and similarly for $ \chi ^{\omega } $.

The proof is along the lines of that of Theorem \[thrm: onechangeofvar\], using the ideas of a similar Proposition in [@dvv:entropy2], and is therefore omitted.

General change of variables formula. 
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Let $ B\subset M $ be a unital subalgebra, and let $ {F_{1},\dots ,F_{n}}$ be non-commutative power series with coefficients from $ B $; i.e., $$F_{i}(t_{1},\dots ,t_{n})=\sum _{k}\sum _{i_{1},\dots ,i_{k}}b_{{i_{1},\dots ,i_{k}}}^{i,0}t_{i_{1}}b^{i,1}_{{i_{1},\dots ,i_{k}}}\dots t_{i_{k}}b_{{i_{1},\dots ,i_{k}}}^{i,k}.$$ Denote by $ B[t_{1},\dots ,t_{n}] $ the set of all such power series, which have the property that if $ F\in B[t_{1},\dots ,t_{n}] $ and $ {X_{1},\dots ,X_{n}}$ are self-adjoint, then $ F({X_{1},\dots ,X_{n}}) $ is also self-adjoint. Given $ F_{i}\in B[t_{1},\dots ,t_{n}] $ as above, denote by $ \hat{F}_{i} $ the power series $$\hat{F}_{i}(z_{1},\dots ,z_{n})=\sum _{k}\sum _{{i_{1},\dots ,i_{k}}}\prod ^{k}_{j=1}\Vert b_{{i_{1},\dots ,i_{k}}}^{i,j}\Vert y_{i_{1}}\dots y_{i_{k}}.$$ We say that $ (R_{1},\dots ,R_{n}) $ is a mutiradius of convergence of $ F_{i} $, if it is the multiradius of convergence of $ \hat{F}_{i} $ (as an ordinary commutative power series).

Let $ F\in B[t_{1},\dots ,t_{n}] $ be such a power series. Then by the derivative of $ F $ with respect to $ t_{i} $ we mean the formal power 