positive element in C* tensor product - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:18:17Z http://mathoverflow.net/feeds/question/58186 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58186/positive-element-in-c-tensor-product positive element in C* tensor product Qingyun 2011-03-11T18:54:23Z 2011-03-12T07:11:20Z <p>Let A, B be two C*-Algebras and $A\otimes B$ denote their minimal tensor product(I don't know whether C*-norm matters or not, but for simplicity we can assume that one of them is nuclear so all C*-norm coincide). Let x be a non-zero positive element in $A\otimes B$, can we always find a single tensor $0\neq x_1\otimes x_2$, where $x_1$ and $x_2$ are positive elements in A and B respectively, such that $x_1\otimes x_2\leq x$?</p> <p>It's fairly easy to see that if both C*-algebras are communicative or one of them is a finite dimensional C*-Algebra(Sorry this is false), then the above assertion is true. So it's tempting to think that more general case should hold.</p> <p>I asked a similar question before, where the stronger assertion that any positive element in tensor algebra is a sum of tensors of positive elements, is false. See the following link:</p> <p><a href="http://mathoverflow.net/questions/43138/positive-elements-in-tensor-products" rel="nofollow">link text</a></p> http://mathoverflow.net/questions/58186/positive-element-in-c-tensor-product/58200#58200 Answer by Jesse Peterson for positive element in C* tensor product Jesse Peterson 2011-03-11T21:43:00Z 2011-03-11T21:43:00Z <p>The same answer as before, the matrix <code>$$a=\begin{bmatrix} 1 &amp; 0 &amp; 0 &amp; 1 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp; 0 &amp; 0 &amp; 0 \\ 1 &amp; 0 &amp; 0 &amp; 1 \end{bmatrix}$$</code> in $M_2(\mathbb{C})\otimes M_2(\mathbb{C})$, also works here since it is twice a rank one projection and so any smaller positive matrix must be a scalar multiple of $a$.</p> http://mathoverflow.net/questions/58186/positive-element-in-c-tensor-product/58240#58240 Answer by Leonel Robert for positive element in C* tensor product Leonel Robert 2011-03-12T07:11:20Z 2011-03-12T07:11:20Z <p>There is a result of Kirchberg that comes close to giving a positive answer to this question. Given $x\geq 0$ as in the question, there exists $z\neq 0$ such that $z^*z=x_1\otimes x_2$ and $zz^* \leq x$. See Lemma 4.1.9 of Rordam's book "Classification of nuclear C*-algebras".</p>