Timeline for Extreme points of unit ball in tensor product of spaces
Current License: CC BY-SA 3.0
7 events
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| Sep 29, 2020 at 19:50 | comment | added | J. van Dobben de Bruyn | Incidentally, Ruess and Stegall point out that this special case of their result had already been settled by I.I. Tseitlin, The extreme points of the unit ball of certain spaces of operators, Matematicheskie Zametki, vol. 20 (1976), issue 4, pp. 521–527. | |
| Sep 29, 2020 at 19:49 | comment | added | J. van Dobben de Bruyn | [...] The injective unit ball does not preserve extreme points. Indeed, if $\mathcal H$ is a Hilbert space with $2 \leq \dim(\mathcal H) < \infty$, then $\mathcal H \mathbin{\otimes_\varepsilon} \mathcal H \cong B(\mathcal H)$ isometrically. But the extreme points of the closed unit ball of $B(\mathcal H)$ are the unitary matrices, whereas the pure tensors $x \mathbin{\otimes} y$ correspond with matrices of rank $\leq 1$. So in fact $\text{ext}(B_{\mathcal H \mathbin{\otimes_\varepsilon} \mathcal H})$ is disjoint from $\text{ext}(B_{\mathcal H}) \mathbin{\otimes} \text{ext}(B_{\mathcal H})$. | |
| Sep 29, 2020 at 19:49 | comment | added | J. van Dobben de Bruyn | @Topology no, their paper deals with the dual of the injective tensor product. Bill Johnson's assumptions on $X$ and $Y$ ensure that $X^* \mathbin{\tilde\otimes_\pi} Y^* = (X \mathbin{\tilde\otimes_\varepsilon} Y)^*$ isometrically (see e.g. Theorem 16.6 in Defant and Floret, Tensor Norms and Operator Ideals, North-Holland, 1993). [...] | |
| Feb 7, 2020 at 17:16 | comment | added | Topology | I think Ruess, Stegall in their paper talk about the extreme point of the injective tensor product. How does the projective case follow from there? | |
| Mar 12, 2012 at 7:04 | vote | accept | Yauhen Radyna | ||
| Mar 11, 2012 at 23:08 | history | edited | Bill Johnson | CC BY-SA 3.0 |
added 1931 characters in body; deleted 1923 characters in body
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| Mar 11, 2012 at 22:17 | history | answered | Bill Johnson | CC BY-SA 3.0 |