By Shoichiro Sakai
From the experiences: "This ebook is a superb and finished survey of the speculation of von Neumann algebras. It comprises the entire basic result of the topic, and is a precious reference for either the newbie and the expert." (Math. stories) "In conception, this e-book might be learn via a well-trained third-year graduate scholar - however the reader had higher have loads of mathematical sophistication. The professional during this and allied parts will locate the wealth of contemporary effects and new techniques through the textual content in particular rewarding." (American Scientist) "The name of this publication straight away indicates comparability with the 2 volumes of Dixmier and the truth that you possibly can heavily make this comparability shows that it's a way more significant paintings that others in this topic that have lately appeared"(BLMSoc)
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Extra info for C*-algebras and W*-algebras (Ergebnisse der Mathematik und ihrer Grenzgebiete)
3) If X is a normal variety, and B = bi Bi ⊂ X is a Q-divisor, the discrepancy of the pair (X , B) is the b-divisor A = A(X , B) with trace AY deﬁned by the formula: KY = f ∗ (KX + B) + AY on models f : Y → X of X . In order for A to be deﬁned, we need to assume that KX + B is Q-Cartier. The language of function algebras and b-divisors 29 (4) For an ordinary divisor D on X , we denote by D the proper transform b-divisor; its trace on models f : Y → X is DY = f∗−1 D. I usually abuse notation and simply write D instead of D.
The (klt, plt) ﬂip of f is a small projective birational morphism f : X → Z such that K + B is Q-Cartier and f -ample. 16 The ﬂip is unique if it exists. Indeed: f∗ OX i(K + B) , X = Proj i≥0 where, by deﬁnition, f∗ OX i(K + B) = f∗ OX iK + iB , provided that the algebra is ﬁnitely generated. The formula makes it clear that the existence of the ﬂip is local on Z in the Zariski topology. For this reason, in this chapter, I almost always assume that Z is afﬁne. 42] If the pair (X , B) has klt (plt) singularities, then so does the pair (X , B ).
5 Given a b-divisor D on X , we say that D descends to X , if D = DX . Surfaces are very special because mobile divisors on a surface are nef. This is the basis of the following theorem, which is the main result of this section. 6 Let (X , B) be a 2-dimensional terminal pair. ) Let f : X → Z be a birational weak Fano contraction to an afﬁne variety Z, in other words, −(K + B) is nef relative to f . If M is a mobile canonically saturated b-divisor on X , then M descends to X . Proof Let f : Y → X be a high enough log resolution of (X , B) such that (1) canonical saturation holds on Y , and (2) M = MY and, therefore, |MY | = |H 0 (Y , M)| is free.