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

By Giovanni Landi (auth.)

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Additional info for An Introduction to Noncommutative Spaces and their Geometries: Characterization of the Shallow Subsurface Implications for Urban Infrastructure and Environmental Assessment

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The maps In are injective -morphisms. The algebra A is the inductive (or direct) limit of the inductive (or direct) system {An , In }n∈N of algebras [124, 153]. As a set, n An is made of coherent sequences, An = {a = (an )n∈N , an ∈ An | ∃N0 , an+1 = In (an ) , ∀ n > N0 }. 38) n Now the sequence (||an ||An )n∈N 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∈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.

8 There is a special class of C ∗ -algebras which have been used in a scheme of approximation by means of topological lattices [7, 8, 12]; they are postliminal 8 If H is finite dimensional, H = Cn say, then B(Cn ) = K(Cn ) = Mn (C), the algebra of n × n matrices with complex entries. Such an algebra has only one irreducible representation (as an algebra), namely the defining one. 5 Real Algebras and Jordan Algebras 19 algebras. For these algebras, a relevant rˆole is again played by the compact operators.

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

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