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The noncommutative torus, deformation of the function algebra of the ordinary torus, can be given the structure of a spectral triple. This class of examples has been studied intensively and still functions as a test case for more complicated situations. Snyder space [10] Noncommutative algebras arising from foliations.
To give an example, let X = P 1 × A 1 and p: X → A 1 the projection. Here X is an algebraic variety since it is a product of varieties. It is not affine since P 1 is a closed subvariety of X (as the zero locus of p ), but an affine variety cannot contain a projective variety of positive dimension as a closed subvariety.
And so certain types of functions can correspond to certain properties of a C*-algebra. For example, self-adjoint elements of a commutative C*-algebra correspond to real-valued continuous functions. Also, projections (i.e. self-adjoint idempotents) correspond to indicator functions of clopen sets. Categorical constructions lead to some examples.
Dirac delta function: everywhere zero except for x = 0; total integral is 1. Not a function but a distribution, but sometimes informally referred to as a function, particularly by physicists and engineers. Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous.
Some common non-elementary antiderivative functions are given names, defining so-called special functions, and formulas involving these new functions can express a larger class of non-elementary antiderivatives. The examples above name the corresponding special functions in parentheses.
The most salient deformation theory in mathematics has been that of complex manifolds and algebraic varieties.This was put on a firm basis by foundational work of Kunihiko Kodaira and Donald C. Spencer, after deformation techniques had received a great deal of more tentative application in the Italian school of algebraic geometry.
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