PlanetPhysics/Category of C Algebras

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Let $𝒜,ℬ$ be two C*-algebras. Then a $*$-homomorphism

${\varphi }_{*}:𝒜⟶ℬ$ is defined as a [[../VonNeumannAlgebra2/|C*-algebra]] [[../TrivialGroupoid/|homomorphism]] $\varphi :𝒜\to ℬ$ which respects involutions, that is:

$\varphi (a^{{*}_{𝒜}})=\varphi (a{)}^{{*}_{ℬ}},\text{ for any }a\in 𝒜.$

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

The [[../Cod/|category]] $𝒞$ whose [[../TrivialGroupoid/|objects]] are $C^{*}$-algebras and whose [[../TrivialGroupoid/|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.