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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.
The generating unitaries are the generator of the group Z and the identity function on the circle z : S 1 → C. [1] Twisted group algebra: The function σ : Z 2 × Z 2 → C; σ((m,n), (p,q)) = e 2πinpθ is a group 2-cocycle on Z 2, and the corresponding twisted group algebra C*(Z 2; σ) is isomorphic to A θ.
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are invariant , under the transformations from ...
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 mathematical physics, where symmetry is of central importance, or even just in multilinear algebra these operations are mostly (multilinear with respect to some vector structures and then) called antisymmetric operations, and when they are not already of arity greater than two, extended in an associative setting to cover more than two arguments.
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).
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