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In this paper we study simple transitive $2$-representations of certain$2$-subcategories of the $2$-category of projective functors over a staralgebra. We show that in the simplest case, which is associated to the Dynkintype $A_2$, simple transitive $2$-representations are classified by cell$2$-representations. In the general case we conjecture that there exist manysimple transitive $2$-representations which are not cell $2$-representationsand provide some evidence for our conjecture.

Simple transitive $2$-representations of left cell $2$-subcategories of projective functors for star algebras