ARITHMETIC HECKE OPERATORS AS COMPLETELY POSITIVE MAPS

We prove that the arithmetic Hecke operators are completely positive m aps with respect to the Berezin's quantization deformation product of functions on H/Gamma. We then show that the associated subfactor defin ed by the Connes's correspondence associated to the completely positiv e map has integer index and graph A(infinity). The same construction f or PSL(3, Z) gives a finite index subfactor of L(PSL(3, Z)) of infinit e depth.