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By M. Aizenman (Chief Editor)

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6. 33 log(N − 1/2) log N − 1 < β( , N ) < , + log ε log ε+ where ε+ = ε+ . Hence, since β is integral, β( , N ) = 0 when N < ε + . Proof. Recall that β( , N ) = |{k ∈ N : tr((ε+ )k ) ≤ N }|. But tr((ε+ )k ) = (ε+ )k + tr((ε + )k ) − (ε + )k , which implies that tr((ε+ )k ) ≤ N k≤ iff log(N − tr((ε + )k ) + (ε + )k ) . 1 establishes the lemma, since then 0 < tr((ε+ )k ) − (ε + )k < 1/2. 7. ve (N ) = β( , N ), ε+

53 56 62 66 69 70 74 76 77 80 81 82 83 84 87 87 89 91 91 92 0. Introduction These notes are a natural outgrowth of our previous work on a local holomorphic formula for the dimension of a superselection sector [47] and were motivated by the purpose to give a geometrical picture to aspects of local quantum physics related to the superselection structure. They may be read from different points of view, in particular, guided by a similarity of the statistical dimension with the Fredholm index and a possible index theorem, already suggested in [15], we shall regard the DHR localized endomorphisms [16] as quantum analogs of elliptic differential operators.

But the present picture is too primitive to be directly applicable because super-KMS functionals fail to exist when the spacetime is non-compact in the most natural situation when the functional is translation invariant and space translations act in an asymptotically abelian fashion [10]. This drawback is entirely caused by the assumption that the superKMS functional is bounded. The structure associated with the unbounded super-KMS functional is a problem for future investigation. 1. First Properties of Holomorphic Cocycles In this section we begin to study holomorphic cocycles and give first formulae for the dimension.

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