Let $X$ denote a complex $C^*$-algebra and $\{Z(t)\}_{t\geq 0}$ is a $C_0$-semigroup of operators on $X$.

Let $x\in X$ satisfy have $x=x^*$ (x is self-adjoint), such that its spectrum satisfies $\sigma(x)\subset [0,\infty)$.

Then under what conditions does it follow that $[Z^*(t)]=[Z(t)]$, and $\sigma(Z(t)x)\subset [0,\infty)$?

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    $\begingroup$ What does it mean for an element of a Banach space to be self-adjoint? $\endgroup$ – Michael Renardy Dec 16 '13 at 8:49
  • $\begingroup$ let me re-phrase it. $X$ is a complex $C^*$-algebra. $\endgroup$ – Shinning Star Dec 16 '13 at 9:02
  • $\begingroup$ What is $Z^\ast(f)$? And what is $f$? $\endgroup$ – András Bátkai Dec 16 '13 at 9:31
  • $\begingroup$ Sorry sir it was typing error. $Z^*(t)$ is adjoint operator of $Z(t)$. $\endgroup$ – Shinning Star Dec 16 '13 at 9:44
  • $\begingroup$ How do you define $Z^\ast (t)x$? $x\in X$ and not from $X^\ast$... $\endgroup$ – András Bátkai Dec 16 '13 at 9:58

Apart from the fact that I do not understand some parts of your question, self-adjoint elements with positive spectrum define a positive cone in your $C^\ast$ algebra. Positivity-preserving semigroups in $C^\ast$ and von Neumann algebras were extensively studies, you should consult the chapter written by Ulrich Groh in the book

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. Lotz, U. Moustakas, R. Nagel (ed.), F. Neubrander, U. Schlotterbeck One-parameter Semigroups of Positive Operators Springer, 1986.

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  • $\begingroup$ yes it actually defines a positive cone in $C^*$-algebra. I want isotonic $Z(t)$. i.e. whenever $x\geq 0$, $Z(t)x\geq 0$. and the order relation is defined above. $\endgroup$ – Shinning Star Dec 16 '13 at 18:02

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