, ▁or ▁of ▁a ▁count able ▁number ▁with ▁states , ▁is ▁a ▁von ▁Ne umann ▁algebra ▁as ▁described ▁in ▁the ▁section ▁above . ▁The ▁crossed ▁product ▁of ▁a ▁von ▁Ne umann ▁algebra ▁by ▁a ▁discrete ▁( or ▁more ▁generally ▁locally ▁compact ) ▁group ▁can ▁be ▁defined , ▁and ▁is ▁a ▁von ▁Ne umann ▁algebra . ▁Special ▁cases ▁are ▁the ▁group - measure ▁space ▁construction ▁of ▁Murray ▁and ▁von ▁Ne umann ▁and ▁Krie ger ▁factors . ▁The ▁von ▁Ne umann ▁al gebras ▁of ▁a ▁meas urable ▁equivalence ▁relation ▁and ▁a ▁meas urable ▁gr ou po id ▁can ▁be ▁defined . ▁These ▁examples ▁general ise ▁von ▁Ne umann ▁group ▁al gebras ▁and ▁the ▁group - measure ▁space ▁construction . ▁ ▁App lications ▁Von ▁Ne umann ▁al gebras ▁have ▁found ▁applications ▁in ▁diverse ▁areas ▁of ▁mathematics ▁like ▁kn ot ▁theory , ▁statistical ▁mechan ics , ▁Quant um ▁field ▁theory , ▁Local ▁quantum ▁physics , ▁Free ▁probability , ▁Non comm ut ative ▁geometry , ▁representation ▁theory , ▁geometry , ▁and ▁probability . ▁ ▁For ▁instance , ▁C * - algebra ▁provides ▁an ▁alternative ▁ax iom at ization ▁to ▁probability ▁theory . ▁In ▁this ▁case ▁the ▁method ▁goes ▁by ▁the ▁name ▁of ▁G elf and – Na im ark – Seg al ▁construction . ▁This ▁is ▁analog ous ▁to ▁the ▁two ▁approaches ▁to ▁measure ▁and ▁integration , ▁where ▁one ▁has ▁the ▁choice ▁to ▁construct ▁measures ▁of ▁sets ▁first ▁and ▁define ▁integr als ▁later , ▁or ▁construct ▁integr als ▁first ▁and ▁define ▁set ▁measures ▁as ▁integr als ▁of ▁characteristic