ament ally ▁different ▁properties . ▁The ▁usual ▁operator ▁trace ▁on ▁the ▁trace - class ▁operators ▁does ▁not ▁extend ▁to ▁the ▁weak ▁trace ▁class . ▁Instead ▁the ▁ideal ▁of ▁weak ▁trace - class ▁operators ▁adm its ▁an ▁infinite ▁number ▁of ▁linear ly ▁independent ▁quasi - continu ous ▁traces , ▁and ▁it ▁is ▁the ▁smallest ▁two - s ided ▁ideal ▁for ▁which ▁all ▁traces ▁on ▁it ▁are ▁singular ▁traces . ▁ ▁We ak ▁trace - class ▁operators ▁feature ▁in ▁the ▁non comm ut ative ▁geometry ▁of ▁French ▁math ematic ian ▁A lain ▁Con nes . ▁ ▁Definition ▁▁ ▁A ▁compact ▁operator ▁A ▁on ▁an ▁infinite ▁dimensional ▁separ able ▁Hil bert ▁space ▁H ▁is ▁weak ▁trace ▁class ▁if ▁ μ ( n , A ) ▁ ▁O ( n − 1 ), ▁where ▁ μ ( A ) ▁is ▁the ▁sequence ▁of ▁singular ▁values . ▁ ▁In ▁mathematical ▁notation ▁the ▁two - s ided ▁ideal ▁of ▁all ▁weak ▁trace - class ▁operators ▁is ▁denoted , ▁ ▁where ▁ ▁are ▁the ▁compact ▁operators . ▁The ▁term ▁weak ▁trace - class , ▁or ▁weak - L 1 , ▁is ▁used ▁because ▁the ▁operator ▁ideal ▁corresponds , ▁in ▁J . ▁W . ▁C alk in ' s ▁correspond ence ▁between ▁two - s ided ▁ide als ▁of ▁bounded ▁linear ▁operators ▁and ▁re arr ang ement ▁invariant ▁sequence ▁spaces , ▁to ▁the ▁weak - l 1 ▁sequence ▁space . ▁ ▁Properties ▁▁▁ ▁the ▁weak ▁trace - class ▁operators ▁admit ▁a ▁quasi - norm ▁defined ▁by ▁ ▁making ▁L 1 ,