decomposition of self-adjoint elements in positive and negative parts



Every real valued function f admits a well-known decomposition into its and parts: f=f+-f-. There is an analogous result for self-adjoint elements in a C*-algebra ( that we will now describe.

Theorem - Let 𝒜 be a C*-algebra and a𝒜 a self-adjoint element. Then there are unique positive elementsMathworldPlanetmath a+ and a- in 𝒜 such that:

  • a=a+-a-

  • a+a-=a-a+=0

  • Both a+ and a- belong to C*-subalgebra generated by a.

  • a=max{a+,a-}

Remark - As a particular case, the result provides a decomposition of each self-adjoint operator T on a Hilbert spaceMathworldPlanetmath as a difference of two positive operators T=T+-T- such that RanT-KerT+ and RanT+KerT-, where Ran and Ker denote, respectively, the range and kernel of an operatorMathworldPlanetmath.


Let us some notation first:

  • σ(a) denotes the spectrum of a𝒜.

  • C*[a] denotes the C*-subalgebra generated by a.

  • C0(σ(a){0}) denotes the algebra of continuous functionsMathworldPlanetmathPlanetmath in σ(a){0} that vanish at infinity.

Let f,f+,f-C0(σ(a){0}) be the functions defined by

f(t):=t    f+(t):={t,ift00,ift0    f-(t):={0,ift0-t,ift0

Since a is , σ(a), so the above functions are well defined. It is clear that

f=f+-f-andf+f-=f-f+=0andf+,f-are both positive (1)

The continuous functional calculus gives an isomorphismPlanetmathPlanetmathPlanetmathPlanetmath C*[a]C0(σ(a){0}) such that the element a corresponds to the function f. Let a+ and a- be the elements corresponding to f+ and f- respectively. From the made in (1) it is now clear that

  • a+ and a- are both positive elements.

  • a=a+-a-

  • a+a-=a-a+=0

  • Both a+ and a- belong to C*[a].

From the fact the every C*-isomorphism is isometric (see this entry ( and f=max{f+,f-} it follows that a=max{a+,a-}.

The uniqueness of the decomposition follows from the uniqueness of the decomposition of real valued functions in its positive and negative parts f=f+-f- (with f+f-=0).

Title decomposition of self-adjoint elements in positive and negative parts
Canonical name DecompositionOfSelfadjointElementsInPositiveAndNegativeParts
Date of creation 2013-03-22 17:51:49
Last modified on 2013-03-22 17:51:49
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 12
Author asteroid (17536)
Entry type Theorem
Classification msc 47C15
Classification msc 47B25
Classification msc 47A60
Classification msc 46L05
Related topic CAlgebra