# Lyapunov Theorems for Operator Algebras by Charles A. Akemann

By Charles A. Akemann

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V) For each s and t, \Mpv(s,t)-(i/2sn)\\°c 0 and {pv(s,t) : £ = 1 , . . 7. 7. • §6. LYAPUNOV T H E O R E M S FOR S I N G U L A R M A P S Introduction Recall that a continuous linear functional / on a von Neumann algebra M. 1]. l28]. We begin this section by studying singular functionals. 8) that, if the center of the finite part of M is finite-dimensional (and modest set theoretic assumptions are made so that M is "essentially countably decomposable"), then we have more precise information about projections in the kernel of a singular state.

5. If s and k are positive integers such that 2s 6, then (**) V ; fc-1 —— < (-} • ~ V3/ P R O O F . The inequality (*) is easily verified for pairs (s, k) with the values (1,3), (1,4) and (2,5). Moreover, (**) is true for (2,6), (2,7) and (2,8). So, to finish the proof, it suffices to show that (**) is true for s > 3 and k > 8. Clearly, it is enough to establish (**) when k — 1 = 2s. Now (**) holds in this case if and only if 3s < 22s~l = (1/2)4*. Since s > 3, we have 3 s = (27)3 5 - 3 < (32)4 S_3 = (1/2)4 S .

REMARK. 1 also holds in the more general case where \I/ is an affine map of a weak* closed face F of (M)i. We now briefly sketch this generalization in the case where F is a positive face. 1, part (1)) that in this case we have F = [r, s] for some projections r < s. We write 6(F, # ) = sup{\\-\$(q) - *(r)||i : q - r has rank one}. 1 may be stated as follows: If M. is atomic, \£ is a non-constant, self-adjoint, affine weak* continuous map from the weak* closed face F of (,M+)i into Cn, then for each a in F there is a projection p in F that commutes with a such that ||tf(p) - *(a)|| < 5(F, tf).