By Giovanni Landi

Those lectures notes are an intoduction for physicists to a number of principles and purposes of noncommutative geometry. the required mathematical instruments are awarded in a fashion which we believe might be obtainable to physicists. We illustrate purposes to Yang-Mills, fermionic and gravity versions, particularly we describe the spectral motion lately brought through Chamseddine and Connes. We additionally current an creation to fresh paintings on noncommutative lattices. The latter were used to build topologically nontrivial quantum mechanical and box concept versions, specifically replacement versions of lattice gauge thought.

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**Example text**

Ii) If F ⊂ P is a closed set which is not the union of two proper closed subset, then F is the closure of a one-point set. (iii) The space P contains at most a countable number of closed sets. (iv) If {Fn }n is a decreasing (Fn+1 ⊂ Fn) sequence of closed subsets of P , then an element in {Fn}n . n Fn is Then, there exists an AF algebra A whose primitive space P rimA is homeomorphic to P . Proof. The proof consists in constructing explicitly the Bratteli diagram D(A) of the algebra A. We shall sketch the main passages while referring to [13] for more details.

N, ∀ n≥N +1 . 96) In the diagram D(AP ), for any n ≥ N, (n, k) (n + 1, j) if and only if {xk } Sj = ∅, namely if and only if xk ∈ Sj . We also sketch the algorithm to construct the algebra limit AP determined by the Bratteli diagram D(AP ) 22 [6, 45]. The idea is to associate to the poset P an infinite dimensional separable Hilbert space H(P ) out of tensor products and direct sums of infinite dimenxj , in the sional (separable) Hilbert spaces Hij associated with each link (xi , xj ), xi 23 poset .

T. xi = πi,i+1 (xi+1) , ∀ i ≥ N0 . 29) As for the map πi∞ , it is just defined by πi∞ (x) = xi . 30) The space Pi∞ is made a T0 topological space by endowing it with the weakest topology (−1) making all maps πi∞ continuous: a basis for it is given by the sets πi∞ (U), for all open sets U ⊂ Pi . The inverse system and its limit are depicted in Fig. 7 It turns out that the limit space P∞ is bigger than the starting M and the latter is contained as a dense subspace. Furthermore, M can be characterized as the set of all closed points of Pi∞ .

### An Introduction to Noncommutative Spaces and their Geometry by Giovanni Landi

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