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An Introduction to Noncommutative Spaces and their Geometry by Giovanni Landi

By Giovanni Landi

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

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37) n An , A = 3 33 The algebra A is the inductive (or direct) limit of the sequence {An, In }n∈IN [102]. As a set, n An is made of coherent sequences, An = {a = (an )n∈IN , an ∈ An | ∃N0 : an+1 = In(an ) , ∀ n > N0 }. 38) n Now the sequence (||an||An )n∈IN is eventually decreasing since ||an+1 || ≤ ||an|| (the maps In are norm decreasing) and therefore convergent. One writes for the norm on A, ||(an)n∈IN|| = n→∞ lim ||an ||An . 39) gives a true norm directly and not simply a seminorm and there is no need to quotient out the zero norm elements.

We start with some machinery from the theory of modules which we take mainly from [15]. 1 Suppose we are given an algebra A over (say) the complex numbers C. 3) for any η, ξ ∈ E and a, b ∈ A . 3 24 In fact, in [95] the correspondence is stated in the continuous category, namely for functions and sections which are continuous. However, it can be extended to the smooth case, see [26]. 2 Given two right A-modules E and F , a morphism of E into E is any linear map ρ : E → F which in addition is A-linear, namely ρ(ηa) = ρ(η)a , ∀ η ∈ E, a ∈ A .

Pj i∞     πij    ?    Pi Figure 7: The inverse system. 2 Let M be T0 and the collection {Ui } of coverings be such that for every m ∈ M and every neighborhood N m, there exists an index i and an element U ∈ τ (Ui ) such that m ∈ U ⊂ N. Then, the map π∞ is injective. Proof. If m1, m2 are two distinct points of M, since the latter is T0, there is an open set V containing m1 (say) and not m2 . By hypothesis, there exists an index i and an open U ∈ τ (Ui ) such that m1 ∈ U ⊂ V . Therefore τ (Ui ) distinguishes m1 from m2.

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