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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.
In mathematics, an injective function (also known as injection, or one-to-one function [1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f(x 1) ≠ f(x 2) (equivalently by contraposition, f(x 1) = f(x 2) implies x 1 = x 2).
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.
In particular when θ = 0, A θ is isomorphic to continuous functions on the 2-torus by the Gelfand transform. Irrational rotation algebra: Let the infinite cyclic group Z act on the circle S 1 by the rotation action by angle 2 π iθ. This induces an action of Z by automorphisms on the algebra of continuous functions C(S 1).
For example, if : [,] is the Dirichlet function that is on irrational numbers and on rational numbers, and [,] is equipped with Lebesgue measure, then the support of is the entire interval [,], but the essential support of is empty, since is equal almost everywhere to the zero function.
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.
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