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%%% Primary Title: category of C*-algebras
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\begin{document}
\begin{definition}
Let $\mathcal{A}, \mathcal{B}$ be two C*-algebras. Then a \emph{$*$-homomorphism}
$\phi_*:\mathcal{A} \longrightarrow \mathcal{B}$ is defined as a \htmladdnormallink{C*-algebra}{http://planetphysics.us/encyclopedia/VonNeumannAlgebra2.html} \htmladdnormallink{homomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\phi:\mathcal{A} \to \mathcal{B}$ which respects involutions, that is:
$$\phi(a^{*_{\mathcal{A}}}) = \phi(a)^{*_{\mathcal{B}}},\quad\mbox{ for any } a \in \mathcal{A}.$$
\end{definition}
{\em Note:}
If `by abuse of notation' one uses $*$ to denote both $*_{\mathcal{A}}$ and $*_{\mathcal{B}}$,
then any $*$-homomorphism $\phi$ \htmladdnormallink{commutes}{http://planetphysics.us/encyclopedia/Commutator.html} with $*$, i.e., $\phi*=*\phi$.
\begin{definition}
The \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} $\mathcal{C}$ whose \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are $C^*$-algebras and whose \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} are $*$-homomorphisms is called the \emph{category of $C^*$-algebras} or the $C^*$-algebra category.
\end{definition}
{\bf Remark:}
Note that homomorphisms between $C^*$-algebras are
automatically continuous.
\begin{thebibliography}{9}
\bibitem{KJ97}
Kustermans, J., C*-algebraic Quantum Groups arising from Algebraic Quantum
Groups, Ph.D. Thesis, K.U.Leuven, 1997.
\bibitem{SAJL97}
Sheu, A.J.L., Compact Quantum Groups and Groupoid C*-Algebras, J. Funct.
Analysis 144 (1997), 371-393.
\end{thebibliography}
\end{document}
