathematics, (1984), Dekker New York · Zbl 0538.46012
[7] Kadison, R.; Pedersen, G., Means and convex combinations of unitary operators, Math. scand., 57, 249-266, (1985) · Zbl 0573.46034
[8] Navarro-Pascual, J.C., Extreme points and retractions in Banach spaces, Israel J. math., (1997) · Zbl 0901.46015
[9] Phelps, R.R., Extreme points in function algebras, Duke math. J., 32, 267-277, (1965) · Zbl 0139.07401
[10] Robertson, A.G., Averages of extreme points in complex functions spaces, J. London math. soc. (2), 19, 345-347, (1979) · Zbl 0393.46022
[11] Rørdam, M., Advances in the theory of unitary rank and regular approximation, Ann. of math., 128, 153-172, (1988) · Zbl 0659.46052
[12] Russo, B.; Dye, H.A., A note on unitary operators inC, Duke math. J., 33, 413-416, (1966) · Zbl 0171.11503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.