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](https://mathoverflow.net/questions/43138/positive-elements-in-tensor-products)



positive element in C* tensor product
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 $a$.


