Convolution Type $$\hbox {C}^*$$-Algebras

Abstract

In this paper, by using the notion of convolution types we introduce symmetric and non-symmetric convolution type $$\hbox {C}^*$$-algebras. It is shown that any (exact) convolution type induces a (an exact) functor on the category of $$\hbox {C}^*$$-algebras. In particular, any group induces a convolution type and a functor on the category of $$\hbox {C}^*$$-algebras. It is also shown that discrete crossed product of $$\hbox {C}^*$$-algebras and discrete inverse semigroup $$\hbox {C}^*$$-algebras can be considered as convolution type $$\hbox {C}^*$$-algebras.