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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 $x_1\otimes x_2\leq x$?

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.

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:

link text

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In view of Jesse Peterson's example, your claim that such elements exist when one of your C*-algebras is finite-dimensional seems to be false... – Yemon Choi Mar 12 '11 at 4:19
Right, I made a mistake – Qingyun Mar 12 '11 at 6:55
In case @DavideGiraudo is reading - could you please stop making trivial edits to questions that were answered over two years ago? The original question seemed perfectly readable to me and it is annoying to see this dead question brought back to the front page for such minor reasons – Yemon Choi Jul 7 '13 at 19:43
@YemonChoi OK, I won't do it anymore. – Davide Giraudo Jul 7 '13 at 20:17
up vote 10 down vote accepted

The same answer as before, the matrix $$ a=\begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{bmatrix} $$ 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 $a$.

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Got it, thanks! – Qingyun Mar 12 '11 at 7:45

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".

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