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Positive Elements and Ideals in C*-Algebras

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Positive Elements

Positive Element

An element in a unital C*-algebra is said to be positive if it is self-adjoint and its spectrum consists of non-negative real numbers, i.e., . The set of all positive elements in is denoted by .

e.g. Let be a locally compact Hausdorff space. Then is positive if and only if either for all (i.e. ) or for some .

Our main goal in this section is to show that the set of positive elements is precisely the set of elements of the form for some .

Proposition

For any element in a C*-algebra , is positive if and only if for some .

Proof We prove this in two steps.

  1. Step 1: We show that if is a positive element, then must be zero. If , its spectrum is contained in . We also know that for any , the spectrum of is contained in . Since , the only way for both conditions to hold is if . Since is self-adjoint, its norm is equal to its spectral radius. Therefore, , which implies . By the C*-identity, , so .
  2. Step 2: Let . As is self-adjoint, we can decompose it into its positive and negative parts: , where and . Our goal is to show that . Consider the element . We compute : Substituting , we get: Since , we have , so the first term vanishes. Thus, . Since is positive, is also positive. We now have that is a positive element. By Step 1, this implies . If , then . Since is positive, implies . Therefore, , which means is a positive element.

Unique Positive Square Root

An element is positive if and only if there exists a unique positive element such that . This element is called the positive square root of , denoted by .

Proof

  • Existence: If , then . The square root function is continuous on . By the continuous functional calculus, we can define .
    • Since is a real-valued function, is self-adjoint.
    • The spectrum of is , so .
    • By the properties of functional calculus, . Since , we have .
  • Uniqueness: Suppose there is another element such that . Since is a polynomial in itself (), it commutes with . The element is a limit of polynomials in , so also commutes with . Now consider the commutative C*-algebra generated by and . By the Gelfand representation theorem, there is a -isomorphism for some compact Hausdorff space . We have and . Since , their Gelfand transforms and are non-negative real-valued functions. From , we can conclude that . Since the Gelfand map is an isomorphism, this implies .

Absolute Value and Positive/Negative Parts

For any self-adjoint element , we define:

  • The absolute value of as .
  • The positive part of as .
  • The negative part of as .

These elements are positive and satisfy and .

The Partial Order on Self-Adjoint Elements

Lemma

If , then .

Proof Since and are positive, they are self-adjoint, so is self-adjoint. The proof that uses the fact that for , if and only if for all . Let and . Then we have and . By the triangle inequality: Since , this shows that is positive.

The Partial Order on

Let be the set of self-adjoint elements in . For , we write if . This defines a partial order on .

Ideals and Quotient Algebras

Ideals and Quotients of C*-Algebras

Let be a closed two-sided ideal in a C*-algebra . Then:

  1. is a self-adjoint ideal (i.e., a -ideal).
  2. The quotient algebra is a C*-algebra with the quotient norm.

To prove this, we first need a result about approximate identities.

Approximate Identity

For any and , there exists an element with such that and . Such a family of elements is called an approximate identity for . If is an ideal, it contains an approximate identity for itself.

Proof of the Theorem

  1. is a -ideal: We need to show that if , then . Since and is a right ideal, . Because is an ideal, it contains an approximate identity . These can be constructed via functional calculus on elements like , and since is closed, these elements lie in . We know that . Then, using the property that the involution is an isometry: So, . Each element is in because and . Since is the limit of a sequence of elements in and is a closed subspace, we must have . Thus, is a -ideal.
  2. is a C*-algebra: Since is a closed ideal in a Banach algebra , the quotient is a Banach algebra with the quotient norm . Since is a -ideal, is a Banach -algebra. We only need to verify the C*-identity: First, for any : Taking the infimum over all gives . For the reverse inequality, let be an approximate identity for . Then . We compute the square: Since , the element is in . Thus, .Therefore, for each : So, for all . Taking the limit as , we get: Combining the two inequalities, we obtain the C*-identity.

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