Hypergroups and invariant complemented subspaces

Let K be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of L∞(K) and C0(K), the class of left translation invariant w⁎-subalgebras of L∞(K) and finally the class of non-zero left translation invariant C⁎-subalgebras of C0(K) in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w⁎-subalgebras of L∞(K), and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C⁎-subalgebras of C0(K). By the help of these two characterizations, we extract some results about invariant complemented subspaces of L∞(K) and C0(K).