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By Giovanni Landi (auth.)

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Extra resources 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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3 From Bratteli Diagrams to Noncommutative Lattices From the Bratteli diagram of an AF-algebra A one can also obtain the (norm closed two-sided) ideals of the latter and determine which ones are primitive. On the set of such ideals the topology is then given by constructing a poset whose partial order is provided by the inclusion of ideals. Therefore, both P rim(A) and its topology can be determined from the Bratteli diagram of A. This is possible thanks to the following results of Bratteli [17].

1 we have associated with each covering Ui a T0 -topological space Pi and a continuous surjection π i : M → Pi . 24) defined whenever i ≤ j and such that πi = πij ◦ πj . 25) These maps are uniquely defined by the fact that the spaces Pi ’s are T0 and that the map πi is continuous with respect to τ (Uj ) whenever i ≤ j. Indeed, (−1) if U is open in Pi , then πi (U ) is open in the Ui -topology by definition, thus it is also open in the finer Uj -topology and πi is continuous in τ (Uj ). 25), it follows that all maps πij are surjective.

1 we have associated with each covering Ui a T0 -topological space Pi and a continuous surjection π i : M → Pi . 24) defined whenever i ≤ j and such that πi = πij ◦ πj . 25) These maps are uniquely defined by the fact that the spaces Pi ’s are T0 and that the map πi is continuous with respect to τ (Uj ) whenever i ≤ j. Indeed, (−1) if U is open in Pi , then πi (U ) is open in the Ui -topology by definition, thus it is also open in the finer Uj -topology and πi is continuous in τ (Uj ). 25), it follows that all maps πij are surjective.

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