ing ▁ ▁makes ▁ ▁self - du al ▁instead . ▁ ▁The ▁group ▁algebra ▁▁ ▁The ▁space ▁of ▁integra ble ▁functions ▁on ▁ ▁a ▁locally ▁compact ▁ab elian ▁group ▁G ▁is ▁an ▁algebra , ▁where ▁multiplication ▁is ▁convolution : ▁the ▁convolution ▁of ▁two ▁integra ble ▁functions ▁f ▁and ▁g ▁is ▁defined ▁as ▁ ▁Theorem . ▁The ▁Ban ach ▁space ▁ ▁is ▁an ▁associ ative ▁and ▁commut ative ▁algebra ▁under ▁convolution . ▁ ▁This ▁algebra ▁is ▁referred ▁to ▁as ▁the ▁Group ▁Al gebra ▁of ▁G . ▁By ▁the ▁F ub ini – T on elli ▁theorem , ▁the ▁convolution ▁is ▁sub multip lic ative ▁with ▁respect ▁to ▁the ▁ ▁norm , ▁making ▁ ▁a ▁Ban ach ▁algebra . ▁The ▁Ban ach ▁algebra ▁ ▁has ▁a ▁multip lic ative ▁identity ▁element ▁if ▁and ▁only ▁if ▁G ▁is ▁a ▁discrete ▁group , ▁namely ▁the ▁function ▁that ▁is ▁ 1 ▁at ▁the ▁identity ▁and ▁zero ▁elsewhere . ▁In ▁general , ▁however , ▁it ▁has ▁an ▁approximate ▁identity ▁which ▁is ▁a ▁net ▁( or ▁generalized ▁sequence ) ▁ ▁indexed ▁on ▁a ▁directed ▁set ▁ ▁such ▁that ▁▁ ▁The ▁Fourier ▁transform ▁takes ▁convolution ▁to ▁multiplication , ▁i . e . ▁it ▁is ▁a ▁hom omorphism ▁of ▁ab elian ▁Ban ach ▁al gebras ▁▁ ▁( of ▁norm ▁ ≤ ▁ 1 ): ▁ ▁In ▁particular , ▁to ▁every ▁group ▁character ▁on ▁G ▁corresponds ▁a ▁unique ▁multip lic ative ▁linear ▁functional ▁on ▁the ▁group ▁algebra ▁defined ▁by ▁ ▁It ▁is ▁an ▁important ▁property ▁of ▁the ▁group ▁algebra ▁that ▁these ▁exhaust ▁the