In this paper, we introduce the concepts of positive implicative [resp. implicative and commutative] $\Gamma$-$KU$-algebras, and obtain their some properties (including characterizations) respectively and some relationships among them. Next, we propose the notions of positive implicative [resp. implicative and commutative] $\Gamma$-ideals of a $\Gamma$-$KU$-algebra, and deal with their some properties (including characterizations) respectively and some relationships among them. Finally, we define a topological $\Gamma$-$KU$-algebra and discuss its various topological structures.