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:

trivialedits to questions that were answered over two years ago? The original question seemedperfectly readableto 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