By Giovanni Landi

An creation to numerous principles & functions of noncommutative geometry. It starts off with a now not unavoidably commutative yet associative algebra that's regarded as the algebra of services on a few digital noncommutative house.

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Let H be an infinite dimensional (separable) Hilbert space. 40) with K(H) the algebra of compact operators, is an AF-algebra [17]. 41) with embedding Mn (C) ⊕ C (Λ, λ) → Λ0 0 λ ,λ ∈ Mn+1 (C) ⊕ C . 42) Indeed, let {ξn }n∈N be an orthonormal basis in H and let Hn be the subspace generated by the first n basis elements, {ξ1 , · · · , ξn }. With Pn the orthogonal projection onto Hn , define An = {T ∈ B(H) | T (I − Pn ) = (I − Pn )T ∈ C(I − Pn )} B(Hn ) ⊕ C Mn (C) ⊕ C . 42). Since each T ∈ An is a sum of a finite rank operator and a multiple of the identity, one has that An ⊆ A = K(H) + CIH and, in turn, n An ⊆ A = K(H) + CIH .

We see that B(x) acts by compact operators on the Hilbert space H(x)u determined by the points which follow x and by multiples of the identity on the Hilbert space H(x)d determined by the points which precede x. These algebras satisfy the rules: B(x)B(y) ⊂ B(x) if x y and B(x)B(y) = 0 if x and y are not comparable. As already mentioned, the algebra A(P ) of the poset P is the algebra of bounded operators on H(P ) generated by all B(x) as x varies 14 15 This algebra is really defined only modulo Morita equivalence.

Furthermore, the ideal {0} ⊂ A is primitive if and only if A is primitive, which means that it has an irreducible faithful representation. This fact can also be inferred 11 In fact, the equivalence of (i) and (ii) is true for any separable C ∗ -algebra [55]. 44 3 Projective Systems of Noncommutative Lattices ❅ I2 ❅ ❅s ❅ ❅ ❅ s s ❅ ❅ ❅ s s ❅ ❅ ❅ s s ❅ .. .. ❅ I3 ❅ s ❅ ❅ ❅ s ❅s ❅ ❅ s ❅s ❅ ❅ s ❅s ❅ ❅ IK ❅ ❅ ❅ ❅ ❅s ❅ ❅ ❅s ❅ ❅ ❅s ❅ (b) (c) .. .. (a) .. .. Fig. 12. The three ideals of the algebra A∨ from the Bratteli diagram.

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