I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I couldn't answer the follow-up question: "Why do we study $C^*$-algebras?". So, the question is why do we study $C^*$-algebras.
I know of their connection to bounded operators on Hilbert spaces and the Wikipedia article mentions something about quantum field theory, but these do not feel satisfactory. In conclusion, is there other and better motivation to study $C^*$-algebras?