Definition 0.1. Let 𝒜,ℬ be two C*-algebras. Then a ∗-homomorphism ϕ∗ : 𝒜→ℬ is defined
as a C*-algebra homomorphism ϕ : 𝒜→ℬ which respects involutions, that is:
Note: If ‘by abuse of notation’ one uses ∗ to denote both ∗𝒜 and ∗ℬ, then any ∗-homomorphism ϕ
commutes with ∗, i.e., ϕ∗ = ∗ϕ.
Definition 0.2. The category 𝒞 whose objects are C∗-algebras and whose morphisms are
∗-homomorphisms is called the category of C∗-algebras or the C∗-algebra category.
Remark: Note that homomorphisms between C∗-algebras are automatically continuous.
References
[1] Kustermans, J., C*-algebraic Quantum Groups arising from Algebraic Quantum
Groups, Ph.D. Thesis, K.U.Leuven, 1997.
[2] Sheu, A.J.L., Compact Quantum Groups and Groupoid C*-Algebras, J. Funct.
Analysis 144 (1997), 371-393.
