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In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. ).



Let be a *-algebra. An element is called self-adjoint if .[1]

The set of self-adjoint elements is referred to as .

A subset that is closed under the involution *, i.e. , is called self-adjoint.[2]

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.

Especially in the older literature on *-algebras and C*-algebras, such elements are often called hermitian.[1] Because of that the notations , or for the set of self-adjoint elements are also sometimes used, even in the more recent literature.


  • Each positive element of a C*-algebra is self-adjoint.[3]
  • For each element of a *-algebra, the elements and are self-adjoint, since * is an involutive antiautomorphism.[4]
  • For each element of a *-algebra, the real and imaginary parts and are self-adjoint, where denotes the imaginary unit.[1]
  • If is a normal element of a C*-algebra , then for every real-valued function , which is continuous on the spectrum of , the continuous functional calculus defines a self-adjoint element .[5]



Let be a *-algebra. Then:

  • Let , then is self-adjoint, since . A similarly calculation yields that is also self-adjoint.[6]
  • Let be the product of two self-adjoint elements . Then is self-adjoint if and commutate, since always holds.[1]
  • If is a C*-algebra, then a normal element is self-adjoint if and only if its spectrum is real, i.e. .[5]



In *-algebras


Let be a *-algebra. Then:

  • Each element can be uniquely decomposed into real and imaginary parts, i.e. there are uniquely determined elements , so that holds. Where and .[1]
  • The set of self-adjoint elements is a real linear subspace of . From the previous property, it follows that is the direct sum of two real linear subspaces, i.e. .[7]
  • If is self-adjoint, then is normal.[1]
  • The *-algebra is called a hermitian *-algebra if every self-adjoint element has a real spectrum .[8]

In C*-algebras


Let be a C*-algebra and . Then:

  • For the spectrum or holds, since is real and holds for the spectral radius, because is normal.[9]
  • According to the continuous functional calculus, there exist uniquely determined positive elements , such that with . For the norm, holds.[10] The elements and are also referred to as the positive and negative parts. In addition, holds for the absolute value defined for every element .[11]
  • For every and odd , there exists a uniquely determined that satisfies , i.e. a unique -th root, as can be shown with the continuous functional calculus.[12]

See also



  1. ^ a b c d e f Dixmier 1977, p. 4.
  2. ^ Dixmier 1977, p. 3.
  3. ^ Palmer 2001, p. 800.
  4. ^ Dixmier 1977, pp. 3–4.
  5. ^ a b Kadison & Ringrose 1983, p. 271.
  6. ^ Palmer 2001, pp. 798–800.
  7. ^ Palmer 2001, p. 798.
  8. ^ Palmer 2001, p. 1008.
  9. ^ Kadison & Ringrose 1983, p. 238.
  10. ^ Kadison & Ringrose 1983, p. 246.
  11. ^ Dixmier 1977, p. 15.
  12. ^ Blackadar 2006, p. 63.


  • Blackadar, Bruce (2006). Operator Algebras. Theory of C*-Algebras and von Neumann Algebras. Berlin/Heidelberg: Springer. p. 63. 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.