On the Structure Topology on the Set of all Extreme Points of the Closed Unit Ball of the Dual of a Banach Space

Let

be a real Banach space, and let

E_{X^*} E X ∗

stand for the set of all extreme points of the closed unit ball of

X^* X ∗

, endowed with the Alfsen-Effros structure topology [see E. M. Alfsen and E. G. Effros, Structure in real Banach spaces I, II, Annals of Math. 96 (1972) 98–128; ibid. 96 (1972) 129–73]. The fact that, for a given

s^* \in E_{X^*} s ∗ ∈ E X ∗

, the set

\{\pm s^* \} { ± s ∗ }

is structurally open can be characterized in many apparently different ways, whenever

X X

is nice. (We recall that

X X

is said to be nice if every extreme operator from any Banach space to

X X

is a nice operator, i.e. its adjoint preserves extreme points.) As a consequence, we obtain new characterizations (as well as new proofs of known characterizations) of those nice Banach spaces which are isometrically isomorphic to

c_0(I) c 0 ( I )

for some set

I I

.