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Positive element

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

Definition

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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

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  • 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

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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 a 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

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In *-algebras

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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

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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

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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

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Citations

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References

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  1. ^ a b c Palmer 2001, 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 2001, p. 800.
  7. ^ a b Blackadar 2006, p. 64.
  8. ^ a b c Palmer 2001, 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 2001, p. 799.
  14. ^ Kadison & Ringrose 1983, p. 249.
  15. ^ a b Kadison & Ringrose 1983, p. 250.
  16. ^ Blackadar 2006, p. 66.

Bibliography

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  • 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. (2001). Banach algebras and the general theory of*-algebras: Volume 2,*-algebras. Cambridge university press. ISBN 0-521-36638-0.