Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector space $(V,\leq )$ is a linear functional $f$ on $V$ so that for all positive elements $v\in V$ , that is $v\geq 0$ , it holds that

f(v)\geq 0.

In other words, a positive linear functional is guaranteed to take nonnegative values for positive elements. The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem.

When $V$ is a complex vector space, it is assumed that for all $v\geq 0$ , $f(v)$ is real. As in the case when $V$ is a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspace $W\subseteq V$ , and the partial order does not extend to all of $V$ , in which case the positive elements of $V$ are the positive elements of $W$ , by abuse of notation. This implies that for a C*-algebra, a positive linear functional sends any $x\in V$ equal to $s^{\ast }s$ for some $s\in V$ to a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of such $x$ . This property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.

Sufficient conditions for continuity of all positive linear functionals

There is a comparatively large class of ordered topological vector spaces on which every positive linear form is necessarily continuous.^[1] This includes all topological vector lattices that are sequentially complete.^[1]

Theorem Let $X$ be an ordered topological vector space with positive cone $C\subseteq X$ and let ${\mathcal {B}}\subseteq {\mathcal {P}}(X)$ denote the family of all bounded subsets of $X$ . Then each of the following conditions is sufficient to guarantee that every positive linear functional on $X$ is continuous:

$C$ has non-empty topological interior (in $X$ ).^[1]
$X$ is complete and metrizable and $X=C-C$ .^[1]
$X$ is bornological and $C$ is a semi-complete strict ${\mathcal {B}}$ -cone in $X$ .^[1]
$X$ is the inductive limit of a family $\left(X_{\alpha }\right)_{\alpha \in A}$ of ordered Fréchet spaces with respect to a family of positive linear maps where $X_{\alpha }=C_{\alpha }-C_{\alpha }$ for all $\alpha \in A$ , where $C_{\alpha }$ is the positive cone of $X_{\alpha}$ .^[1]

Continuous positive extensions

The following theorem is due to H. Bauer and independently, to Namioka.^[1]

Theorem:^[1] Let

X

be an ordered topological vector space (TVS) with positive cone

C

, let

M

be a vector subspace of

E

, and let

f

be a linear form on

M

. Then

f

has an extension to a continuous positive linear form on

X

if and only if there exists some convex neighborhood

U

of

0\in X

such that

\operatorname {Re} f

is bounded above on

M\cap \left(U-C\right)

.

Corollary:^[1] Let

X

be an ordered topological vector space with positive cone

C

, let

M

be a vector subspace of

E

. If

C\cap M

contains an interior point of

C

then every continuous positive linear form on

M

has an extension to a continuous positive linear form on

X

.

Corollary:^[1] Let

X

be an ordered vector space with positive cone

C

, let

M

be a vector subspace of

E

, and let

f

be a linear form on

M

. Then

f

has an extension to a positive linear form on

X

if and only if there exists some convex absorbing subset

W

in

X

containing

0\in X

such that

\operatorname {Re} f

is bounded above on

M\cap \left(W-C\right)

.

Proof: It suffices to endow $X$ with the finest locally convex topology making $W$ into a neighborhood of $0\in X$ .

Examples

Consider, as an example of $V$ , the C*-algebra of complex square matrices with the positive elements being the positive-definite matrices. The trace function defined on this C*-algebra is a positive functional, as the eigenvalues of any positive-definite matrix are positive, and so its trace is positive.
Consider the Riesz space $\mathrm {C} _{\mathrm {c} }(X)$ of all continuous complex-valued functions of compact support on a locally compact Hausdorff space $X$ . Consider a Borel regular measure $\mu$ on $X$ , and a functional $\psi$ defined by

\psi(f) = \int_X f(x) d \mu(x) \quad

for all

f

in

\mathrm {C} _{\mathrm {c} }(X)

. Then, this functional is positive (the integral of any positive function is a positive number). Moreover, any positive functional on this space has this form, as follows from the Riesz–Markov–Kakutani representation theorem.