▁al gebras , ▁so ▁that ▁the ▁Pon try agin ▁du ality ▁fun ctor ▁ ▁turns ▁into ▁the ▁operation ▁ ▁of ▁taking ▁the ▁dual ▁vector ▁space ▁( which ▁is ▁a ▁du ality ▁fun ctor ▁in ▁the ▁category ▁of ▁finite ▁dimensional ▁H opf ▁al gebras ). ▁ ▁In ▁ 1 9 7 3 ▁Leon id ▁I . ▁V ain erman , ▁George ▁I . ▁K ac , ▁Michel ▁En ock , ▁and ▁Jean - Marie ▁Schw art z ▁built ▁a ▁general ▁theory ▁of ▁this ▁type ▁for ▁all ▁locally ▁compact ▁groups . ▁From ▁the ▁ 1 9 8 0 s ▁the ▁research ▁in ▁this ▁area ▁was ▁res umed ▁after ▁the ▁discovery ▁of ▁quantum ▁groups , ▁to ▁which ▁the ▁constructed ▁theories ▁began ▁to ▁be ▁act ively ▁transferred . ▁These ▁theories ▁are ▁form ulated ▁in ▁the ▁language ▁of ▁C * - al gebras , ▁or ▁Von ▁Ne umann ▁al gebras , ▁and ▁one ▁of ▁its ▁variants ▁is ▁the ▁recent ▁theory ▁of ▁locally ▁compact ▁quantum ▁groups . ▁ ▁One ▁of ▁the ▁draw back s ▁of ▁these ▁general ▁theories , ▁however , ▁is ▁that ▁in ▁them ▁the ▁objects ▁general izing ▁the ▁concept ▁of ▁group ▁are ▁not ▁H opf ▁al gebras ▁in ▁the ▁usual ▁algebraic ▁sense . ▁This ▁def ic iency ▁can ▁be ▁corrected ▁( for ▁some ▁classes ▁of ▁groups ) ▁within ▁the ▁framework ▁of ▁du ality ▁theories ▁constructed ▁on ▁the ▁basis ▁of ▁the ▁notion ▁of ▁en velope ▁of ▁topological ▁algebra . ▁ ▁See ▁also ▁ ▁Peter – W ey l ▁theorem ▁ ▁Cart ier ▁du ality ▁ ▁S tere ot