PlanetPhysics/Category of C Algebras

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Let ${\mathcal {A}},{\mathcal {B}}$ be two C*-algebras. Then a $*$-homomorphism

$\phi _{*}:{\mathcal {A}}\longrightarrow {\mathcal {B}}$ is defined as a C*-algebra homomorphism $\phi :{\mathcal {A}}\to {\mathcal {B}}$ which respects involutions, that is:

$\phi (a^{*_{\mathcal {A}}})=\phi (a)^{*_{\mathcal {B}}},\quad {\mbox{ for any }}a\in {\mathcal {A}}.$

Note: If `by abuse of notation' one uses $*$ to denote both $*_{\mathcal {A}}$ and $*_{\mathcal {B}}$, then any $*$-homomorphism $\phi$ commutes with $*$, i.e., $\phi *=*\phi$.

The category ${\mathcal {C}}$ whose objects are $C^{*}$-algebras and whose morphisms are $*$-homomorphisms is called the category of $C^{*}$-algebras or the $C^{*}$-algebra category.

{\mathbf Remark:} Note that homomorphisms between $C^{*}$-algebras are automatically continuous.