Extreme points of unit ball in tensor product of spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:38:27Z http://mathoverflow.net/feeds/question/90875 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90875/extreme-points-of-unit-ball-in-tensor-product-of-spaces Extreme points of unit ball in tensor product of spaces Yauhen Radyna 2012-03-11T03:44:35Z 2012-08-25T22:39:27Z <p>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.</p> <p>Consider the unit ball $B$ in tensor product $X_1\otimes X_2$ with the largest (projective) cross-norm on it. </p> <blockquote> <p>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. $e(B)=\{u\otimes w: u\in e(B_1), w\in e(B_2)\}$? </p> </blockquote> <p>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. $(-u)\otimes (-w) = u\otimes w$. </p> http://mathoverflow.net/questions/90875/extreme-points-of-unit-ball-in-tensor-product-of-spaces/90942#90942 Answer by Bill Johnson for Extreme points of unit ball in tensor product of spaces Bill Johnson 2012-03-11T22:17:29Z 2012-03-11T23:08:57Z <p>See</p> <p>[11] Ruess, W.M. and Stegall, C.P., Extreme points in duals of operator spaces, Math. Ann., 261 (1982), 535–546. </p> <p>They prove what you want in a more general context: If $X$, $Y$ are Banach spaces s.t. either <code>$X^*$</code> or <code>$Y^*$</code> has the approximation property and either <code>$X^*$</code> or <code>$Y^*$</code> has the Radon-Nikodyn property, then the extreme points of the unit ball of the projective tensor product of <code>$X^*$</code> and <code>$Y^*$</code> are the tensor products of extreme points of the respective unit balls.</p> http://mathoverflow.net/questions/90875/extreme-points-of-unit-ball-in-tensor-product-of-spaces/105485#105485 Answer by Yauhen Radyna for Extreme points of unit ball in tensor product of spaces Yauhen Radyna 2012-08-25T22:39:27Z 2012-08-25T22:39:27Z <p>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 </p> <p>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).</p> <p>The corresponding statement is true for denting points as shown in </p> <p>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).</p> <p>Now, it's enough to note that notions of extreme point and denting point are equivalent in my context.</p>