Enroute Controller described a loss of separation event apparently a result of confused and incomplete coordination between sectors.

Narrative

I was working Sector X and had given an E120 the restriction to cross ZZZZZ Intersection at 140. Sector Y called me and told me that a C172 wanted the RNAV approach to the airport and that he was my control. By the time the E120 was over ZZZZZ; Sector Y had taken the hand off and I had transferred communications of the E120. Sector Y has control for descent but since the C172 was over a nearby fix at the time I knew that they were responsible for separating the two aircraft in their sector. When I noticed the E120 [was] descending and converging with the C172; I initiated a call to Sector Y. After 3 attempts; Sector Y answered and I asked them if it was their intention to descend the E120 below the C172. I had already issued an approach clearance to the C172. The Sector Y Controller said he would turn the E120. The E120 was turned behind the C172 and I issued traffic to the C172. The closest proximity appeared to be 2.6 miles and 500 FT. I believe that the Sector Y Controller was not aware that control of the C172 was given to Sector 7 and did not know that the C172 was not flying on the flight plan route. Communication needed to be better.