▁L CA ▁to ▁L CA op . ▁ ▁The ▁du ality ▁inter changes ▁the ▁sub categories ▁of ▁discrete ▁groups ▁and ▁compact ▁groups . ▁If ▁R ▁is ▁a ▁ring ▁and ▁G ▁is ▁a ▁left ▁R - module , ▁the ▁dual ▁group ▁ ▁will ▁become ▁a ▁right ▁R - module ; ▁in ▁this ▁way ▁we ▁can ▁also ▁see ▁that ▁discrete ▁left ▁R - modules ▁will ▁be ▁Pon try agin ▁dual ▁to ▁compact ▁right ▁R - modules . ▁The ▁ring ▁End ( G ) ▁of ▁end omorph isms ▁in ▁L CA ▁is ▁changed ▁by ▁du ality ▁into ▁its ▁opposite ▁ring ▁( change ▁the ▁multiplication ▁to ▁the ▁other ▁order ). ▁For ▁example , ▁if ▁G ▁is ▁an ▁infinite ▁cy clic ▁discrete ▁group , ▁ ▁is ▁a ▁circle ▁group : ▁the ▁former ▁has ▁ ▁so ▁this ▁is ▁true ▁also ▁of ▁the ▁latter . ▁ ▁General izations ▁ ▁General izations ▁of ▁Pon try agin ▁du ality ▁are ▁constructed ▁in ▁two ▁main ▁directions : ▁for ▁commut ative ▁topological ▁groups ▁that ▁are ▁not ▁locally ▁compact , ▁and ▁for ▁non comm ut ative ▁topological ▁groups . ▁The ▁theories ▁in ▁these ▁two ▁cases ▁are ▁very ▁different . ▁ ▁D ual ities ▁for ▁commut ative ▁topological ▁groups ▁▁ ▁When ▁ ▁is ▁a ▁Haus dor ff ▁ab elian ▁topological ▁group , ▁the ▁group ▁ ▁with ▁the ▁compact - open ▁topology ▁is ▁a ▁Haus dor ff ▁ab elian ▁topological ▁group ▁and ▁the ▁natural ▁mapping ▁from ▁ ▁to ▁its ▁double - du al ▁▁ ▁makes ▁sense . ▁ ▁If ▁this ▁mapping ▁is ▁an ▁isomorphism , ▁it ▁is