Glivenko theorem for $N^*$-extensions

Logic $N^*$ was defined semantically via combination of Kripke frames for intuitionistic logic with Routley's $*$-operator, which is used to interpret the negation operation. In this notice, we find out the least logic in the class of $N^*$-extensions, which satisfy Glivenko's theorem, and describe the Kripke semantics of this logic.