By Giovanni Landi

ISBN-10: 3540635092

ISBN-13: 9783540635093

An advent to a number of principles & functions of noncommutative geometry. It starts off with a now not inevitably commutative yet associative algebra that's considered the algebra of capabilities on a few digital noncommutative area.

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**Additional info for An introduction to noncommutative spaces and their geometry**

**Sample text**

3 Projective Systems of Noncommutative Lattices 34 10 3 20 4 s ✏s ✏ ❅ ✏✏ ❅ ❅ ❅ ✏✏ ✏ ❅ ❅✏✏ ❅ ✏ ❅ ✏✏✏ ❅ ❅ ❅ ❅ ❅ ✏✏ ✏ s ✏ ✏ ❅s ❅s ❅s 12 11 21 s ❅ s ❅ 22 π23 ❄ 3 s ❅ 20 4 s ❅ ✟s ✟ ❅✟✟ ✟ ❅ ✟ ❅ ✟ ❅✟ ❅ ✟ ❅ ✟✟ ❅ s ✟ ❅s ❅s ❅ 1 22 21 π12 ❄ 3 s ❅ 4 ❅ ❅ ❅ ❅ s 1 s ❅s 2 Fig. 9. The projective system of posets for S 1 from one level to the next one, only one of the bottom points x is ‘split’ in three {x0 , x1 , x1 } while the others are not changed. The projection from one level to the previous one is the map which sends the triple {x0 , x1 , x1 } to the parent x while acting as the identity on the remaining points.

The projection is easily seen to be order preserving (and then continuous). As in the general case, the limit space P∞ consists of S 1 together with extra points. 4 Noncommutative Lattices 35 extra points come in couples anyone of which is ‘glued’ (in the sense of being inﬁnitely close) to a point in a numerable collection of points. This collection is dense in S 1 and could be taken as the collection of all points of the form {m/2n , m, n ∈ N} of the interval [0, 1] with endpoints identiﬁed. In [12] a somewhat diﬀerent interpretation of the approximation and of the limiting procedure in terms of simplicial decompositions has been proposed.

The projection from one level to the previous one is the map which sends the triple {x0 , x1 , x1 } to the parent x while acting as the identity on the remaining points. The projection is easily seen to be order preserving (and then continuous). As in the general case, the limit space P∞ consists of S 1 together with extra points. 4 Noncommutative Lattices 35 extra points come in couples anyone of which is ‘glued’ (in the sense of being inﬁnitely close) to a point in a numerable collection of points.

### An introduction to noncommutative spaces and their geometry by Giovanni Landi

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