2 Preliminaries

The natural numbers are and , and the fields of real and complex numbers are denoted by $\mathbb{R}$ and $\mathbb{C}$ , respectively. If X and Y are partially ordered sets (i.e., sets together with a reflexive, transitive, and antisymmetric relation $\le $ ), then a map $\Phi \colon X \to Y$ is called increasing if $\Phi (x) \le \Phi (x')$ holds for all $x,x'\in X$ with $x\le x'$ . It is called an order embedding if it is increasing and if additionally $x\le x'$ holds for all $x,x'\in X$ for which $\Phi (x) \le \Phi (x')$ . Note that an order embedding is injective. A partially ordered set X is called directed if, for all $x,x' \in X$ , there exists a $y\in X$ such that $x\le y$ and $x'\le y$ . Similarly, a subset S of a partially ordered set X is called directed if it is directed with respect to the order inherited from X. For two vector spaces V and W over the same field of scalars $\mathbb{F}$ , the vector space of all linear maps from V to W is denoted by $\mathcal {L}(V,W)$ . In the special case that $W = \mathbb{F}$ , we write , the elements of $V^*$ are called linear functionals on V, and the evaluation of a linear functional $\omega \in V^*$ on a vector $v\in V$ is denoted by means of the bilinear dual pairing ${\langle \,{ {\,\cdot \,} } \,,\, { {\,\cdot \,} }\,\rangle } \colon V^*\times V \to \mathbb{F}$ , $(\omega ,v) \mapsto {\langle \,{\omega } \,,\, {v}\,\rangle }$ .

The main technical tools needed in the following are some basic theorems from locally convex analysis. A filter on a set X is a nonempty set $\mathcal {F}$ of subsets of X with the following two properties:

• • If $S,T \in \mathcal {F}$ , then $S\cap T \in \mathcal {F}$ .

• • If $S \in \mathcal {F}$ and if a subset T of X fulfils $T\supseteq S$ , then $T\in \mathcal {F}$ .

Similarly, a basis of a filter on X is a nonempty set $\mathcal {E}$ of subsets of X such that for all $S,T\in \mathcal {E}$ , there exists an $R\in \mathcal {E}$ with $R\subseteq S\cap T$ . In this case,

(2.1)

is a filter on X, called the filter generated by $\mathcal {E}$ . A (real) topological vector space is a real vector space V endowed with a (not necessarily Hausdorff) topology under which addition $V\times V \to V$ as well as scalar multiplication $\mathbb{R} \times V \to V$ are continuous. Then it follows from the continuity of addition that a subset U of V is a neighborhood of a vector $v\in V$ if and only if is a neighborhood of $0$ , so the topology of V is completely described by the $0$ -neighborhoods. The set $\mathcal {N}_0$ of all $0$ -neighborhoods of V is a filter on V, and $\mathcal {N}_{0,\mathrm {conv}}$ , the set of all convex $0$ -neighborhoods of V, is a basis of a filter on V. In general, ${\langle \!\langle \,{\mathcal {N}_{0,\mathrm {conv}}}\,\rangle \!\rangle _{\mathrm {fltr}}} \subseteq \mathcal {N}_0$ holds, and V is called a (real) locally convex vector space if even ${\langle \!\langle \,{\mathcal {N}_{0,\mathrm {conv}}}\,\rangle \!\rangle _{\mathrm {fltr}}} = \mathcal {N}_0$ , i.e., if the filter of $0$ -neighborhoods of V is generated by the convex $0$ -neighborhoods. A locally convex topology on V is a topology with which V becomes a locally convex vector space.

There is an easy procedure to construct locally convex topologies on a real vector space V: A subset S of V is called absorbing if, for every $v\in V$ , there is a $\lambda \in {]0,\infty [}$ such that $\lambda v \in S$ , and it is called balanced if $\lambda s \in S$ for all $s\in S$ and all $\lambda \in {[-1,1]}$ . If $\mathcal {N}_{0,\mathrm {abc}}$ is a basis of a filter on V consisting of only absorbing balanced convex subsets of V, then its generated filter ${\langle \!\langle \,{\mathcal {N}_{0,\mathrm {abc}}}\,\rangle \!\rangle _{\mathrm {fltr}}}$ is the filter of $0$ -neighborhoods of a locally convex topology on V. Note that every absorbing balanced convex subset S of V yields a seminorm ${\lVert { {\,\cdot \,} } \rVert }_{S}$ on V by setting for all $v\in V$ , which is the Minkowski functional of S. Conversely, for every seminorm ${\lVert { {\,\cdot \,} } \rVert }$ on V, the unit ball is an absorbing balanced convex subset of V whose Minkowski functional is again the original seminorm ${\lVert { {\,\cdot \,} } \rVert }$ . However, the map from absorbing balanced convex subsets of V to seminorms on V is only surjective, but not injective in general. Because of this, there is more freedom in describing locally convex vector spaces via a basis of the filter of $0$ -neighborhoods than via the corresponding seminorms.

An important example of Hausdorff locally convex vector spaces is the dual space $V^*$ of an arbitrary real vector space V with the weak- $^*$ -topology, i.e., with the weakest topology on $V^*$ under which all the evaluation maps $V^* \ni \omega \mapsto {\langle \,{\omega } \,,\, {v}\,\rangle } \in \mathbb{R}$ with vectors $v\in V$ are continuous. This is the locally convex topology whose filter of $0$ -neighborhoods is generated by the intersections of unit balls of finitely many seminorms of the form $V^* \ni \omega \mapsto {\lvert {{\langle \,{\omega } \,,\, {v}\,\rangle }} \rvert } \in {[0,\infty [}$ with $v\in V$ .

The following classical theorems will be crucial for the various representation theorems. As usual, for such results, their proofs typically make use of the axiom of choice.

Theorem 2.2 (Banach–Alaoglu)

Let V be a real vector space, and let U be an absorbing subset of V, then

(2.2)

is a convex subset of $V^*$ and compact in the weak- $^*$ -topology.

If C is a convex subset of a real vector space V, then an extreme point of C is an element $e\in C$ with the following property: Whenever $e = \lambda c_1 + (1-\lambda ) c_2$ holds for some $c_1,c_2\in C$ and $\lambda \in {]0,1[}$ , then $e=c_1 = c_2$ . The set of all extreme points of C will be denoted by $\mathrm {ex}(C)$ . Moreover, if V is a locally convex vector space and S a nonempty subset of V, then ${\langle \!\langle \,{S}\,\rangle \!\rangle _{\mathrm {cl\text {-}conv}}}$ will denote the closed convex hull of S, i.e., the closure of the convex hull of S, or equivalently, the intersection of all closed convex subsets of V which contain S. In the special case that V is a real vector space and S a nonempty subset of $V^*$ , then ${\langle \!\langle \,{S}\,\rangle \!\rangle _{\mathrm {cl\text {-}conv}}}$ is understood to be the closed convex hull of S with respect to the weak- $^*$ -topology and

(2.3) $$ \begin{align} {\langle\!\langle\,{S}\,\rangle\!\rangle_{\mathrm{cl\text{-}conv}}} = {\big\{\,{ \omega \in V^* } \;\big|\; { \forall_{v\in V}: {\langle\,{\omega} \,,\, {v}\,\rangle} \in {\langle\!\langle\,{{\{\,{{\langle\,{\rho} \,,\, {v}\,\rangle}} \;|\; {\rho \in S}\,\}}}\,\rangle\!\rangle_{\mathrm{cl\text{-}conv}}} }\,\big\}} \end{align} $$

holds. This can be seen either by elementary linear algebra or as a consequence of the Hahn–Banach Theorem and the fact that all weak- $^*$ -continuous linear functionals on $V^*$ can be expressed as maps $V^* \ni \omega \mapsto {\langle \,{\omega } \,,\, {v}\,\rangle } \in \mathbb{R}$ with a suitable vector $v\in V$ .

The tools needed to prove the generalized Gelfand–Naimark theorems are essentially results about the existence of many (extremal) positive linear functionals on ordered vector spaces.

An ordered vector space is a real vector space V endowed with a partial order $\le $ such that the conditions

(2.4) $$ \begin{align} u+v\le u+w \quad\quad\text{and}\quad\quad \lambda v\le \lambda w \end{align} $$

hold for all $u,v,w\in V$ with $v\le w$ and all $\lambda \in {[0,\infty [}$ . Then is the set of positive elements of V, which uniquely determines the order on V because $v \le w$ is equivalent to $w-v\in V^+$ for all $v,w\in V$ . An ordered vector space V is called Archimedean if it has the following property: Whenever $v\le \epsilon w$ holds for two vectors $v\in V$ , $w\in V^+$ , and all $\epsilon \in {]0,\infty [}$ , then $v\le 0$ . Note also that an ordered vector space V is directed if and only if every $v\in V$ can be decomposed as $v = v_{(+)} - v_{(-)}$ with $v_{(+)}, v_{(-)} \in V^+$ . Of course, such a decomposition is not uniquely determined in general.

If V and W are ordered vector spaces, then a linear map $\Phi \colon V\to W$ is increasing if and only if $\Phi (v) \in W^+$ for all $v\in V^+$ . We write $\mathcal {L}(V,W)^+$ for the set of all increasing linear maps from V to W. If V is directed, then there actually exists a (unique) partial order on $\mathcal {L}(V,W)$ such that $\mathcal {L}(V,W)$ becomes an ordered vector space whose positive elements are precisely the increasing linear functions. However, for simplicity, a linear map $\Phi \colon V \to W$ will always be called positive if it is increasing, also in the case that V is not necessarily directed. Note that a positive linear map $\Phi \colon V\to W$ is an order embedding if and only if $\Phi (v) \in W\setminus W^+$ for all $v\in V \setminus V^+$ . The case that $W=\mathbb{R}$ will be especially interesting: Then the set of positive linear functionals is denoted by

(2.5)

If V is directed, then $V^*$ is an ordered vector space itself and a positive linear functional $\omega $ on V is said to be extremal if, for every $\rho \in V^{*,+}$ with $\rho \le \omega $ , there is an $\mu \in {[0,1]}$ such that $\rho = \mu \omega $ . The set of all extremal positive linear functionals on V will then be denoted by $V^{*,+,\mathrm {ex}}$ , and one can check that $0\in V^{*,+,\mathrm {ex}}$ and $\lambda \omega \in V^{*,+,\mathrm {ex}}$ for all $\omega \in V^{*,+,\mathrm {ex}}$ and all $\lambda \in {]0,\infty [}$ . Note that this definition of extremal positive linear functionals only makes sense on a directed ordered vector space V because it refers to the partial order on $V^*$ . There is an extension theorem for (extremal) positive linear functionals from a sufficiently large linear subspace of a directed ordered vector space to the whole space (see [Reference Schmüdgen12, Lemma 1.3.2] for details)

The question of existence of many (extremal) positive linear functionals is nontrivial in general. More precisely, one asks whether or not the following two properties are fulfilled.

These two conditions are equivalent to demanding that, for every $v\in V$ , the inequality $0 \le v$ holds if and only if $0\le {\langle \,{\omega } \,,\, {v}\,\rangle }$ for all $\omega \in V^{*,+}$ or for all $\omega \in V^{*,+, \mathrm {ex}}$ , respectively.

We will see shortly that the above conditions of existence of “many” (extremal) positive linear functionals are closely related to the properties of certain locally convex topologies: Given two elements $\ell $ and u of an ordered vector space V, then the order interval between $\ell $ and u is defined as . Moreover, a subset S of V is called saturated if $[\ell , u] \subseteq S$ is fulfilled for all $\ell , u\in S$ with $\ell \le u$ . For example, every order interval is saturated. It is not hard to see that the intersection of finitely many (or arbitrarily many) saturated subsets of an ordered vector space V is again saturated. As a consequence, the set of all absorbing balanced convex and saturated subsets of V is a basis of the filter of $0$ -neighborhoods of a locally convex topology.

This topology is not unknown in the theory of ordered vector spaces and $^*$ -algebras. For example, the normal topology is essentially the topology $\tau _n$ in [Reference Schmüdgen12, Section 1.5].

3 Existence of extremal positive linear functionals

Lemma 3.1 is a standard application of the Hahn–Banach theorem, and a proof is given for convenience of the reader.

For finding a sufficient condition under which the order of a directed ordered vector space is induced by its extremal positive linear functionals, it will be helpful to be able to decompose a positive linear functional into extremal ones. We follow essentially the argument of [Reference Schmüdgen12, Lemmas 12.4.3 and 12.4.4].

Proof First, note that $K = U^\circ \cap V^{*,+}$ with $U^\circ $ like in (2.2) because U is balanced. As $U^\circ $ is convex and weak- $^*$ -compact by the Banach–Alaoglu theorem, and as $V^{*,+}$ is convex and weak- $^*$ -closed in $V^*$ , it follows that K is also convex and weak- $^*$ -compact. The Krein–Milman theorem then shows that $K = {\langle \!\langle \,{\mathrm {ex}(K)}\,\rangle \!\rangle _{\mathrm {cl\text {-}conv}}}$ .

In order to complete the proof, it is sufficient to show that $\mathrm {ex}(K) \subseteq V^{*,+,\mathrm {ex}}$ . Denote the linear span of K in $V^*$ by W. Then the map $h \colon W \to {[0,\infty [}$

is a seminorm on W and $K = {\{\,{\omega \in W\cap V^{*,+}} \; |\; {h(\omega ) \le 1}\,\}}$ . Moreover, as U is balanced and directed, $h(\omega + \omega ') \ge h(\omega ) + h(\omega ')$ , hence $h(\omega + \omega ') = h(\omega ) + h(\omega ')$ , holds for all $\omega ,\omega ' \in W \cap V^{*,+}$ .

Now, let $\omega \in \mathrm {ex}(K)$ be given. If $h(\omega ) = 0$ , then $\omega = 0$ because U is absorbing, so $\omega \in V^{*,+,\mathrm {ex}}$ is trivially fulfilled. Otherwise, $h(\omega ) = 1$ because on the one hand, $h(\omega ) \le 1$ is clear, and on the other, $\omega = h(\omega ) \big ( h(\omega )^{-1} \omega \big ) + \big (1-h(\omega )\big ) 0$ is a representation of $\omega $ as a convex combination of the two elements $h(\omega )^{-1} \omega $ and $0$ of K, which excludes the possibility that $h(\omega ) \in {]0,1[}$ . In this second case that $h(\omega )=1$ , consider some $\rho \in V^{*,+}$ that fulfils $\rho \le \omega $ . Then, for all $u\in U$ , there exists $v \in U \cap V^+$ such that $u \le v$ because U is directed, and hence ${\langle \,{\rho } \,,\, {u}\,\rangle } \le {\langle \,{\rho } \,,\, {v}\,\rangle } \le {\langle \,{\omega } \,,\, {v}\,\rangle }\le 1$ . This means $\rho \in K$ , and the same estimate with $\omega - \rho $ in place of $\rho $ shows that $\omega -\rho \in K$ as well. If $h(\rho ) = 0$ or $h(\omega -\rho ) = 0$ , then $\rho = \mu \omega $ with $\mu = 0$ or $\mu = 1$ , respectively. Otherwise, $\omega = h(\rho ) \big ( h(\rho )^{-1} \rho \big ) + h(\omega -\rho ) \big ( h(\omega -\rho )^{-1} (\omega -\rho ) \big )$ is a representation of $\omega $ as a nontrivial convex combination of the two elements $h(\rho )^{-1} \rho $ and $h(\omega -\rho )^{-1} (\omega -\rho )$ of K; therefore, $\rho = \mu \omega $ with $\mu = h(\rho )$ . We conclude that $\omega \in V^{*,+,\mathrm {ex}}$ in this case as well.

There is a rather large class of ordered vector spaces for which the normal topology cannot only be described explicitly, but also allows to apply Corollary 3.4.

Note that being $\sigma $ -bounded can be seen as the combination of two properties: First, it is required that V is directed, and then, additionally, that there exists an increasing sequence $(\hat {v}_n)_{n\in \mathbb{N}}$ in $V^+$ which has the property that, for every $v\in V^+$ , there exists an $n\in \mathbb{N}$ such that $v\le \hat {v}_n$ .

Definition 3.2 Let V be a $\sigma $ -bounded ordered vector space with a dominating sequence $(\hat {v}_n)_{n\in \mathbb{N}}$ , and let $(\delta _n)_{n\in \mathbb{N}}$ be a sequence in ${]0,\infty [}$ . Then the subset $U_\delta $ of V is defined as the union of increasing order intervals

(3.1)

Of course, $U_\delta $ depends not only on the sequence $(\delta _n)_{n\in \mathbb{N}}$ , but also on the choice of the dominating sequence $(\hat {v}_n)_{n\in \mathbb{N}}$ and of V itself, which will always be clear from the context.

The above description of the normal topology on a $\sigma $ -bounded ordered vector space has essentially already been given in [Reference Schmüdgen12, Propositions 4.1.2 and 4.1.3] and has also been applied indirectly to the decomposition of positive linear functionals into extremal ones in [Reference Schmüdgen12, Theorem 12.4.7]. Even though these results where stated for $O^*$ -algebras or $^*$ -algebras, their above generalization to ordered vector spaces does not require any new techniques. Combining them with a completely order-theoretic characterization of those $\sigma $ -bounded ordered vector spaces, whose set of positive elements is closed, gives our first important result.

One special class of $\sigma $ -bounded ordered vector spaces are ordered vector spaces V with a strong order unit e, i.e., an element $e\in V^+$ with the property that, for all $v\in V$ , there exists $\lambda \in {[0,\infty [}$ such that $v \le \lambda e$ . In this case, $(ne)_{n\in \mathbb{N}}$ is a dominating sequence, and the existence of positive linear functionals on Archimedean ordered vector spaces with a strong order unit was already proved in [Reference Kadison8, Lemma 2.5]. Theorem 3.7 generalizes this classical result to $\sigma $ -bounded ordered vector spaces and applies the decomposition of positive linear functionals into extremal ones from [Reference Schmüdgen12, Theorem 12.4.7].

It should be unnecessary to point out that there are many examples of important $\sigma $ -bounded ordered vector spaces which do not have a strong order unit. These can be as ordinary as the space of real-valued polynomial functions on $\mathbb{R}$ with the pointwise order.

Before closing this section, we prove a new representation theorem for ordered vector spaces. Given any set X, then $\mathbb{R}^X$ denotes the ordered vector space of all $\mathbb{R}$ -valued functions on X with the pointwise operations and the pointwise order. As a consequence of Theorem 3.7, we obtain the following corollary.

Corollary 3.8 Let V be an Archimedean $\sigma $ -bounded ordered vector space, then the map $\pi \colon V \to \mathbb{R}^{V^{*,+,\mathrm {ex}}}$ , $v\mapsto \pi (v)$ with $\pi (v) \colon V^{*,+,\mathrm {ex}} \to \mathbb{R}$ ,

(3.2)

is linear and an order embedding.

If V is a Riesz space, i.e., an ordered vector space in which all finite infima and suprema exist, then Corollary 3.8 essentially is the representation theorem for Riesz spaces from [Reference Buskes and van Rooij4]: One can show that, in this case, the extremal positive linear functionals on V are Riesz morphisms, i.e., compatible with finite infima and suprema, so that the representation $\pi $ from (3.2) is also a Riesz morphism. We will not discuss the details here, but rather turn our attention in the next section to the case that V is an ordered vector space that carries a multiplication.

4 Representation of ordered $^*$ -algebras

We now come to the main section, which develops the generalized Gelfand–Naimark theorems for ordered $^*$ -algebras. The definition of $^*$ -algebras has already been given in the introduction. An element a of a $^*$ -algebra $\mathcal {A}$ is called Hermitian if $a=a^*$ and the real linear subspace of all Hermitian elements in $\mathcal {A}$ is denoted by $\mathcal {A}_{\mathrm {H}}$ . An ordered $^*$ -algebra is a $^*$ -algebra $\mathcal {A}$ together with a partial order $\le $ on $\mathcal {A}_{\mathrm {H}}$ such that $\mathcal {A}_{\mathrm {H}}$ becomes an ordered vector space fulfilling and $a^*b\,a \in \mathcal {A}_{\mathrm {H}}^+$ for all $a\in \mathcal {A}$ , $b\in \mathcal {A}^+_{\mathrm {H}}$ . This ordered vector space of Hermitian elements is automatically directed because and hold for all $a\in \mathcal {A}_{\mathrm {H}}$ . A unital $^*$ -subalgebra $\mathcal {B}$ of an ordered $^*$ -algebra $\mathcal {A}$ , i.e., a linear subspace $\mathcal {B}$ of $\mathcal {A}$ fulfilling as well as $b^* \in \mathcal {B}$ and $bb' \in \mathcal {B}$ for all $b,b'\in \mathcal {B}$ , is again an ordered $^*$ -algebra with the order on $\mathcal {B}_{\mathrm {H}}$ inherited from $\mathcal {A}_{\mathrm {H}}$ . Ordered $^*$ -algebras have already been used for understanding representations of $^*$ -algebras, e.g., in [Reference Powers9] or [Reference Schmüdgen12]. The set of positive Hermitian elements of an ordered $^*$ -algebra is an “m-admissible cone” in the language of [Reference Schmüdgen12], or a “quadratic module” in the language of (noncommutative) real algebraic geometry.

An ordered $^*$ -algebra $\mathcal {A}$ is called $\sigma $ -bounded if the ordered vector space $\mathcal {A}_{\mathrm {H}}$ is $\sigma $ -bounded. Similarly, an ordered $^*$ -algebra $\mathcal {A}$ is said to be closed if $\mathcal {A}_{\mathrm {H}}$ is an Archimedean ordered vector space. Note that this is not a topological property but rather an order property; only for $\sigma $ -bounded ordered $^*$ -algebras, it follows from Theorem 3.7 that $\mathcal {A}$ is closed as an ordered $^*$ -algebra if and only if the quadratic module $\mathcal {A}^+_{\mathrm {H}}$ is closed in $\mathcal {A}_{\mathrm {H}}$ with respect to a certain locally convex topology. Note also that this property is not at all related to the notion of Archimedean quadratic modules in real algebraic geometry. It is unfortunate that the term “Archimedean” is being used in several different ways. Because of this, the term “Archimedean” should be avoided in the context of ordered $^*$ -algebras. There are some important examples of ordered $^*$ -algebras that will be relevant in the following, and these also illustrate that the notion of ordered $^*$ -algebras used here is rather minimalistic.

Example 4.5 Let $\mathcal {A}$ be a $^*$ -algebra, and define the set

of algebraically positive elements in $\mathcal {A}$ . If $\mathcal {A}^{++}_{\mathrm {H}}$ does not contain a real linear subspace of $\mathcal {A}_{\mathrm {H}}$ in addition to the trivial one $\{0\}$ , then $\mathcal {A}$ can be turned into an ordered $^*$ -algebra such that $\mathcal {A}^+_{\mathrm {H}} = \mathcal {A}^{++}_{\mathrm {H}}$ . Otherwise, i.e., if there is $a\in \mathcal {A}_{\mathrm {H}} \setminus \{0\}$ such that both a and $-a$ are algebraically positive, there is no possibility to turn $\mathcal {A}$ into an ordered $^*$ -algebra. So the existence of a suitable order on a $^*$ -algebra (especially the antisymmetry of the order) is a nontrivial condition. We will see that this, together with two or three further conditions, allows to prove representation theorems similar to, but more general than those known for $C^*$ -algebras.

Note also that there are many different types of ordered algebras that have been examined in the last decades. Motivations to do so can be rather divers, e.g., generalizing results for ordered vector spaces (especially Riesz spaces) to ordered algebras, generalizing results from (commutative) real algebraic geometry to the noncommutative case, or studying operator algebras by purely algebraic means. Consequently, one can find conflicting definitions in the literature.

If $\mathcal {A}$ is a $^*$ -algebra, then its complex dual vector space $\mathcal {A}^*$ carries an antilinear involution $ {\,\cdot \,} ^* \colon \mathcal {A}^* \to \mathcal {A}^*$ , $\omega \mapsto \omega ^*$ , given by for all $a\in \mathcal {A}$ . We say again that an element $\omega \in \mathcal {A}^*$ is Hermitian if $\omega ^*=\omega $ and write $\mathcal {A}_{\mathrm {H}}^*$ for the set of all Hermitian linear functionals on $\mathcal {A}$ , which is a real linear subspace of $\mathcal {A}^*$ . Note that a linear functional $\omega $ on $\mathcal {A}$ is Hermitian if and only if ${\langle \,{\omega } \,,\, {a}\,\rangle } \in \mathbb{R}$ holds for all $a\in \mathcal {A}_{\mathrm {H}}$ . Thus, every $\omega \in \mathcal {A}^*_{\mathrm {H}}$ can be restricted to an $\mathbb{R}$ -linear functional on $\mathcal {A}_{\mathrm {H}}$ , and one can check that this restriction describes an $\mathbb{R}$ -linear isomorphism between the vector spaces $\mathcal {A}^*_{\mathrm {H}}$ and $(\mathcal {A}_{\mathrm {H}})^*$ . An (extremal) positive Hermitian linear functional on an ordered $^*$ -algebra $\mathcal {A}$ is then defined as a Hermitian linear functional on $\mathcal {A}$ whose restriction to a (real) linear functional on the ordered vector space $\mathcal {A}_{\mathrm {H}}$ is an (extremal) positive linear functional. The sets of these (extremal) positive Hermitian linear functionals are denoted by $\mathcal {A}^{*,+}_{\mathrm {H}}$ and $\mathcal {A}^{*,+,\mathrm {ex}}_{\mathrm {H}}$ , respectively, and we say that the order on $\mathcal {A}$ is induced by its (extremal) positive Hermitian linear functionals if the order on $\mathcal {A}_{\mathrm {H}}$ is induced by its (extremal) positive linear functionals. Note that positivity of a Hermitian linear functional $\omega $ on an ordered $^*$ -algebra $\mathcal {A}$ is in general a stronger condition than just the requirement that ${\langle \,{\omega } \,,\, {a^*a}\,\rangle } \ge 0$ for all $a\in \mathcal {A}$ , which is used quite often in the literature when linear functionals on general $^*$ -algebras are discussed. However, if $\mathcal {A}$ is an ordered $^*$ -algebra in which only the algebraically positive Hermitian linear functionals are positive, i.e., if $\mathcal {A}$ is of the type of Example 4.5, then ${\langle \,{\omega } \,,\, {a^*a}\,\rangle } \ge 0$ for all $a\in \mathcal {A}$ is also sufficient for a Hermitian linear functional $\omega $ on $\mathcal {A}$ to be positive.

Positive Hermitian linear functionals on an ordered $^*$ -algebra $\mathcal {A}$ have some nice properties: Given $\omega \in \mathcal {A}^{*,+}_{\mathrm {H}}$ and $a,b\in \mathcal {A}$ , then the Cauchy–Schwarz inequality

(4.1) $$ \begin{align} {\big\lvert {{\langle\,{\omega} \,,\, {b^*a}\,\rangle}} \big\rvert}^2 &\le {\langle\,{\omega} \,,\, {b^*b}\,\rangle} {\langle\,{\omega} \,,\, {a^*a}\,\rangle} \end{align} $$

holds, as well as

(4.2)

in the special case that . This has an important consequence: If a positive Hermitian linear functional $\omega $ on $\mathcal {A}$ fulfils , then $\omega = 0$ . A state on $\mathcal {A}$ is a positive Hermitian linear functional $\omega $ on $\mathcal {A}$ that fulfils . So every $\tilde {\omega } \in \mathcal {A}^{*,+}_{\mathrm {H}} \setminus \{0\}$ is a multiple of a unique state $\omega $ on $\mathcal {A}$ , namely of . The set of all states on $\mathcal {A}$ will be denoted by $\mathcal {S}(\mathcal {A})$ and is clearly a convex (possibly empty) subset of the real vector space $\mathcal {A}^*_{\mathrm {H}}$ . Again, note that by this definition, states are positive on whole $\mathcal {A}^+_{\mathrm {H}}$ , not just on squares.

A map $\Phi \colon \mathcal {A} \to \mathcal {B}$ between two $^*$ -algebras is said to be multiplicative if $\Phi (aa') = \Phi (a) \Phi (a')$ holds for all $a,a' \in \mathcal {A}$ . Furthermore, it is called a unital $^*$ -homomorphism if it is linear and multiplicative, maps the unit of $\mathcal {A}$ to the unit of $\mathcal {B}$ and fulfils $\Phi (a^*) = \Phi (a)^*$ for all $a\in \mathcal {A}$ . This last condition is equivalent to $\Phi (a) \in \mathcal {B}_{\mathrm {H}}$ for all $a\in \mathcal {A}_{\mathrm {H}}$ . If $\mathcal {A}$ and $\mathcal {B}$ are ordered $^*$ -algebras, then such a unital $^*$ -homomorphism $\Phi $ is called positive or an order embedding if its restriction to an $\mathbb{R}$ -linear map between the ordered vector spaces $\mathcal {A}_{\mathrm {H}}$ and $\mathcal {B}_{\mathrm {H}}$ is positive or an order embedding, respectively.

For ordered $^*$ -algebras, we are going to discuss two different types of representations, which correspond to the first two examples mentioned above.

Of course, representations as functions are especially interesting for commutative ordered $^*$ -algebras. As we will see, the existence of faithful representations of an ordered $^*$ -algebra $\mathcal {A}$ is closely linked to the question of whether or not the order on $\mathcal {A}$ is induced by its (extremal) positive Hermitian linear functionals.

4.1 Representation as operators

The well-known construction of the Gelfand–Naimark–Segal (GNS) representation yields a representation as operators of an ordered $^*$ -algebra $\mathcal {A}$ out of any positive Hermitian linear functional.

Proposition 4.7 Let $\mathcal {A}$ be an ordered $^*$ -algebra, and let $\omega \in \mathcal {A}^{*,+}_{\mathrm {H}}$ . Then the map ${\langle \,{ {\,\cdot \,} } \,|\, { {\,\cdot \,} }\,\rangle }_\omega \colon \mathcal {A} \times \mathcal {A} \to \mathbb{C}$ ,

(4.3)

is sesquilinear (antilinear in the first, and linear in the second argument) and fulfils $\overline {{{\langle \,{a} \,|\, {b}\,\rangle }_\omega }} = {\langle \,{b} \,|\, {a}\,\rangle }_\omega $ as well as ${\langle \,{a} \,|\, {a}\,\rangle }_\omega \in {[0,\infty [}$ for all $a,b\in \mathcal {A}$ . Write ${{\lVert { {\,\cdot \,} } \rVert }_{\omega }}$ for the corresponding seminorm on $\mathcal {A}$ , defined as for all $a\in \mathcal {A}$ , and write for its kernel. Then ${\langle \,{ {\,\cdot \,} } \,|\, { {\,\cdot \,} }\,\rangle }_\omega $ remains well defined on the quotient vector space $\mathcal {A} / \mathrm {ker}{{\lVert { {\,\cdot \,} } \rVert }_{\omega }}$ on which it describes an inner product. Now, write $\mathcal {D}_\omega $ for the pre-Hilbert space of $\mathcal {A} / \mathrm {ker}{{\lVert { {\,\cdot \,} } \rVert }_{{\omega }}}$ with inner product ${\langle \,{ {\,\cdot \,} } \,|\, { {\,\cdot \,} }\,\rangle }_\omega $ , and $[b]_\omega \in \mathcal {A} / \mathrm {ker}{{\lVert { {\,\cdot \,} } \rVert }_{\omega }}$ for the equivalence class of an element $b\in \mathcal {A}$ . Then, for every $a\in \mathcal {A}$ , the map $\pi _\omega (a) \colon \mathcal {D}_\omega \to \mathcal {D}_\omega $ ,

(4.4)

is a well-defined linear endomorphism of $\mathcal {D}_\omega $ , it is adjointable with adjoint $\pi _\omega (a^*)$ , and the resulting map $\mathcal {A} \ni a \mapsto \pi _\omega (a) \in \mathcal {L}^*(\mathcal {D}_\omega )$ is a positive unital $^*$ -homomorphism. Altogether, $\big (\mathcal {D}_\omega ,\pi _\omega \big )$ is a representation as operators of the ordered $^*$ -algebra $\mathcal {A}$ .

Proof The only detail which is not completely part of the classical GNS construction for $^*$ -algebras as described, e.g., in [Reference Schmüdgen12, Section 8.6], is the observation that $\pi _\omega $ is not only a unital $^*$ -homomorphism, but also positive, because ${\langle \,{[b]_\omega } \,|\, {\pi _\omega (a)\,[b]_\omega }\,\rangle } = {\langle \,{\omega } \,,\, {b^*a\,b}\,\rangle } \ge 0$ for all $a\in \mathcal {A}^+_{\mathrm {H}}$ .

Definition 4.2 Let $\mathcal {A}$ be an ordered $^*$ -algebra, and let $\omega \in \mathcal {A}^{*,+}_{\mathrm {H}}$ , then the representation as operators $\big ( \mathcal {D}_\omega , \pi _\omega \big )$ from Proposition 4.7 is called the GNS representation of $\mathcal {A}$ with respect to $\omega $ .

The problem of existence of representations as operators of ordered $^*$ -algebras can be treated completely analogous to the case of general $^*$ -algebras:

Proposition 4.8 Let $\mathcal {A}$ be an ordered $^*$ -algebra, then there exists a faithful representation as operators of $\mathcal {A}$ if and only if the order on $\mathcal {A}$ is induced by its positive Hermitian linear functionals.

Proof Assume that there exists a faithful representation as operators $(\mathcal {D},\pi )$ of $\mathcal {A}$ . Given $a\in \mathcal {A}_{\mathrm {H}} \setminus \mathcal {A}_{\mathrm {H}}^+$ , then there exists $\phi \in \mathcal {D}$ such that ${\langle \,{\phi } \,|\, {\pi (a)(\phi )}\,\rangle } < 0$ . However, $\mathcal {A} \ni b \mapsto {\langle \,{\phi } \,|\, {\pi (b)(\phi )}\,\rangle } \in \mathbb{C}$ is a positive Hermitian linear functional. So we conclude that the order on $\mathcal {A}$ is induced by its positive Hermitian linear functionals.

Conversely, assume that the order on $\mathcal {A}$ is induced by its positive Hermitian linear functionals. Using the GNS representations of $\mathcal {A}$ , define the orthogonal sum of pre-Hilbert spaces

with inner product denoted by ${\langle \,{ {\,\cdot \,} } \,|\, { {\,\cdot \,} }\,\rangle }_{\mathrm {tot}}$ , as well as for every element $a\in \mathcal {A}$ the linear endomorphism

of $\mathcal {D}_{\mathrm {tot}}$ , i.e.,

. Then it is easy to check that $\pi _{\mathrm {tot}}(a)$ is adjointable with adjoint $\pi _{\mathrm {tot}}(a^*)$ and that the resulting map $\pi _{\mathrm {tot}} \colon \mathcal {A} \to \mathcal {L}^*\big ( \mathcal {D}_{\mathrm {tot}} \big )$ , $a\mapsto \pi _{\mathrm {tot}}(a)$ is a positive unital $^*$ -homomorphism. Moreover, $\pi _{\mathrm {tot}}$ is an order embedding: Indeed, for every $a\in \mathcal {A}_{\mathrm {H}} \setminus \mathcal {A}_{\mathrm {H}}^+$ , there exists an $\omega \in \mathcal {A}^{*,+}_{\mathrm {H}}$ such that ${\langle \,{\omega } \,,\, {a}\,\rangle } < 0$ and thus

. It follows that $\big ( \mathcal {D}_{\mathrm {tot}}, \pi _{\mathrm {tot}} \big )$ is a faithful representation as operators.

Application of Theorem 3.7 to Proposition 4.8 immediately yields the following generalization of the (noncommutative) Gelfand–Naimark theorem.

Theorem 4.9 Let $\mathcal {A}$ be a $\sigma $ -bounded ordered $^*$ -algebra, then $\mathcal {A}$ has a faithful representation as operators if and only if $\mathcal {A}$ is closed.

The original (noncommutative) Gelfand–Naimark theorem is essentially the special case of this Theorem 4.9 for ordered $^*$ -algebras $\mathcal {A}$ in which the multiplicative unit is also a strong order unit. As discussed under Theorem 3.7, $\mathcal {A}$ is automatically $\sigma $ -bounded in this case. Note also that the image of a $\sigma $ -bounded ordered $^*$ -algebra $\mathcal {A}$ under a faithful representation as operators is an $O^*$ -algebra with metrizable graph topology in the language of [Reference Schmüdgen12], which, conversely, are always $\sigma $ -bounded. So Theorem 4.9 yields an order-theoretic characterization of the $O^*$ -algebras with metrizable graph topology. This goes in a similar direction as [Reference Schmüdgen11], where a topological characterization of a large class of $O^*$ -algebras has been given.

Despite being valid only in the $\sigma $ -bounded case, Theorem 4.9 still implies that, heuristically, every closed ordered $^*$ -algebra “behaves essentially like an $O^*$ -algebra”. In order to make this more precise, we need the following lemma.

Lemma 4.10 Let $\mathcal {A}$ be a $^*$ -algebra that is generated by a countable subset of $\mathcal {A}$ . If $\mathcal {A}_{\mathrm {H}}$ is endowed with any partial order $\le $ such that $\mathcal {A}$ becomes an ordered $^*$ -algebra, then $\mathcal {A}$ with this order is $\sigma $ -bounded.

Proof As $\mathcal {A}$ is generated by a countable subset, it has at most countable dimension, so there exists a sequence $(a_n)_{n\in \mathbb{N}}$ in $\mathcal {A}_{\mathrm {H}}$ such that the $\mathbb{R}$ -linear span of ${\{\,{a_n} \;|\; {n\in \mathbb{N}}\,\}}$ is whole $\mathcal {A}_{\mathrm {H}}$ . Moreover, from , it follows that for all $n\in \mathbb{N}$ . So define the increasing sequence $(\hat {v}_n)_{n\in \mathbb{N}}$ in $\mathcal {A}_{\mathrm {H}}^+$ as , then $(\hat {v}_n)_{n\in \mathbb{N}}$ is a dominating sequence because for every $b \in \mathcal {A}_{\mathrm {H}}$ there exists an $n\in \mathbb{N}$ such that b can be expressed as $b = \sum _{k=1}^n 2\beta _n a_n$ with coefficients $\beta _1,\dots ,\beta _n \in [{-n},n]$ , and hence $b \le \hat {v}_n$ .

Proposition 4.11 Let $\mathcal {A}$ be a closed ordered $^*$ -algebra, and let $(a_n)_{n\in \mathbb{N}}$ be any sequence in $\mathcal {A}$ . Then the unital $^*$ -subalgebra of $\mathcal {A}$ that is generated by ${\{\,{a_n} \;|\; {n \in \mathbb{N}}\,\}}$ has a faithful representation as operators.

Proof By Lemma 4.10, the unital $^*$ -subalgebra of $\mathcal {A}$ that is generated by ${\{\,{a_n} \;|\; {n \in \mathbb{N}}\,\}}$ is $\sigma $ -bounded, so Theorem 4.9 applies.

4.2 Representation as functions

A slight modification of the well-known Gelfand transformation yields a representation as functions of any ordered $^*$ -algebra.

Definition 4.3 Let $\mathcal {A}$ be an ordered $^*$ -algebra, then the set of all multiplicative states on $\mathcal {A}$ , i.e., of all positive unital $^*$ -homomorphisms from $\mathcal {A}$ to $\mathbb{C}$ , will be denoted by $\mathcal {S}_{\mathrm {m}}(\mathcal {A})$ .

Proposition 4.12 Let $\mathcal {A}$ be an ordered $^*$ -algebra. Then the map $\pi _{\mathrm {Gelfand}} \colon \mathcal {A} \to \mathbb{C}^{\mathcal {S}_{\mathrm {m}}(\mathcal {A})}$ , $a \mapsto \pi _{\mathrm {Gelfand}}(a)$ with $\pi _{\mathrm {Gelfand}}(a) \colon \mathcal {S}_{\mathrm {m}}(\mathcal {A}) \to \mathbb{C}$ ,

(4.5)

is a positive unital $^*$ -homomorphism and $\big ( \mathcal {S}_{\mathrm {m}}(\mathcal {A}), \pi _{\mathrm {Gelfand}} \big )$ is a representation as functions of $\mathcal {A}$ .

Proof Immediate consequence of the properties of the elements in $\mathcal {S}_{\mathrm {m}}(\mathcal {A})$ .

In order to guarantee the existence of many multiplicative states, we have to examine states which are at the same time extremal positive Hermitian linear functionals.

Definition 4.4 Let $\mathcal {A}$ be an ordered $^*$ -algebra. Then a state $\omega $ on $\mathcal {A}$ is called pure if $\omega $ is also an extremal positive Hermitian linear functional on $\mathcal {A}$ . The set of all pure states on $\mathcal {A}$ will be denoted by .

The above definition of pure states is equivalent to the more common one as extreme points of the convex set of states.

Proposition 4.13 Let $\mathcal {A}$ be an ordered $^*$ -algebra, then $\mathcal {S}_{\mathrm {p}}(\mathcal {A}) = \mathrm {ex}\big ( \mathcal {S}(\mathcal {A})\big )$ .

Proof If $\omega $ is an extreme point of $\mathcal {S}(\mathcal {A})$ , then it is also an extremal positive Hermitian linear functional, hence a pure state: Indeed, given $\rho \in \mathcal {A}^{*,+}_{\mathrm {H}}$ such that $\rho \le \omega $ , then, as a consequence of the Cauchy–Schwarz inequality, either both

and

are in ${]0,1[}$ , or $\rho = \mu \omega $ with $\mu \in \{0,1\}$ . In the former case,

is a representation of $\omega $ as a nontrivial convex combination of two elements of $\mathcal {S}(\mathcal {A})$ , which implies that $\rho = \mu \omega $ with

.

Conversely, if $\omega $ is pure state on $\mathcal {A}$ , then it is an extreme point of $\mathcal {S}(\mathcal {A})$ : If $\omega = \lambda \rho + (1-\lambda )\rho '$ with $\rho ,\rho ' \in \mathcal {S}(\mathcal {A})$ and $\lambda \in {]0,1[}$ , then $\lambda \rho \le \omega $ and $(1-\lambda ) \rho ' \le \omega $ . Consequently, there are $\mu ,\mu '\in {[0,1]}$ such that $\lambda \rho = \mu \omega $ and $(1-\lambda ) \rho ' = \mu '\omega $ . Evaluation on shows that $\lambda = \mu $ and $(1-\lambda ) = \mu '$ , and hence $\rho = \omega = \rho '$ .

The sets of pure states and of multiplicative states on an ordered $^*$ -algebra are closely related. In order to see this, the following concept will be helpful.

Definition 4.5 Let $\mathcal {A}$ be an ordered $^*$ -algebra, let $\omega $ be a state on $\mathcal {A}$ , and let $a\in \mathcal {A}$ . The variance of $\omega $ on a is defined as

(4.6)

Note that $\mathrm {Var}_\omega (a) \in {[0,\infty [}$ and that $\mathrm {Var}_\omega (a) = {\langle \,{\omega } \,,\, {a^*a}\,\rangle } - {\lvert {{\langle \,{\omega } \,,\, {a}\,\rangle }} \rvert }^2$ holds for every state $\omega $ on every ordered $^*$ -algebra $\mathcal {A}$ and all $a\in \mathcal {A}$ .

Proposition 4.14 If $\mathcal {A}$ is an ordered $^*$ -algebra and $\omega $ is a multiplicative state on $\mathcal {A}$ , then $\omega $ is a pure state on $\mathcal {A}$ .

Proof By Proposition 4.13, the pure states are precisely the extreme points of the set of all states. So assume that $\rho ,\rho ' \in \mathcal {S}(\mathcal {A})$ and $\lambda \in {]0,1[}$ fulfil $\omega = \lambda \rho + (1-\lambda )\rho '$ , then one can check that the identity

$$ \begin{align*} \mathrm{Var}_\omega(a) = \mathrm{Var}_{\lambda\rho + (1-\lambda)\rho'}(a) = \lambda \mathrm{Var}_{\rho}(a) + (1-\lambda) \mathrm{Var}_{\rho'}(a) + \lambda(1-\lambda){\lvert {{\langle\,{\rho-\rho'} \,,\, {a}\,\rangle}} \rvert}^2 \end{align*} $$

holds for all $a\in \mathcal {A}$ . Moreover, $\mathrm {Var}_\omega (a) = 0$ because $\omega $ is multiplicative. It follows that ${\lvert {{\langle \,{\rho -\rho '} \,,\, {a}\,\rangle }} \rvert }^2 = 0$ for all $a\in \mathcal {A}$ because $\mathrm {Var}_{\rho }(a)$ and $\mathrm {Var}_{\rho '}(a)$ are nonnegative, so $\rho = \rho ' = \omega $ .

In order to be able to obtain a converse statement, we need some more assumptions.

Definition 4.6 An ordered $^*$ -algebra $\mathcal {A}$ is called radical if it has the following property: Whenever $a, b \in \mathcal {A}_{\mathrm {H}}$ commute and fulfil and $0 \le ab$ , then $0 \le b$ .

There are some important examples of radical commutative ordered $^*$ -algebras.

Proposition 4.15 Let $\mathcal {A}$ be a commutative ordered $^*$ -algebra. If $\mathcal {A}$ has a faithful representation as functions, then $\mathcal {A}$ is radical and closed.

Proof Let $(X,\pi )$ be a faithful represent as functions of $\mathcal {A}$ . It is easy to check that $\mathbb{C}^X$ is radical and closed, essentially because $\mathbb{C}$ is radical and closed. Using that $\pi \colon \mathcal {A} \to \mathbb{C}^X$ is a positive unital $^*$ -homomorphism and an order embedding, it follows immediately that $\mathcal {A}$ has to be radical and closed as well.

Proposition 4.16 Let be the complexification of a $\Phi $ -algebra $\mathcal {R}$ , then $\mathcal {A}$ is a radical and closed commutative ordered $^*$ -algebra.

Proof As discussed in Example 4.3, $\mathcal {A}$ is a closed commutative ordered $^*$ -algebra. Consider $a, b \in \mathcal {A}_{\mathrm {H}}$ such that and $0 \le ab$ . We can express b as the difference $b = b_+ -b_-$ of its positive and negative components, , and . From $b_+ \wedge b_- = 0$ , it follows that $b_+ b_- = 0$ . As $\mathcal {A}^+_{\mathrm {H}}$ is closed under products by definition of $\Phi $ -algebras, multiplication with $b_-$ yields $0 \le b_-\,ab = - b_-\,a\,b_- \le 0$ , so $b_-\,a\,b_- = 0$ . From , it now follows that $0 \le (b_-)^2 \le b_-\,a\,b_- = 0$ , so $(b_-)^2 = 0$ . As a consequence, , and hence for all $\epsilon \in {]0,\infty [}$ , which implies that $b_- \le 0$ because $\mathcal {A}$ is closed by assumption. This finally shows that $b = b_+-b_- \ge 0$ .

It is also worthwhile to mention an important nonexample.

Example 4.17 Consider the $^*$ -algebra $\mathbb{C}[x,y]$ of complex polynomials in two variables x and y with the $^*$ -involution given by complex conjugation of coefficients, and thus $\mathbb{C}[x,y]_{\mathrm {H}} \cong \mathbb{R}[x,y]$ . On $\mathbb{C}[x,y]_{\mathrm {H}}$ , choose the partial order that turns $\mathbb{C}[x,y]$ into an ordered $^*$ -algebra with cone of positive elements given by sums of Hermitian squares, i.e.,

Note that $\mathbb{C}[x,y]_{\mathrm {H}}^+ \cap (- \mathbb{C}[x,y]_{\mathrm {H}}^+) = \{0\}$ because every sum of Hermitian squares of polynomials is pointwise positive on $\mathbb{R}^2$ . Moreover, the product of two elements of $\mathbb{C}[x,y]_{\mathrm {H}}^+$ is again in $\mathbb{C}[x,y]_{\mathrm {H}}^+$ . It is well known that there exist polynomials $p \in \mathbb{C}[x,y]_{\mathrm {H}} \setminus \mathbb{C}[x,y]_{\mathrm {H}}^+$ that are pointwise positive on $\mathbb{R}^2$ . An explicit example from [Reference Berg, Christensen and Jensen2] is

Now, consider

, then

As

, we conclude that $\mathbb{C}[x,y]$ (with this choice for the order) is not a radical ordered $^*$ -algebra and especially does not have a faithful representation as functions due to Proposition 4.15. Closer inspection shows that this is indeed because of the existence of ill-behaved pure states (see, e.g., [Reference Schmüdgen12, Corollary 11.6.4] and the discussion there for details).

We proceed with examining the relation between pure states and multiplicative states.

Lemma 4.18 Let $\mathcal {A}$ be an ordered $^*$ -algebra, let $\omega $ be a state on $\mathcal {A}$ , and let $a\in \mathcal {A}$ with $\mathrm {Var}_\omega (a) = 0$ , then

(4.7) $$ \begin{align} {\langle\,{\omega} \,,\, {b^*a}\,\rangle} = {\langle\,{\omega} \,,\, {b^*}\,\rangle} {\langle\,{\omega} \,,\, {a}\,\rangle} \ \text{and}\ {\langle\,{\omega} \,,\, {a^*b}\,\rangle} = {\langle\,{\omega} \,,\, {a^*}\,\rangle} {\langle\,{\omega} \,,\, {b}\,\rangle} \end{align} $$

hold for all $b\in \mathcal {A}$ . A state $\omega $ on $\mathcal {A}$ thus is multiplicative if (and only if) $\mathrm {Var}_\omega (a) = 0$ for all $a\in \mathcal {A}$ .

Proof The Cauchy–Schwarz inequality yields

for all $a,b\in \mathcal {A}$ . If $\mathrm {Var}_\omega (a) = 0$ , then this implies that (4.7) holds.

Lemma 4.19 Let $\mathcal {A}$ be a commutative ordered $^*$ -algebra, and let $\omega \in \mathcal {S}(\mathcal {A})$ , then the subset ${\big \{\,{a\in \mathcal {A}} \;\big |\; {\mathrm {Var}_\omega (a)=0}\,\big \}}$ of $\mathcal {A}$ is a unital $^*$ -subalgebra and this is the largest (with respect to inclusion) unital $^*$ -subalgebra of $\mathcal {A}$ on which the restriction of $\omega $ is multiplicative. In the special case that holds for all $a\in \mathcal {A}_{\mathrm {H}}$ , it follows that $\omega $ is multiplicative on whole $\mathcal {A}$ .

Proof First assume that $\omega $ is an arbitrary state on $\mathcal {A}$ . Then it is easy to check that $\mathrm {Var}_\omega (\lambda a) = {\lvert {\lambda } \rvert }^2 \mathrm {Var}_\omega (a)$ and also (using the commutativity of $\mathcal {A}$ ) $\mathrm {Var}_\omega (a^*) = \mathrm {Var}_\omega (a)$ hold for all $a\in \mathcal {A}$ and all $\lambda \in \mathbb{C}$ . Moreover, if $a,b\in \mathcal {A}$ fulfil $\mathrm {Var}_\omega (a) = \mathrm {Var}_\omega (b) = 0$ , then one can check with the help of Lemma 4.18 that

$$ \begin{align*} \mathrm{Var}_\omega(a+b) &= {\big\langle\,{\omega} \,,\, {a^*a+a^*b +b^*a+b^*b}\,\big\rangle} - {\big\lvert {{\langle\,{\omega} \,,\, {a}\,\rangle} + {\langle\,{\omega} \,,\, {b}\,\rangle}} \big\rvert}^2 = 0 \end{align*} $$

and

$$ \begin{align*} \mathrm{Var}_\omega(ab) &= {\big\langle\,{\omega} \,,\, {b^*a^*a\,b}\,\big\rangle} - {\big\lvert {{\big\langle\,{\omega} \,,\, {ab}\,\big\rangle}} \big\rvert}^2 = 0\,. \end{align*} $$

As is clearly fulfilled as well, one sees that ${\{\,{a\in \mathcal {A}} \;|\; {\mathrm {Var}_\omega (a)=0}\,\}}$ is a unital $^*$ -subalgebra of $\mathcal {A}$ . From Lemma 4.18, it also follows that the restriction of $\omega $ to ${\{\,{a\in \mathcal {A}} \;|\; {\mathrm {Var}_\omega (a)=0}\,\}}$ is multiplicative. Conversely, if $\mathcal {B}$ is another unital $^*$ -subalgebra of $\mathcal {A}$ such that the restriction of $\omega $ to $\mathcal {B}$ is multiplicative, then it follows that $\mathrm {Var}_\omega (b) = {\langle \,{\omega } \,,\, {b^*b}\,\rangle } - {\lvert {{\langle \,{\omega } \,,\, {b}\,\rangle }} \rvert }^2 = 0$ for all $b\in \mathcal {B}$ and therefore the inclusion $\mathcal {B} \subseteq {\{\,{a\in \mathcal {A}} \;|\; {\mathrm {Var}_\omega (a)=0}\,\}}$ holds.

Now, assume that holds for all $a\in \mathcal {A}_{\mathrm {H}}$ . As holds for all $b\in \mathcal {A}_{\mathrm {H}}$ , it follows from that $\mathrm {Var}_\omega (b) = 0$ for all $b\in \mathcal {A}_{\mathrm {H}}$ , and hence $\mathrm {Var}_\omega (c) = 0$ for all $c\in \mathcal {A}$ because $c = c_r+\mathrm {i} c_i$ with and . Application of Lemma 4.18 shows that $\omega $ is multiplicative on whole $\mathcal {A}$ .

Theorem 4.20 Let $\mathcal {A}$ be a radical commutative ordered $^*$ -algebra, then $\mathcal {S}_{\mathrm {p}}(\mathcal {A}) = \mathcal {S}_{\mathrm {m}}(\mathcal {A})$ , i.e., a state on $\mathcal {A}$ is a pure state if and only if it is multiplicative.

Proof Let $\mathcal {\omega }$ be a state on $\mathcal {A}$ . If $\omega $ is multiplicative, then Proposition 4.14 shows that $\omega $ is also a pure state. Conversely, if $\omega $ is a pure state, then it remains to show that the identity holds for all $a\in \mathcal {A}_{\mathrm {H}}$ , which, by Lemma 4.19, already implies that $\omega $ is multiplicative.

So let $a\in \mathcal {A}_{\mathrm {H}}$ be given and define the subset

of $\mathcal {A}_{\mathrm {H}}$ . It is clear that $S_a$ is a (real) linear subspace of $\mathcal {A}_{\mathrm {H}}$ . If

with $b,b' \in \mathcal {A}_{\mathrm {H}}$ , then $b-b'=0$ because

and because $\mathcal {A}$ is radical. So every element of $S_a$ is of the form

with a unique $b\in \mathcal {A}_{\mathrm {H}}$ and the map $\tilde {\rho } \colon S_{a} \to \mathbb{R}$ ,

is well defined and is clearly $\mathbb{R}$ -linear. Moreover, for every $c\in \mathcal {A}_{\mathrm {H}}$ , there exists an element

with $b\in \mathcal {A}_{\mathrm {H}}$ such that

and

, e.g.,

. Using again that $\mathcal {A}$ is radical one sees that $\tilde {\rho }$ is positive with respect to the order on $S_a$ inherited from $\mathcal {A}_{\mathrm {H}}$ , so the extension theorem for positive linear functionals applies and shows that there exists a positive linear functional $\rho $ on $\mathcal {A}_{\mathrm {H}}$ fulfilling

for all $b\in \mathcal {A}_{\mathrm {H}}$ . Using the isomorphism between $(\mathcal {A}_{\mathrm {H}})^*$ and $\mathcal {A}_{\mathrm {H}}^*$ , we can also treat $\rho $ as a positive Hermitian linear functional on $\mathcal {A}$ .

As

holds for all $b\in \mathcal {A}_{\mathrm {H}}$ , it follows that $\rho \le \omega $ , and hence there exists $\mu \in {[0,1]}$ such that $\rho = \mu \omega $ because $\omega $ is a pure state by assumption. From evaluation on

and

, one gets

and

which yields $\mu \neq 0$ and

as well as

, and thus

.

Similar results about the relation between pure and multiplicative states on certain $^*$ -algebras have already occurred before, e.g., [Reference Bucy and Maltese3, Theorem 2] for Banach $^*$ -algebras or [Reference Schmüdgen12, Proposition 11.3.9] for general $^*$ -algebras endowed with a special choice of a (pre-)order.

Corollary 4.21 Let X be a set, and let $\mathcal {A}$ be a unital $^*$ -subalgebra of $\mathbb{C}^X$ , endowed with the pointwise order inherited from $\mathbb{C}^X$ , then the pure states on $\mathcal {A}$ are precisely the multiplicative ones.

Note, however, that there might be more multiplicative states on $\mathcal {A}$ than just evaluation functionals at points of X.

Corollary 4.22 Let $\mathcal {A}$ be a radical commutative ordered $^*$ -algebra, then

(4.8)

and the set $\mathcal {A}^{*,+,\mathrm {ex}}_{\mathrm {H}}$ of extremal positive Hermitian linear functionals on $\mathcal {A}$ is weak- $^*$ -closed in $\mathcal {A}_{\mathrm {H}}^*$ .

Proof Let $\omega $ be a positive Hermitian linear functional on $\mathcal {A}$ . Then either $\omega = 0$ , in which case $\omega \in \mathcal {A}^{*,+,\mathrm {ex}}_{\mathrm {H}}$ and for all $a\in \mathcal {A}$ hold, or . In this second case, is a state on $\mathcal {A}$ and the following chain of equivalences holds: $\omega $ is an extremal positive Hermitian linear functional if and only if $\tilde {\omega }$ is a pure state, which is equivalent to $\tilde {\omega }$ being multiplicative by Theorem 4.20, and Lemma 4.18 shows that this holds if and only if $\mathrm {Var}_{\tilde {\omega }}(a) = 0$ for all $a\in \mathcal {A}$ . As , identity (4.8) is proved.

Finally, as $\mathcal {A}^{*,+}_{\mathrm {H}}$ is weak- $^*$ -closed in $\mathcal {A}^*_{\mathrm {H}}$ and as is a weak- $^*$ -continuous function, one sees that the right-hand side of (4.8) is weak- $^*$ -closed in $\mathcal {A}^*_{\mathrm {H}}$ .

Corollary 4.23 Let $\mathcal {A}$ be a radical commutative ordered $^*$ -algebra, then the following are equivalent:

1. (1) The Gelfand transformation $\big ( \mathcal {S}_{\mathrm {m}}(\mathcal {A}), \pi _{\mathrm {Gelfand}} \big )$ of $\mathcal {A}$ discussed in Proposition 4.12 is a faithful representation as functions.

2. (2) There exists a faithful representation as functions of $\mathcal {A}$ .

3. (3) The order on $\mathcal {A}$ is induced by its extremal positive Hermitian linear functionals.

Proof The implication (1) $\Longrightarrow $ (2) is trivial.

Assume that there exists a faithful representation as functions $(X,\pi )$ of $\mathcal {A}$ , and let $a\in \mathcal {A}_{\mathrm {H}} \setminus \mathcal {A}_{\mathrm {H}}^+$ be given. Then there exists $x\in X$ fulfilling $\pi (a)(x) < 0$ . However, the linear functional $\mathcal {A} \ni b \mapsto \pi (b)(x) \in \mathbb{C}$ is a multiplicative state, hence a pure state by Proposition 4.14, and thus especially an extremal positive Hermitian linear functional on $\mathcal {A}$ . This proves the implication (2) $\Longrightarrow $ (3).

Finally, if the order on $\mathcal {A}$ is induced by its extremal positive Hermitian linear functionals, then for every $a\in \mathcal {A}_{\mathrm {H}} \setminus \mathcal {A}_{\mathrm {H}}^+$ , there exists an extremal positive Hermitian linear functional $\tilde {\omega }$ on $\mathcal {A}$ such that ${\langle \,{\tilde {\omega }} \,,\, {a}\,\rangle } < 0$ . From $\tilde {\omega } \neq 0$ , it follows that and therefore is a well-defined pure state on $\mathcal {A}$ , hence also a multiplicative state by Theorem 4.20. So we see that $\pi _{\mathrm {Gelfand}}(a)$ cannot be positive because $\pi _{\mathrm {Gelfand}}(a)(\omega ) < 0$ , and conclude that the Gelfand transformation of $\mathcal {A}$ is faithful, i.e., (3) $\Longrightarrow $ (1) holds.

There are some representation theorems for commutative $^*$ -algebras endowed with a locally convex topology defined by submultiplicative seminorms, i.e., seminorms fulfilling the estimate ${\lVert {ab} \rVert } \le {\lVert {a} \rVert } {\lVert {b} \rVert }$ for all elements a and b of the algebra. One example is of course the commutative Gelfand–Naimark theorem. However, for radical commutative ordered $^*$ -algebras, Corollary 4.23 combined with Corollary 3.4 yields an approach using a rather different type of locally convex topologies.

Application of Theorem 3.7 to Corollary 4.23 immediately yields the following generalization of the commutative Gelfand–Naimark theorem.

Theorem 4.24 Let $\mathcal {A}$ be a $\sigma $ -bounded radical commutative ordered $^*$ -algebra, then $\mathcal {A}$ has a faithful representation as functions if and only if $\mathcal {A}$ is closed.

Like before, this implies that, heuristically, every closed radical commutative ordered $^*$ -algebra “behaves essentially like a $^*$ -algebra of functions.”

Corollary 4.25 Let $\mathcal {A}$ be a closed radical commutative ordered $^*$ -algebra, and let $(a_n)_{n\in \mathbb{N}}$ be any sequence in $\mathcal {A}$ . Then the unital $^*$ -subalgebra of $\mathcal {A}$ that is generated by ${\{\,{a_n} \;|\; {n \in \mathbb{N}}\,\}}$ has a faithful representation as functions.

Proof By Lemma 4.10, the unital $^*$ -subalgebra of $\mathcal {A}$ that is generated by ${\{\,{a_n} \;|\; {n \in \mathbb{N}}\,\}}$ is $\sigma $ -bounded, so Theorem 4.24 applies.

Theorems 4.9 and 4.24 allow to develop a theory of $^*$ -algebras of (possibly unbounded) operators in a representation-independent way. While the assumption of being $\sigma $ -bounded might be replaced by others, e.g., the existence of a well-behaved locally convex topology, the property of being radical is necessary for a commutative ordered $^*$ -algebra to have a faithful representation as functions (see Proposition 4.15).