Perturbation Formulas for Traces on $C^*$-algebras

We introduce the Frechet differential of operator functions on C*-aIgebras obtained via spectral theory from ordinary differentiate functions. In the finite-dimensional case this differential is expressed in terms of Hadamard products of matrices. A perturbation formula with applications to traces is given. §1. The Frechet Differentia! Definition 1.1. // & and ty are Banach spaces, and Q) is an open subset of &, we say that a function F: & -+& is Frechet differentiate, if for each x in 2 there is a bounded linear operator F^. in B(2£9 <&) such that lim Ufeir^FOc + h) F(x) Fx (h)) = 0 . If the differential map x -» F^ is continuous from & to B(3E9 $/\ we say that F is continuously Frechet differ entiable. Straightforward computations give the following result, which we list for easy reference. Proposition 1.2. // F: X -» %/ and G: ty -> 3£ are continuously Frechet differentiable maps between Banach spaces 3£9 exp (A) is continuously Frechet differentiable with f 1 exp^ (B) = exp (sA)B exp ((1 s)A)ds Jo for all A, B in <$/. Proof. By elementary calculus we have 11 k s (l s)ds = Jo v ' (fc + m+1)! and we can prove either by direct calculation or by induction that (A + B) -A = ^(A + B)BA~v . k=0 Combining these two expressions we establish the Dyson formula (*) exp (A + B)exp (A) = £ V —(A + n=i fc=o nl f 1 = exp (s(A + B))B exp ((1 s)A)ds , Jo where we rearranged the sums by setting m = n — k— 1. It is clear that the proposed expression for exp^ is a bounded linear operator that depends continuously on A, and by subtraction we get from (*) that Hexp (A + B) exp (A) exp^ (B)|| 1 (exp (s(A + B)) exp (sA))B exp ((1 s)A)ds \ J ||exp (s(A + B) exp (sA)\\ds . o PERTURBATION FORMULAS FOR TRACES ON C*-ALGEBRAS 171 From Lebesgues theorem of dominated convergence we see that the last integral converges to zero as B -» 0. We can thus conclude that exp is continuously Frechet differentiable with the desired differential. QED Definition 1.4. We denote by C£(R) the set of real ^-functions f of the form