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By Giovanni Landi (auth.)

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Extra info for An Introduction to Noncommutative Spaces and their Geometries: Characterization of the Shallow Subsurface Implications for Urban Infrastructure and Environmental Assessment

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By identifying points in P∞ which cannot be separated one recovers M . The interpretation that emerges is that the top points of a poset P (M ) (which are always closed) approximate the points of M and give all of M in the limit. The rˆ ole of the remaining points is to ‘glue’ the top points together so as to produce a topologically nontrivial approximation to M . They also give the extra points in the limit. 8 shows the 2N -poset approximation to S 1 obtained with a covering consisting of N open sets.

Then the following statements are equivalent. (i) The algebra A is AF. (ii) The algebra A is generated in the norm topology by a sequence of projectors {Pi }, with P0 = I. 10 Proof. The equivalence of (i) and (ii) is clear. To prove that (iii) implies (ii), let X be a second-countable, totally disconnected, compact Hausdorff space. Then X has a countable basis {Xn } of open-closed sets. Let Pn be the characteristic function of Xn . The ∗ -algebra generated by the projector {Pn } is dense in C(X): since P rimC(X) = X, (iii) implies (ii).

Let A = n Un be any AF-algebra with associated Bratteli diagram D(A). Let I be an ideal of A. 62) with the subset ΛI ⊂ D(A) satisfying the following two properties: i) if (n, k) ∈ ΛI and (n, k) p (n + 1, j) , p > 0, then (n + 1, j) belongs to ΛI ; ii) if all factors (n + 1, j) , j = 1, . . , nn+1 , in which (n, k) is partially embedded belong to ΛI , then (n, k) belongs to ΛI . 62) (with Λ substituted for ΛI ) is an ideal in A such that I An = ⊕k;(n,k)∈ΛI Md(n) (C). k Proposition 18. Let A = n Un , let I be an ideal of A and let ΛI ⊂ D(A) be the associated subdiagram.

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