Fringe is left off the shelf (4)
I believe the answer is:
edge
'fringe' is the definition.
(synonyms)
'left off the shelf' is the wordplay.
'left' becomes 'l' (common abbreviation).
'off' indicates named letters should be taken away (some letters taken off).
'the shelf' becomes 'ledge' (synonyms).
'ledge' with 'l' taken away is 'EDGE'.
'is' is the link.
(Other definitions for edge that I've seen before include "Steal" , "Final line" , "Sharp side of a blade" , "Skirt" , "Sharp ridge" .)