ARTCC RADAR CTLR DOES NOT CHALLENGE WEAK READBACK; AND OBSERVES A C525 DSND; WHEN THE CLRNC WAS INTENDED FOR ANOTHER ACFT. ACTIONS TAKEN WERE INSUFFICIENT TO MAINTAIN SEPARATION.

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A C525 (X) CAME OVER DSNDING TO FL270. A B737 (Y) WAS CLBING TO FL250. BOTH ACFT WERE IN LEVEL FLT AT FL270 AND FL250 RESPECTIVELY. I THEN ISSUED A CLB TO FL260 TO THE B737. THEN I CALLED TFC TO BOTH ACFT. THE C525 REPLIED HE HAD TFC IN SIGHT. I THEN TOLD THE C525 I HAD SOME AMENDED ROUTING WHEN HE WAS READY TO COPY. IT WAS TAKING HIM A WHILE TO RESPOND AND I WAS BUSY; SO I WENT TO MY NEXT MOVE WHICH WAS TO CLR ACFT ACR (Z) TO DSND TO FL200. THE C525 READ BACK HIS DSCNT CLRNC. THE READBACK WAS WEAK AND GARBLED AND HE RESPONDED WITH A PARTIAL CALL SIGN. I DID NOT CATCH THIS READBACK. A FEW MOMENTS LATER; I NOTICED THE C525 START WHAT APPEARED TO BE A DSCNT. I QUESTIONED HIM AS TO HIS ASSIGNED ALT. WHEN I REALIZED HE WAS DSNDING; I TOOK CORRECTIVE ACTION TO RE-ESTABLISH SEPARATION. I FEEL THERE WERE A FEW REASONS FOR THIS ERROR; I DID NOT CATCH THE BAD READBACK; AND THE C525 HAD A BAD RADIO AND DID NOT USE HIS FULL CALL SIGN FOR THE READBACK. THE OTHER ACR WAS ALSO NOT PAYING ATTN AND MISSED HIS CLRNC. ALSO THE OTHER ACR MISSED THE FACT THAT ANOTHER ACFT TOOK HIS CLRNC; THE RADAR COMPUTER; CDC; BLINKED; CAUSING A DISTR AND INCREASED WORKLOAD.