Positive linear functional - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. A positive linear functional is a complex-linear map $\varphi:A\to\mathbb C$ such that

\varphi(a)\geq0\qquad(a\in A_+),

where $A_+$ is the positive cone. Equivalently, $\varphi(b^*b)\geq0$ for every $b\in A$ . These inequalities force $\varphi(a^*)=\overline{\varphi(a)}$ , and a positive functional is automatically bounded. If $A$ is unital, then

\lVert\varphi\rVert=\varphi(1).

A state is a positive functional of norm one, an additional normalization rather than part of positivity.

Cauchy–Schwarz inequality

Positivity of the quadratic form $a\mapsto\varphi(a^*a)$ yields

|\varphi(b^*a)|^2\leq\varphi(a^*a)\varphi(b^*b).

This inequality controls continuity and identifies the null space used in the GNS construction. In the unital case, a bounded functional $\varphi$ is positive exactly when $\lVert\varphi\rVert=\varphi(1)$ ; the equality includes the assertion that $\varphi(1)$ is real and nonnegative.

GNS construction

The form $\langle a,b\rangle_\varphi=\varphi(b^*a)$ is positive semidefinite. Quotienting $A$ by its null left ideal and completing gives a Hilbert space $H_\varphi$ . Left multiplication descends to a $*$ -representation $\pi_\varphi:A\to B(H_\varphi)$ , with a cyclic vector when $A$ is unital, such that

\varphi(a)=\langle\pi_\varphi(a)\xi_\varphi,\xi_\varphi\rangle.

Thus positive functionals are exactly the scalar matrix coefficients arising from cyclic representations in this way.

Nearby notions

A positive functional need not be tracial, multiplicative, faithful, or normalized. A character of a unital $C^*$ -algebra is positive because it is a nonzero $*$ -homomorphism to $\mathbb C$ , but most positive functionals are not characters. On a von Neumann algebra, normality is a further continuity condition; positivity alone does not imply it.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. Elsevier DOI record. Relevant: §3.2 on positive functionals and the GNS construction.
Gert K. Pedersen, C-Algebras and Their Automorphism Groups*, 2nd ed., Academic Press, 2018. Elsevier DOI record. Relevant: §3.3 on positive forms and representations.