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Let $A$ be a unital C*-algebra. A positive block matrix in $M_2(A)$ must have the form $$ \begin{pmatrix} a & a^{1/2} x b^{1/2} \\ b^{1/2} x^* a^{1/2} & b \end{pmatrix}, $$ where $a,b$ are positive, and $x$ is a contraction.

Suppose now I look at $A\otimes A$, and I have a positive matrix of the form $$ T=\begin{pmatrix} a\otimes 1+1\otimes c & (a\otimes 1+1\otimes c)^{1/2} x (b\otimes 1+1\otimes d)^{1/2} \\ (b\otimes 1+1\otimes d)^{1/2} x^* (a\otimes 1+1\otimes c)^{1/2} & b\otimes 1+1\otimes d\end{pmatrix}, $$ where now $a,b,c,d$ are positive (it turns out that a positive element of the form $a\otimes 1+1\otimes c$, can always be written with $a,c\geq0$). It's not possible to write $$ T = \begin{pmatrix} a\otimes 1 & y \\ y^* & b\otimes 1\end{pmatrix} + \begin{pmatrix} 1\otimes c & z \\ z^* & 1\otimes d\end{pmatrix}, $$ for any $y,z$ (in general: you can get a counter-example with $A=M_2$).

What sort of decomposition, of this form, might I be able to get for $T$?

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