Extreme Points and Linear Isometries of the Banach Space of Lipschitz Functions

Abstract

Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such thatThe smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,

Lipschitz spaces

“…There are still many outstanding conjectures concerning these spaces, some of which seem to be fairly difficult; especially certain questions about extreme points. The known results along these lines can be found in [2], [3], [4], [10], and [11].…”

“…Some of the main references are [1], [2], [3], [4], [10], [11], and [12]. There are still many outstanding conjectures concerning these spaces, some of which seem to be fairly difficult; especially certain questions about extreme points.…”

A note on Banach spaces of Lipschitz functions

“…The same argument as above shows / is an extreme point of the unit ball. It is not difficult to see that the function defined in [11,Lemma 1.3] is in 35 (P = 0 in this case) and that si C$.…”

“…The only reason we include it is that, as mentioned in the introduction, there have to our knowledge been no characterizations of extreme Lipschitz functions, other than that of Roy [11], and no real clues as to how to extend his characterization. We only hope that the next proposition, as well as Proposition 1.1, may give some ideas as to possible directions to take.…”

Isometries Between Function Spaces

“…Therefore to prove that any isometry from Lipa(X) onto LipQ(Y) or from lipQ(X) onto lipQ(Y) is canonical we have to define a property which "separates" X = {a6x: x e X,a e S} from the rest of the extreme points of the unit ball of (Lipa(X))* or (lipQ(X))\ We have The isometries of the complex Lip: (X) spaces with the above M-norm were considered by Roy [21], when X is connected with diameter at most 1, and by Vasavada [22], when X satisfies certain separation conditions.…”