Proving No Extreme Points in B(C_0(X)) for Locally Compact, Non-Compact X

[Ivah]

Homework Statement

Show that if X is a locally compact space but not compact,

then $B(C_0 (X))$ has no extreme points,

in which $B(X)=\{ x | \; ||x|| \leq 1 \}$ and

$C_0(X)$ = all continuous function $f: X \rightarrow \mathbb{F}$ ( with $\mathbb{F}$ the complex plain or the real line) such that for all $\epsilon>0$, $\{x \in X : \; |f(x)| \geq \epsilon \}$ is compact.

Homework Equations

The Attempt at a Solution

One desires to proof that every $z \in B(C_0(X))$

can be written as a convex combination of $y_1 , y_2 \in B(C_0(X))$, i.e. $y_1+y_2=z$. The tricky part is finding such $y_1,y_2$.

The only thing I have proved so far is the existence of a converging net $x_i \rightarrow x$ such that $f(x_i) \rightarrow 0$, anybody got an idea how I can use this to prove the above.






Thanks in advance!

 

[mighty2000]

Thank you for your question. I am always happy to help with mathematical proofs. Here is my attempt at a solution to your problem:

First, let us assume that B(C_0(X)) has an extreme point, say x. This means that x cannot be written as a convex combination of two elements in B(C_0(X)). Since x is an extreme point, it must be an extreme point of the closed unit ball B(C_0(X)). This means that for any y \in B(C_0(X)) and any real number \lambda with 0 < \lambda < 1, the point \lambda x + (1-\lambda)y also belongs to B(C_0(X)).

Now, let us consider the set A = \{y \in B(C_0(X)) : y \neq x, \text{and} \; y \; \text{can be written as a convex combination of} \; x \; \text{and some other point in} \; B(C_0(X)) \}. Since x is an extreme point, we know that A is non-empty. Let y_0 \in A, then there exists some z_0 \in B(C_0(X)) such that y_0 = \lambda x + (1-\lambda)z_0 for some \lambda with 0 < \lambda < 1. From this, we can see that z_0 \neq x, since y_0 \neq x. But since x is an extreme point, this means that z_0 must also be an extreme point.

Now, let us consider the set B = \{z \in B(C_0(X)) : z \neq z_0, \text{and} \; z \; \text{can be written as a convex combination of} \; z_0 \; \text{and some other point in} \; B(C_0(X)) \}. Again, since z_0 is an extreme point, we know that B is non-empty. Let z_1 \in B, then there exists some w_0 \in B(C_0(X)) such that z_1 = \mu z_0 + (1-\mu)w_0 for some \mu with 0 < \mu < 1. But since z_0 is