Order on self-adjoint elements - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra, and write $A_{\mathrm{sa}}$ for its self-adjoint part. The $C^*$ -algebra order on $A_{\mathrm{sa}}$ is defined by

a\leq b\quad\Longleftrightarrow\quad b-a\in A_+.

Here $A_+$ is the positive cone of $A$ . Equivalently, $a\leq b$ when $b-a=c^*c$ for some $c\in A$ . This is a partial order compatible with addition and multiplication by nonnegative real scalars. Positivity therefore supplies the order intrinsically; no external cone is chosen. The order is defined on self-adjoint elements; arbitrary elements of $A$ are not compared unless a separate convention is supplied.

Order-preserving operations

If $a\leq b$ , then $x^*ax\leq x^*bx$ for every $x\in A$ . Consequently, every positive linear map is order-preserving on self-adjoint elements, and every $*$ -homomorphism is positive. The continuous functional calculus also makes increasing operator-monotone functions order preserving on their domains Pedersen, treatment of positivity and order.

Examples and incomparability

For $A=C_0(X)$ , the order is pointwise: $f\leq g$ exactly when $f(x)\leq g(x)$ for all $x\in X$ . In $M_2(\mathbb C)$ , the projections $\operatorname{diag}(1,0)$ and $\operatorname{diag}(0,1)$ are incomparable, since their difference has both a positive and a negative eigenvalue. Thus this order is usually not total.

Noncommutative caution

Multiplication need not preserve order. Even if $0\leq a\leq b$ , the product $xa$ need not be self-adjoint and hence need not be comparable with $xb$ . The conjugated inequality $x^*ax\leq x^*bx$ is the valid replacement. The order should also not be confused with an ordering of the complex vector space $A$ ; it lives on the real vector space $A_{\mathrm{sa}}$ .

References

Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: the introductory chapters on positive elements and the order of the self-adjoint part.
Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: the chapter on positive elements and positive maps.