Lemma 40 ([Spectral-projection comparison]). Let T,X be positive affiliated units in M , and let t>0 . For p₊=1_ (t,∞) (T) and p_-=1_ (0,t) (T) , τ(p₊F(T,X)) , ≥τ(p₊F(t1,X)), ; τ(p_-F(T,X)) , ≤τ(p_-F(t1,X)).

First assume that T,X and their inverses are bounded. Put p=p_+ and q=1-p , and decompose with respect to p+q : T= A, 0; 0, B , X= X_ 11 , X_ 12 ; X_ 21 , X_ 22 . Here A≥tp and B≤tq . The positive Schur complements K_B=X_ 11 -X_ 12 (B+X_ 22 )^ -1 X_ 21 , K_t=X_ 11 -X_ 12 (tq+X_ 22 )^ -1 X_ 21 satisfy K_B≤K_t . Block inversion and cyclicity of the trace give τ(pF(T,X))=τ(A(A+K_B)^ -1 ). For positive invertible A,K in p Mp , set ψ(A,K)=τ(A(A+K)^ -1 ). This function is increasing in A , because ψ(A,K)=τ(p) -τ(K^ 1/2 (A+K)^ -1 K^ 1/2 ), and is decreasing in K , by its defining expression with A^ 1/2 on the two sides of the inverse. Consequently ψ(A,K_B)≥ψ(tp,K_B)≥ψ(tp,K_t) =τ(pF(t1,X)), which proves. For p=p_- , the comparisons A≤tp , B≥tq , and K_B≥K_t reverse both inequalities, giving. The cases p=0 or q=0 follow from the same argument with the empty block omitted. For affiliated units, clip T and X spectrally into [1/R,R] , with 1/R<t<R . The high and low spectral projections of the clipped T at t are the original projections. The clipped operators converge in measure to T,X , and their inverses converge to the corresponding inverses. Lemma 39 ([Measure calculus]) and show that the resulting positive contractions converge in L^1 . Both inequalities pass to the limit.