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.