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 C 0 ( 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 C 0 ( 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 C 0 ( K ) . By the help of these two characterizations, we extract some results about invariant complemented subspaces of L ∞ ( K ) and C 0 ( K ) .