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Since λ ∈ Lη , we have λ η < ∞. Using the Novikov property of λ and the injectivity of ω : Γ → R again, we find t0 ∈ R so that 1 − λ η +tω < 1 for all t t0 . Since L1η +tω is a Banach algebra, k 0 (1 − λ)k will converge and λ−1 ∈ Λω ∩ L1η +tω for all t t0 . + Recall that we have a bijection exp : Λ+ ω → 1 + Λω . 3. Suppose λ ∈ Λ+ ω and exp(λ) ∈ Lη . Then, there exists t0 ∈ R so that λ ∈ 1 Λω ∩ Lη +tω for all t t0 . Proof. 2 using log(1 − µ) = − µk k>0 k . 6. 6) in terms of closed trajectories under some assumptions.

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D. Burghelea and S. Haller, ‘Torsion as a function on the space of representations’, Proceedings of the a user, Basel), to appear. Conference on C ∗ -algebra and elliptic theory II, Trends in Mathematics (Birkh¨ 7. D. Fried, ‘Lefschetz formulas for flows’, Proceedings of the Lefschetz Centennial Conference, Part III, Mexico City, Mexico, 1984, Contemporary Mathematics 58 (American Mathematical Society, Providence, RI, 1987) 19–69. 8. E. Hille and R. S. Phillips, Functional analysis and semi-groups, Third printing of the revised edition of 1957, American Mathematical Society Colloquium Publications XXXI (American Mathematical Society, Providence, RI, 1974).

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