Question

Let $B_1, B_2$ be unit balls in finite-dimensional normed spaces $X_1, X_2$ respectively. Let $e(B_1), e(B_2)$ be corresponding extreme points sets.

Consider the unit ball $B$ in tensor product $X_1\otimes X_2$ with the largest (projective) cross-norm on it.

Can we say that extreme points of $B$ in tensor product are exactly tensor products of extreme points for $B_1, B_2$, i.e. $e(B)=\{u\otimes w: u\in e(B_1), w\in e(B_2)\}$?

This seems plausible, but things are not looking very straightforward. In particular, opposit pairs of extreme points produce the same point in tensor product, i.e. $(-u)\otimes (-w) = u\otimes w$.

Answer 1

See

[11] Ruess, W.M. and Stegall, C.P., Extreme points in duals of operator spaces, Math. Ann., 261 (1982), 535–546.

They prove what you want in a more general context: If $X$, $Y$ are Banach spaces s.t. either $X^*$ or $Y^*$ has the approximation property and either $X^*$ or $Y^*$ has the Radon-Nikodyn property, then the extreme points of the unit ball of the projective tensor product of $X^*$ and $Y^*$ are the tensor products of extreme points of the respective unit balls.

Answer 2

There is a hierarchy of notions related to the notion of extreme point, among them strong extreme point, point of continuity, denting point, as sketched in

BOR-LUH LIN, PEI-KEE LIN AND S. L. TROYANSKI. CHARACTERIZATIONS OF DENTING POINTS. PROC. OF AMS. Volume 102, Number 3, March 1988, p.526-528 (http://www.ams.org/journals/proc/1988-102-03/S0002-9939-1988-0928972-1/S0002-9939-1988-0928972-1.pdf).

The corresponding statement is true for denting points as shown in

DIRK WERNER. DENTING POINTS IN TENSOR PRODUCTS OF BANACH SPACES. PROC. OF AMS. Volume 101, Number 1, September 1987, p. 122-126 (http://www.ams.org/journals/proc/1987-101-01/S0002-9939-1987-0897081-1/S0002-9939-1987-0897081-1.pdf).

Now, it's enough to note that notions of extreme point and denting point are equivalent in my context.