证明
Lemma 4. Suppose P=P^*=P^2∈_m([G]) , an odd D>0 clears its denominators, and…Lemma 3. Let G be a group, m,r≥1 , D an odd positive integer, and P∈_m([G])…观点
Put N=M^*M . Continuity of M and its inverse on the compact interval gives constants 0<δ<c with δI≤N(x)≤cI for all x . Write W=I-N/c , so \|W\|≤ρ=1-δ/c<1 . Functional calculus gives the uniformly norm-convergent series N=( c)I-∑_ j≥1 (W^j)/(j). Its derivative series is also uniformly norm-convergent, since \|(d)/(dx)(W^j)/(j)\| ≤_ y∈[0,1] \|W'(y)\|ρ^ j-1 . In the derivative of W^j , cyclicity makes the traces of its j terms equal; commutation of N and N' is not assumed. Using W'=-N'/c therefore gives (d)/(dx)τ_n( N) =1c∑_ j≥1 τ_n(W^ j-1 N') =τ_n(N^ -1 N'). Since N'=M'^*M+M^*M' , the second summand contributes τ_n(M'M^ -1 ) . Cyclicity and MN^ -1 =(M^*)^ -1 turn the first into τ_n((M^*)^ -1 M'^*) = τ_n(M'M^ -1 ) . Their sum is the asserted real part. This is the tracial logarithmic derivative of~, written with the unnormalized trace used here.