In mathematics, an element of a *-algebra is called positive if it is the sum of elements of the form .[1]

Definition edit

Let   be a *-algebra. An element   is called positive if there are finitely many elements  , so that   holds.[1] This is also denoted by  .[2]

The set of positive elements is denoted by  .

A special case from particular importance is the case where   is a complete normed *-algebra, that satisfies the C*-identity ( ), which is called a C*-algebra.

Examples edit

  • The unit element   of an unital *-algebra is positive.
  • For each element  , the elements   and   are positive by definition.[1]

In case   is a C*-algebra, the following holds:

  • Let   be a normal element, then for every positive function   which is continuous on the spectrum of   the continuous functional calculus defines a positive element  .[3]
  • Every projection, i.e. every element   for which   holds, is positive. For the spectrum   of such an idempotent element,   holds, as can be seen from the continuous functional calculus.[3]

Criteria edit

Let   be a C*-algebra and  . Then the following are equivalent:[4]

  • For the spectrum   holds and   is a normal element.
  • There exists an element  , such that  .
  • There exists a (unique) self-adjoint element   such that  .

If   is an unital *-algebra with unit element  , then in addition the following statements are equivalent:[5]

  •   for every   and   is a self-adjoint element.
  •   for some   and   is a self-adjoint element.

Properties edit

In *-algebras edit

Let   be a *-algebra. Then:

  • If   is a positive element, then   is self-adjoint.[6]
  • The set of positive elements   is a convex cone in the real vector space of the self-adjoint elements  . This means that   holds for all   and  .[6]
  • If   is a positive element, then   is also positive for every element  .[7]
  • For the linear span of   the following holds:   and  .[8]

In C*-algebras edit

Let   be a C*-algebra. Then:

  • Using the continuous functional calculus, for every   and   there is a uniquely determined   that satisfies  , i.e. a unique  -th root. In particular, a square root exists for every positive element. Since for every   the element   is positive, this allows the definition of a unique absolute value:  .[9]
  • For every real number   there is a positive element   for which   holds for all  . The mapping   is continuous. Negative values for   are also possible for invertible elements  .[7]
  • Products of commutative positive elements are also positive. So if   holds for positive  , then  .[5]
  • Each element   can be uniquely represented as a linear combination of four positive elements. To do this,   is first decomposed into the self-adjoint real and imaginary parts and these are then decomposed into positive and negative parts using the continuous functional calculus.[10] For it holds that  , since  .[8]
  • If both   and   are positive   holds.[5]
  • If   is a C*-subalgebra of  , then  .[5]
  • If   is another C*-algebra and   is a *-homomorphism from   to  , then   holds.[11]
  • If   are positive elements for which  , they commutate and   holds. Such elements are called orthogonal and one writes  .[12]

Partial order edit

Let   be a *-algebra. The property of being a positive element defines a translation invariant partial order on the set of self-adjoint elements  . If   holds for  , one writes   or  .[13]

This partial order fulfills the properties   and   for all   with   and  .[8]

If   is a C*-algebra, the partial order also has the following properties for  :

  • If   holds, then   is true for every  . For every   that commutates with   and   even   holds.[14]
  • If   holds, then  .[15]
  • If   holds, then   holds for all real numbers  .[16]
  • If   is invertible and   holds, then   is invertible and for the inverses   holds.[15]

See also edit

Citations edit

References edit

  1. ^ a b c Palmer 1977, p. 798.
  2. ^ Blackadar 2006, p. 63.
  3. ^ a b Kadison & Ringrose 1983, p. 271.
  4. ^ Kadison & Ringrose 1983, pp. 247–248.
  5. ^ a b c d Kadison & Ringrose 1983, p. 245.
  6. ^ a b Palmer 1977, p. 800.
  7. ^ a b Blackadar 2006, p. 64.
  8. ^ a b c Palmer 1977, p. 802.
  9. ^ Blackadar 2006, pp. 63–65.
  10. ^ Kadison & Ringrose 1983, p. 247.
  11. ^ Dixmier 1977, p. 18.
  12. ^ Blackadar 2006, p. 67.
  13. ^ Palmer 1977, p. 799.
  14. ^ Kadison & Ringrose 1983, p. 249.
  15. ^ a b Kadison & Ringrose 1983, p. 250.
  16. ^ Blackadar 2006, p. 66.

Bibliography edit

  • Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. ISBN 3-540-28486-9.
  • Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969.
  • Kadison, Richard V.; Ringrose, John R. (1983). Fundamentals of the Theory of Operator Algebras. Volume 1 Elementary Theory. New York/London: Academic Press. ISBN 0-12-393301-3.
  • Palmer, Theodore W. (1994). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0.