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0 i= 1. Recall that a refinement of the topology on the range space or a coarsening of the topology on the domain space may destroy the continuity of a mapping.

20; 3)). 1 The Ordering of the States 7. If a and b are Hermitian, then Tr(ab)2 :s; Tr a 2b 2, a = a*, b- I = b*: ITr(ab)2 I :s; Tra 2 . By iterating this, ITr(ab)2P I :s; Tr(abb*a*)2 P - 1 = Tr(1 a 121 b2 1)2P - 1 :s; ... :s; Tr Ia 12P Ib 12P • Because of the Trotter product formula exp(a + b) = s-lim n .... 9)), ITr exp(o:a + f3b) I :s; Tr Iexp(o:a) II exp(f3b) I, for 0:, f3 E C, and initially for Hermitian operators of finite rank. It then extends to exp(o:a + f3b) E ge I (£'), exp(o:a) E gel (£'), exp(f3b) E gee£') and thereby yields a generalization of Corollary 3 known as the Golden-Thompson-Symanzik inequality [6], exp( -f3

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