LOSS OF SEPARATION ON APCH TO HNL; HI. TCASII RA IGNORED.

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WE WERE TOLD TO MAINTAIN 4000 FT AND LOOK FOR TFC AT OUR 9 O'CLOCK POS APPROX 4-5 MI AWAY ALSO AT 4000 FT; A B737. WITH THE TFC IN SIGHT AND ARPT IN SIGHT WE REQUESTED A VISUAL APCH FOR RWY 8L. WE THOUGHT WE'D BE CLRED FOR THE VISUAL BEHIND THE B737 BECAUSE OF OUR REDUCED AIRSPD. APCH TOLD US TO MAINTAIN 4000 FT BECAUSE OF A GA ACFT (OR A LIGHT TWIN?) DOING APCHS ON RWY 4R. AT THIS TIME I ALSO NOTICED THE LACK OF MOVEMENT BTWN OUR ACFT AND THE B737; WHICH MUST HAVE MEANT THAT IT WAS SPD RESTR TO ABOUT 170 KTS AS WELL. OUR APPROXIMATE VECTOR AT THIS TIME WAS 310 DEG HDG. ABOUT 6 OR 7 DME; APCH GAVE US A 270 DEG HDG FOR SEQUENCING; WHICH TOOK OUR ACFT ON A CONVERGING COURSE WITH THE B737. SHORTLY BEFORE OUR HEADING ASSIGNMENT; THE B737 SAID IT HAD THE ARPT IN SIGHT AND APCH ASSIGNED THAT ACFT A VISUAL APCH FOR RWY 8L. APCH TOLD THE B737 TO MAINTAIN VISUAL SEPARATION WITH OUR ACFT AND THEY SAID THEY HAD US IN SIGHT. I NOTICED THE ACFT BEGAN TURNING TOWARD US; BUT WAS STILL AT 4000 FT (LOOKING AT THEIR TARGET ON THE TCASII). MY FO WAS FLYING AT THE TIME AND I ADVISED HIM THAT I WAS WATCHING THE CONFLICTING TFC. AS THE ACFT BEGAN A DSCNT; WE GOT A 'TFC; TFC' ADVISORY ON OUR TCASII AND SOON AFTER AN RA THAT TOLD US TO CLB. I HAD BEEN WATCHING THE B737 THE WHOLE TIME AND REALIZED THE TFC WAS NO FACTOR FOR US AND TOLD MY FO TO MAINTAIN 4000 FT. THE LOSS OF SEPARATION AS THE ACFT PASSED BELOW US WAS ABOUT 200-300 FT. WE WERE SOON AFTER CLRED FOR THE VISUAL APCH BEHIND THE B737 AND LANDED UNEVENTFULLY.