C * – algebra 

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C * – algebra . It is a structure that arises in functional analysis; Before proposing it directly, another structure, previously called asterisk – algebra, will be presented beforehand.

Summary

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  • 1 Asterisk – algebra
  • 2 names
  • 3 Properties
  • 4 Definition
    • 1 C * Properties – Algebras
    • 2 Examples
  • 5 Source
  • 6 Notes and references

Asterisk – algebra

An algebra A is called * – algebra if in A there is an application * of A in A, called involution , it fulfills the following being syt element is A and l, complex number:

  • The involution of the involution of an element returns this: (s *) * = s
  • the involution of a sum is the sum of the respective involutions: (s + t) * = s * + t *
  • the involution of a product is the product of the respective involutions (st) * = s * t *
  • the involution of a scalar multiple is equal to the product of the conjugate by the involution of the element. (lsT * = l’s * where l ‘= complex conjugate l.

Names

  • an element is hermeticif it coincides with its involution: s = s *
  • if the product of an element with its involution commutes is called the normal element: ss * = s * s
  • if tt * = t * t = 1 is true for an element t of A, t is said to be a unit element.

Properties

  • In all * -algebra elements 0 and 1 (the second if any) are hermetic
  • Every element s of a * -algebra can be expressed in a univocal way as:

s = h + ik where h and k are hermetic, i = imaginary unit.

  • Let B and D be two * -algebras. A morphism mof B in D is called * – ‘ morphism if m (s *) = m (s) *. In particular, two * -algebras are isomorphic if there is an isomorphism of B over D which is an * -morphism.

Definition

A Banach algebra is called C * – algebra if it is a * – algebra where it satisfies that the norm of t times t * is the square of the norm of t: || tt * || = || t || 2 .

C * Properties – Algebras

  • In any C * – algebra the norm of an element coincides with the norm of its involution: || t || = || t * ||
  • Involution is continuous.
  • In all C * – commutative algebra it is true that || t 2|| = || t || 2

Examples

  • Let H be a locally compact Hausdorff space. The set C 0(H) of all the functions of H in C of the complexes, continuous equal to zero at infinity is a closed subalgebra of the algebra C b (H) and therefore a commutative Banach algebra. C 0 (H), with involution x * (t) equal to conjugate of x (t)) is a C * – algebra.
  • In particular, C 0(N, M 2 ) [1] is a simple example of an infinitely dimensioned C * – non-commutative algebra.

 

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