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  1. Abstract Let$$\phi $$ ϕ be a positive map from the$$n\times n$$ n × n matrices$$\mathcal {M}_n$$ M n to the$$m\times m$$ m × m matrices$$\mathcal {M}_m$$ M m . It is known that$$\phi $$ ϕ is 2-positive if and only if for all$$K\in \mathcal {M}_n$$ K M n and all strictly positive$$X\in \mathcal {M}_n$$ X M n ,$$\phi (K^*X^{-1}K) \geqslant \phi (K)^*\phi (X)^{-1}\phi (K)$$ ϕ ( K X - 1 K ) ϕ ( K ) ϕ ( X ) - 1 ϕ ( K ) . This inequality is not generally true if$$\phi $$ ϕ is merely a Schwarz map. We show that the corresponding tracial inequality$${{\,\textrm{Tr}\,}}[\phi (K^*X^{-1}K)] \geqslant {{\,\textrm{Tr}\,}}[\phi (K)^*\phi (X)^{-1}\phi (K)]$$ Tr [ ϕ ( K X - 1 K ) ] Tr [ ϕ ( K ) ϕ ( X ) - 1 ϕ ( K ) ] holds for a wider class of positive maps that is specified here. We also comment on the connections of this inequality with various monotonicity statements that have found wide use in mathematical physics, and apply it, and a close relative, to obtain some new, definitive results. 
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  2. The Golden–Thompson trace inequality, which states that Tr  e H+ K ≤ Tr  e H e K , has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or [Formula: see text] and K = potential, Tr  e H+(1− u) K e uK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr  X s Y t X 1− s Y 1− t ≤ Tr  XY for positive operators X, Y and for [Formula: see text], and [Formula: see text]. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the [Formula: see text] range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem. 
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