It is well known that a matrix which all entries are positive real numbers, has a positive eigenvalue.(see algebraic topology, by Allen Hatcher). Now is the following generalization, true?
Let A be a C\* algebra and X is a matrix which entries are positive elements of A.
Does sp(X) has nonempty intersection with positive real line?



the spectrum of matrix with positive entries
The answer is no. For example, the 4-by-4 matrix 
$\begin{pmatrix} p & e \\ f & q \end{pmatrix}$,
with $p=\begin{pmatrix} 1 & 0\\ 0 & 0\end{pmatrix}$, $q=1-p$, $e=\begin{pmatrix} 1/2 & 1/2\\ 1/2 & 1/2\end{pmatrix}$, and $f=1-e$ has the characteristic polynomial $x^4-2x^3+x^2+1/4=(x^2-x)^2+1/4\geq1/4$ and has no real eigenvalues.


