An Introduction to Noncommutative Spaces and their by Giovanni Landi (auth.)

By Giovanni Landi (auth.)

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26 3 Projective Systems of Noncommutative Lattices a s ❅ b c s ❅ x3 ✟s ✟ ❅✟✟ ✟ ❅ ✟ ❅ ✟ ❅✟ ❅ ✟ ✟ ❅ ❅ ✟ ✟ ❅s ❅s s ❅ α β x s4 s ❅ s ❅s ❅ ❅ ❅ ❅ x1 γ x2 Fig. 1. The Hasse diagrams for P6 (S 1 ) and for P4 (S 1 ) Ui+1 Ui−1 ... ( ) ) ( ( ) Ui ) ( ( Ui+2 ) ... π ❄ yi−2 ··· ❅ s ❅ yi−1 s ❅ ❅ xi−2 yi s ❅ s ❅ xi−1 yi+1 s ❅ ❅ s ❅ xi ❅ s ❅ s ❅ ❅ xi+1 s ❅ ··· xi+2 Fig. 2. The finitary poset of the line R The generic finitary poset P (R) associated with the real line R is shown in Fig. 2. The corresponding projection π : R → P (R) is given by Ui+1 \ {Ui ∩ Ui+1 Ui ∩ Ui+1 −→ xi , i ∈ Z , Ui+1 ∩ Ui+2 } −→ yi , i ∈ Z .

45) The partial order given by the inclusions I1 ⊂ I2 produces the two point poset shown in Fig. 10. As we shall see, this space is really the fundamental building block for all posets. A comparison with the poset of the line in Fig. 2, shows that it can be thought of as a two point approximation of an interval. 38 3 Projective Systems of Noncommutative Lattices s I2 ✁ ✁ ✁ ✁ ✁ s I1 ✁ Fig. 10. 46) k where Md (C) is the algebra of d × d matrices with complex coefficients. In order to study the embedding A1 → A2 of any two such algebras A1 = n1 n2 j=1 Md(1) (C) and A2 = k=1 Md(2) (C), the following proposition [68, 153] is useful.

An } of elements of A and ε > 0, there exists a finite dimensional C ∗ -algebra B ⊆ A and elements b1 , . . , bn ∈ B such that ||ak − bk || < ε, k = 1, . . , n. 40 3 Projective Systems of Noncommutative Lattices Given a set D of ordered pairs (n, k), k = 1, · · · , kn , n = 0, 1, · · ·, with k0 = 1, and a sequence { p }p=0,1,··· of relations on D, the latter is the diagram D(A) of an AF-algebras when the following conditions are satisfied, (i) If (n, k), (m, q) ∈ D and m = n + 1, there exists one and only one nonnegative (or equivalently, at most a positive) integer p such that (n, k) p (n + 1, q).

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