Algebraic Probability (continued)

Continuing on from the previous post, I look at cases where the abstract concept of states on algebras correspond to classical probability measures. Up until now, we have considered commutative real algebras but, before going further, it will help to look instead at algebras over the complex numbers {{\mathbb C}}. In the commutative case, we will see that this is equivalent to using real algebras, but can be more convenient, and in the non-commutative case it is essential. When using complex algebras, we will require the existence of an involution, which can be thought of as a generalisation of complex conjugation.

Recall that, by an algebra {\mathcal A} over a field {K}, we mean that {\mathcal A} is a {K}-vector space together with a binary product operation satisfying associativity, distributivity over addition, compatibility with scalars, and which has a multiplicative identity.

Definition 1 A *-algebra {\mathcal A} is an algebra over {{\mathbb C}} together with an involution, which is a unary operator {\mathcal A\rightarrow\mathcal A}, {a\mapsto a^*}, satisfying,

  1. Anti-linearity: {(\lambda a+\mu b)^*=\bar\lambda a^*+\bar\mu b^*}.
  2. {(ab)^*=b^*a^*}.
  3. {a^{**}=a}

for all {a,b\in\mathcal A} and {\lambda,\mu\in{\mathbb C}}.

Here, I am using {\bar\lambda} to denote the complex conjugate of {\lambda\in{\mathbb C}}. Examples of *-algebras include:

  • {{\mathbb C}} is a *-algebra, with involution given by complex conjugation.
  • The algebra {{\mathbb C}[X_1,X_2,\ldots,X_n]} of complex polynomials over indeterminates {X_1,\ldots,X_n} is a commutative *-algebra. Involution, {f\mapsto f^*} is given by taking complex conjugates of the polynomial coefficients.
  • For a topological space {X}, the collection {C(X)} of continuous functions {f\colon X\rightarrow{\mathbb C}} is a commutative *-algebra, where addition and multiplication are defined pointwise, and involution is pointwise complex conjugation, {f^*(x)=\overline{f(x)}}.
  • For a (complex) Hilbert space {\mathcal H}, the space {B(\mathcal H)} of bounded linear operators {a\colon\mathcal H\rightarrow\mathcal H} is a *-algebra. Involution is defined as the operator adjoint, so {\langle a^*x,y\rangle=\langle x,ay\rangle} for all {x,y\in\mathcal H}.

A homomorphism {f\colon\mathcal A\rightarrow\mathcal B} of *-algebras is a map from {\mathcal A} to {\mathcal B} preserving the algebra operations,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle f(\lambda a+\mu b)=\lambda f(a)+\mu f(b),\smallskip\\ &\displaystyle f(ab)=f(a)f(b),\smallskip\\ &\displaystyle f(1)=1,\smallskip\\ &\displaystyle f(a^*)=f(a)^*, \end{array}

for all {a,b\in\mathcal A} and {\lambda,\mu\in{\mathbb C}}. Sometimes, such maps are called *-homomorphisms to distinguish them from algebra homomorphisms which need not preserve the involution. Also, we will occasionally want to consider non-unitial *-algebras, for which the existence of the unit {1} is not required. For homomorphisms of non-unitial *-algebras, we simply drop the requirement that {f(1)=1}.

An element {a} of a *-algebra is called self-adjoint or real, if {a^*=a}. Using {\mathcal A_{\rm sa}} to denote the set of self-adjoint elements of {\mathcal A}, anti-linearity implies that {\mathcal A_{\rm sa}} is closed under taking real linear combinations, so is a vector subspace of {\mathcal A} over the real numbers. The product of two self-adjoint elements need not be self adjoint. Considering {a,b\in\mathcal A_{\rm sa}}, the identity {(ab)^*=ba} shows that the product {ab} is in {\mathcal A_{\rm sa}} if and only if {a} and {b} commute. Finding self-adjoint elements of an algebra is easy enough. For an element {a} then {a^*+a} and {a^*a} are both self-adjoint,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle(a^*+a)^*=a^{**}+a^*=a^*+a,\smallskip\\ &\displaystyle (a^*a)^*=a^*a^{**}=a^*a. \end{array}

Also, the identity {1\in\mathcal A} is automatically self-adjoint,

\displaystyle  1^*=1^*1=(1^*1)^*=1^{**}=1.

Every {a\in\mathcal A} can be uniquely decomposed into its real and imaginary parts,

\displaystyle  a=x+iy, (1)

for self-adjoint {x,y}, which are given by

\displaystyle  x=\frac12(a^*+a),\ y=\frac{i}2(a^*-a).

Commutative *-algebras are equivalent to commutative real algebras (precisely, they are equivalent categories). For any commutative *-algebra {\mathcal A}, the subset {\mathcal A_{\rm sa}} of self-adjoint elements is a commutative real algebra. In the opposite direction, if {\mathcal A} is a commutative real algebra, then we can consider its complexification {\mathcal A^{{\mathbb C}}=\mathcal A\otimes_{{\mathbb R}}{\mathbb C}}. Every element of {\mathcal A^{\mathbb C}} can be written uniquely as {x\otimes1+y\otimes i} for {x,y\in\mathcal A}, which is written more simply as {x+iy}. Multiplication and involution are given by

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle(x+iy)(x^\prime+iy^\prime)=(xx^\prime-yy^\prime)+i(xy^\prime+yx^\prime),\smallskip\\ &\displaystyle(x+iy)^*=x-iy, \end{array}

which makes {\mathcal A^{\mathbb C}} into a commutative *-algebra. The self-adjoint elements of {\mathcal A^{\mathbb C}} are precisely {x+i0} for {x\in\mathcal A}. Hence, {x\mapsto x+i0} gives an isomorphism of real algebras from {\mathcal A} to {\mathcal A^{\mathbb C}_{\rm sa}}. Hence, up to isomorphism, any commutative real algebra can be considered as the set of self-adjoint elements of a commutative *-algebra. Conversely, (1) shows that any commutative *-algebra {\mathcal A} is isomorphic to the complexification of the real algebra {\mathcal A_{\rm sa}}.

States on *-algebras are defined as follows.

Definition 2 Let {\mathcal A} be a *-algebra. Then, a linear map {p\colon\mathcal A\rightarrow{\mathbb C}} is

  1. real, or self-adjoint, if {p(a^*)=\overline{p(a)}} for all {a\in\mathcal A}.
  2. positive if {p(a^*a)\ge0} for all {a\in\mathcal A}.
  3. a state if it is positive and {p(1)=1}.

It can be seen that {p\colon\mathcal A\rightarrow{\mathbb C}} is self-adjoint if and only if {p(a)} is real for all {a\in\mathcal A_{\rm sa}}. If {p} is positive then it must also be self-adjoint. Noting that, for {a\in\mathcal A_{\rm sa}}, the value {p(a^2)=p(a^*a)} is a nonnegative real number so, writing {a} as a combination of squares,

\displaystyle  a=\frac12((1+a)^2-1^2-a^2), (2)

shows that {p(a)} is real.

We previously defined the concept of states on commutative real algebras, and have noted above that there is an equivalence between commutative real algebras and commutative *-algebras. The definitions of states on real and *-algebras are consistent with this equivalence. For a state {p} on a commutative *-algebra {\mathcal A}, its restriction, {q}, to {\mathcal A_{\rm sa}} is a state. Conversely, any state {q\colon\mathcal A_{\rm sa}\rightarrow{\mathbb R}} extends uniquely to a state on {\mathcal A}. By writing {a\in\mathcal A} in terms if its real and imaginary parts (1) then, by linearity, we have {p(a)=q(x)+iq(y)}. The linear map {p} satisfies positivity,

\displaystyle  p(a^*a)=p(x^2)+p(y^2)=q(x^2)+q(y^2)\ge0

so is a state. This means that the theorems stated in the previous post concerning states on commutative real algebras can be translated to statements for *-algebras. We start with theorem 3 of that post.

Theorem 3 Let {p} be a state on *-algebra {\mathcal A}, and {a\in\mathcal A} be self-adjoint. Then, there exists a probability measure {\mu} on {{\mathbb R}} such that

\displaystyle  p(f(a))=\int f(x)d\mu(x)

for all polynomials {f\in{\mathbb C}[X]}.

We do not need to assume that {\mathcal A} is commutative here, as it only depends only the restriction of {p} to the subalgebra generated by {a}, which is always commutative. We also translate theorem 5 of the previous post to the language of *-algebras.

Theorem 4 Let {p} be a state on *-algebra {\mathcal A} and {a_1,a_2,\ldots,a_m} be commuting self-adjoint elements of {\mathcal A} satisfying Carleman’s condition

\displaystyle  \sum_{n=1}^\infty p(a_k^{2n})^{-\frac1{2n}}=\infty.

Then, there exists a unique probability measure {\mu} on {{\mathbb R}^m} such that

\displaystyle  p(f(a_1,\ldots,a_m))=\int f(x_1,\ldots,x_m)d\mu(x_1,\ldots,x_m)

for all polynomials {f\in{\mathbb C}[X_1,\ldots,X_m]}.

Again, only the restriction of {p} to the subalgebra generated by {a_1,\ldots,a_m} is important for the statement of this theorem, and this subalgebra is necessarily a commutative *-algebra.

Before moving on, we note a simple but extremely useful inequality for states and positive linear maps on a *-algebra. If {p\colon\mathcal A\rightarrow{\mathbb C}} is positive, then {(a,b)\mapsto p(a^*b)} defines a semi-inner product on {\mathcal A}. Hence, the Cauchy–Schwarz inequality applies,

\displaystyle  \lvert p(a^*b)\rvert^2\le p(a^*a)p(b^*b). (3)

Bochner’s Theorem

An alternative method by which states on *-algebras lead to classical probability measures is via groups of unitary elements. An element {u} of a *-algebra {\mathcal A} is called unitary if it satisfies

\displaystyle  u^*u=uu^*=1.

States on a *-algebra automatically imply probability distributions on the unit circle {\mathbb S^1=\{z\in{\mathbb C}\colon\lvert z\rvert=1\}} associated to each unitary element.

Theorem 5 Let {p} be a state on *-algebra {\mathcal A}, and {u\in\mathcal A} be unitary. Then, there is a unique probability measure {\mu} on {\mathbb S^1} satisfying

\displaystyle  p(u^n)=\int x^nd\mu(x)

for all {n\in{\mathbb Z}}.

Groups of unitary elements also lead to probability measures on more general spaces, although continuity requirements are imposed. As an example, we have an equivalence between probability measures on {{\mathbb R}^n} and states defined on certain spaces of unitary algebra elements.

Theorem 6 Let {p} be a state on *-algebra {\mathcal A}, and {\{u_x\colon x\in{\mathbb R}^n\}} be unitary elements of {\mathcal A} satisfying {u_xu_y=u_{x+y}}. If {x\mapsto p(u_x)} is continuous, then there is a unique probability measure {\mu} on {{\mathbb R}^n} satisfying

\displaystyle  p(u_x)=\int e^{ix\cdot y}d\mu(y)

for all {x\in{\mathbb R}^n}.

Theorems 5 and 6 are both instances of a much more general form of Bochner’s theorem. For a locally compact abelian group {G}, we denote its dual group by {\widehat G}. This is the set of continuous group homomorphisms {\chi\colon G\rightarrow\mathbb S^1}, and is itself a locally compact abelian group under the multiplication rule {(\phi\cdot\chi)(x)=\phi(x)\chi(x)} and topology of uniform convergence on compact sets. The elements {x\in G} can also be considered as maps {\widehat G\rightarrow\mathbb S^1} under the action {\chi\mapsto\chi(x)}. This identifies elements of {G} with elements of the dual of {\widehat G} which, by Pontryagin duality, is an isomorphism. That is, the natural map from {G} to its double-dual is an isomorphism. So, really, we have a pair {(G,\widehat G)} of locally compact abelian groups and a pairing {G\times\widehat G\rightarrow\mathbb S^1}, {(x,\chi)\mapsto\chi(x)}, with respect to which {\widehat G} is the dual of {G} and {G} is the dual of {\widehat G}. Important examples of dual pairs of locally compact abelian groups are:

  • {{\mathbb Z}} and {\mathbb S^1} are dual groups under the pairing {(n,\omega)\mapsto\omega^n}.
  • {{\mathbb R}^n} is self-dual under the pairing {(x,y)\mapsto e^{2\pi ix\cdot y}}.
  • For any {n\ge1}, the quotient {{\mathbb Z}/(n)} under addition is self-dual under the pairing {(x,y)\mapsto e^{2\pi ixy/n}}.
  • if {(G,\widehat G)} and {(H,\widehat H)} are pairs of dual groups then so is {(G\times H,\widehat G\times\widehat H)}.

We now give the general statement of Bochner’s theorem.

Theorem 7 Let {p} be a state on *-algebra {\mathcal A} and {G} be a locally compact abelian group. If {\{u_x\colon x\in G\}} are unitary elements of {\mathcal A} satisfying {u_xu_y=u_{x+y}}, and {x\mapsto p(u_x)} is continuous, then there is a unique regular probability measure {\mu} on {\widehat G} satisfying

\displaystyle  p(u_x)=\int \chi(x)d\mu(\chi) (4)

for all {x\in X}.

Regularity of the probability measure simply means that, for any Borel {S\subseteq\widehat G}, {\mu(S)} is the supremum of {\mu(K)} over compact {K\subseteq S}. This is a technicality which is not needed for for spaces with a countable base for the topology as, then, all probability measures are regular.

In the opposite direction, start with any group {G}, which I will express multiplicatively, so the group composition of two elements {x,y} will be written as {xy}. We construct the group algebra {{\mathbb C}[G]}. Elements of {a\in{\mathbb C}[G]} can be defined as formal sums

\displaystyle  a=\sum_{x\in G}\lambda_xu_x (5)

where {\lambda_x\in{\mathbb C}} is zero outside of a finite subset of {G}, so (5) can be thought of as a finite sum. Multiplication and involution are given by

\displaystyle  u_xu_y=u_{xy},\ u_x^*=u_{x^{-1}}.

This makes {{\mathbb C}[G]} into a *-algebra, with {u_x} being unitary elements and {u_1} being the identity, and is commutative if {G} is commutative (i.e., abelian). If {G} is a locally compact abelian group, then elements of the dual {\widehat G} can be extended to operate on all of {{\mathbb C}[G]} by,

\displaystyle  \chi\left(\sum_{x\in G}\lambda_xu_x\right)=\sum_{x\in G}\lambda_x\chi(x).

Any probability measure {\mu} on {\widehat G} defines a state {p} on {{\mathbb C}[G]} as,

\displaystyle  p(a)=\int\chi(a)d\mu(\chi),

so that (4) holds. Noting that {\chi(a^*a)=\overline{\chi(a)}\chi(a)\ge0}, we see that {p} is a state. Hence, Bochner’s theorem provides a one-to-one mapping between states on {{\mathbb C}[G]} such that {x\mapsto p(u_x)} is continuous and regular probability measures on {\widehat G}.

The Riesz–Markov Representation Theorem

I will now look at the Riesz–Markov representation theorem, and consider how states on certain *-algebras correspond naturally to probability measures on locally compact topological spaces. For any space {X}, we use {C(X)} to denote the space of continuous functions {X\rightarrow{\mathbb C}}, with the algebra operations of addition and multiplication defined pointwise, and involution being pointwise complex conjugation. This expresses {C(X)} as a *-algebra. We start with compact spaces which, by convention, I take to be Hausdorff. For a compact space {X}, the theorem can be stated as follows.

Theorem 8 Let {X} be a compact space and {p} be a state on {C(X)}. Then, there is a unique regular probability measure {\mu} on {X} satisfying

\displaystyle  p(f)=\int f(x)d\mu(x) (6)

for all {f\in C(X)}.

A few comments are in order. In most statements of the theorem, nonnegativity of of the state is required in the sense that {p(f)\ge0} whenever {f\ge0}. However, this is implied by definition 2 for states on a *-algebra, since {p(f)=p(\sqrt f^2)\ge0}. Sometimes, continuity of {p} is also required but, again, this is implied by our definition of a state. By Cauchy–Schwarz (3), and using the fact that {f^*f\le\lVert f\rVert^2},

\displaystyle  \lvert p(f)\rvert=\lvert p(1^*f)\rvert\le\sqrt{p(1^*1)p(f^*f)}\le\sqrt{p(\lVert f\rVert^2)}=\lVert f\rVert.

In the opposite direction, every probability measure {\mu} on {X} defines a state on {C(X)}, given by (6). So, theorem 8 gives a one-to-one correspondence between states on {C(X)} and regular probability measures on {X}.

There is a more general form of the Riesz-Markov theorem which applies to locally compact spaces. This involves linear functions on the space {C_0(X)} of continuous functions {f\colon X\rightarrow{\mathbb C}} which vanish at infinity. To be precise, this means that for each {\epsilon > 0}, there exists a compact {K\subseteq X} such that {\lvert f\rvert < \epsilon} on {X\setminus K}. If {X} is not itself compact, then the constant function {1} is not in {C_0(X)}. This results in {C_0(X)} being a non-unitial *-algebra.

Recall that, so far, we are requiring algebras to contain a multiplicative identity, or unit. When we do not assume that an identity exists, then I will refer to the algebra as non-unitial. Note that this does not mean that the algebra does not contain a multiplicative identity, it just means that we do not require it. So, a non-unitial algebra may or may not be unitial.

As non-unitial algebras need not contain a multiplicative identity, definition 2 cannot be used, as the equality {p(1)=1} is undefined. So, we need to extend the definition of a state to include non-unitial *-algebras. For the algebra {C_0(X)}, we have the norm {\lVert f\rVert} equal to the supremum of {\lvert f(x)\rvert} over {x\in X} which, in fact, gives {C_0(X)} the structure of a C*-algebra. In this case, the operator norm can be used, which I will denote by {\lVert p\rVert_{\rm op}}, and is the smallest nonnegative real number satisfying

\displaystyle  \lvert p(f)\rvert\le\lVert p\rVert_{\rm op}\lVert f\rVert

for all {f\in C_0(X)}, and is infinite if there is no such real number. States on C*-algebras can be defined using the condition {\lVert p\rVert_{\rm op}=1} in place of {p(1)=1}. However, definition (2) for the unitial case did not require any additional structure on the *-algebra, such as a C*-norm. We would prefer to not require any such structure when considering the non-unitial case either.

Suppose that we have a non-unitial *-algebra {\mathcal A}. This can always be embedded in the unitial *-algebra {{\mathbb C}\oplus\mathcal A}, elements of which can be uniquely expressed as {\lambda+a} for {\lambda\in{\mathbb C}} and {a\in\mathcal A}. Multiplication and involution in {{\mathbb C}\oplus\mathcal A} are defined as,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle(\lambda+a)(\mu+b)=\lambda\mu+(\lambda b+\mu a+ab),\smallskip\\ &\displaystyle(\lambda+a)^*=\bar\lambda+a^*. \end{array}

This makes {{\mathbb C}\oplus\mathcal A} into a unitial *-algebra, with unit {1+0}. In the case where {\mathcal A=C_0(X)} for a locally compact space {X}, it can be seen that {{\mathbb C}\oplus\mathcal A} is isomorphic to {C(X^*)}, where {X^*} is the compactification of {X} formed by adding a `point at infinity’. Next, consider a linear map {p\colon\mathcal A\rightarrow{\mathbb C}} which is self-adjoint and positive, so that {p(a^*)=\overline{p(a)}} and {p(a^*a)\ge0} for all {a\in\mathcal A}. For each real {K}, there is a unique extension of {p} to a self-adjoint linear map {p^\prime\colon{\mathbb C}\oplus\mathcal A\rightarrow{\mathbb C}} satisfying {p^\prime(1+0)=K}, given by,

\displaystyle  p^\prime(\lambda+a)=\lambda K+ p(a).

We would like {p^\prime} to be a state on {{\mathbb C}\oplus\mathcal A}, which requires it to be positive and also that {K=1}. We ask, for what values of {K} is this positive? To answer this, we compute

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle p^\prime\left((\lambda+ a)^*(\lambda+ a)\right)&\displaystyle=p^\prime\left(\bar\lambda\lambda+(\bar\lambda a+\lambda a^*+a^*a)\right)\smallskip\\ &\displaystyle=\bar\lambda\lambda K+2\Re[\bar\lambda p(a)]+p(a^*a). \end{array}

This is positive for all {\lambda\in{\mathbb C}} if and only if

\displaystyle  Kt^2+2\lvert p(a)\rvert t+p(a^*a)\ge0

for all {t\in{\mathbb R}}. By the standard result for quadratics, this is equivalent to

\displaystyle  \lvert p(a)\rvert^2\le Kp(a^*a).

This suggests defining a norm of the positive linear map {p} by

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle\lVert p\rVert &\displaystyle= \inf\left\{K\in{\mathbb R}^+\colon \lvert p(a)\rvert^2\le Kp(a^*a){\rm\ for\ all\ }a\in\mathcal A\right\}\smallskip\\ &\displaystyle=\sup\left\{\lvert p(a)\rvert^2\colon a\in\mathcal A, p(a^*a)\le1\right\}. \end{array} (7)

So, {p} can be extended to a state on {{\mathbb C}\oplus\mathcal A} if and only if {\lVert p\rVert\le1}. We would want equality to hold since, otherwise, the state on {{\mathbb C}\oplus\mathcal A} assigns nonzero probability to the `point at infinity’. We arrive at the following definition of states on non-unitial *-algebras.

Definition 9 Let {\mathcal A} be a (non-unitial) *-algebra. Then, a linear map {p\colon\mathcal A\rightarrow{\mathbb C}} is

  1. real, or self-adjoint, if {p(a^*)=\overline{p(a)}} for all {a\in\mathcal A}.
  2. positive if it is self-adjoint and {p(a^*a)\ge0} for all {a\in\mathcal A}.
  3. a state if it is positive and {\lVert p\rVert=1}.

Note that, here, positive maps are required to also be self-adjoint. In the unitial case, (2) was used to express {a\in\mathcal A_{\rm sa}} as a combination of squares and imply that {p} is self-adjoint, so it was not required as part of the definition. However, that made use of the unit, so does not apply in the non-unitial case.

We had better check that this definition of a state coincides with definition 2 when it applies. Suppose that the algebra {\mathcal A} does, in fact, contain a unit. The Cauchy–Schwarz inequality (3) can be applied,

\displaystyle  \lvert p(a)\rvert^2=\lvert p(1^*a)\rvert^2\le p(1^*1)p(a^*a)=p(1)p(a^*a),

with equality holding at {a=1}. So, {\lVert p\rVert=p(1)}. Hence, definition 9 coincides with 2 for the unitial case. We now state the extension of the Riesz–Markov representation theorem to locally compact spaces.

Theorem 10 Let {X} be a locally compact space and {p\colon C_0(X)\rightarrow{\mathbb C}} be a state. Then, there is a unique regular probability measure {\mu} on {X} such that

\displaystyle  p(f)=\int f(x)d\mu(x)

for all {f\in C_0(X)}.

Since statements of the theorem, such as the one given at Wikipedia, usually make use of the operator norm of {p} rather than the norm defined by (7) above, we check that they agree.

Lemma 11 Let {X} be a locally compact space and {p\colon C_0(X)\rightarrow{\mathbb C}} be positive. Then, {\lVert p\rVert=\lVert p\rVert_{\rm op}}.

Proof: Consider {f\in C_0(X)} satisfying {0\le f\le 1}. Then {f^2\le f} giving,

\displaystyle  p(f)^2\le \lVert p\rVert p(f^2)\le \lVert p\rVert p(f).

So, {p(f)\le\lVert p\rVert}. Next, suppose that {\lvert f\rvert\le 1}. Then, by what we have just shown, {p(f^*f)\le\lVert p\rVert} so,

\displaystyle  \lvert p(f)\rvert^2\le\lVert p\rVert p(f^*f)\le\lVert p\rVert^2.

Taking the supremum over all such {f} shows that {\lVert p\rVert_{\rm op}\le\lVert p\rVert}. To prove the reverse inequality, consider any {g\in C_0(X)} with {\lvert g\rvert\le1} and apply the Cauchy–Schwarz inequality (3),

\displaystyle  \lvert p(g^*f)\rvert^2\le p(g^*g)p(f^*f)\le \lVert p\rVert_{\rm op}p(f^*f).

Now, for each {n\ge1}, by the definition of {C_0(X)}, there exists a compact {K\subseteq X} such that {\lvert f\rvert\le1/n} outside of {K}. By standard properties of locally compact spaces, there exists continuous {g_n\colon X\rightarrow[0,1]} with compact support such that {g_n=1} on {K}. Then, {\lVert g_nf-f\rVert\le1/n}. Assuming that {\lVert p\rVert_{\rm op}} is finite, so that {p} is norm-continuous,

\displaystyle  \lvert p(f)\rvert^2=\lim_{n\rightarrow\infty}\lvert p(g_n^*f)\rvert^2\le\lVert p\rVert_{\rm op}p(f^*f),

giving {\lVert p\rVert\le\lVert p\rVert_{\rm op}} as required. ⬜

The Commutative Gelfand–Naimark theorem

The commutative Gelfand–Naimark theorem shows that general commutative C*-algebras can be represented as spaces of continuous functions on a locally compact space. This enables the Riesz–Markov theorem described above to be applied, so that states on such algebras are in one-to-one correspondence with regular probability measures.

For any (non-unitial) *-algebra {\mathcal A}, we define a character to be a *-homomorphism {\chi\colon\mathcal A\rightarrow{\mathbb C}} which does not vanish everywhere. The collection of all characters is denoted {\widehat{\mathcal A}}, and can be given the weak topology, which is the weakest topology making the maps {\chi\mapsto\chi(a)} continuous, for each {a\in\mathcal A}. It can be seen that this makes {\widehat{\mathcal A}} into a locally compact space and, in the case that {\mathcal A} has a unit, a compact space.

In the case where {\mathcal A=C_0(X)} for a locally compact space, then we have a natural map {X\rightarrow\widehat{\mathcal A}} given by {x\mapsto\chi_x}, where {\chi_x(f)=f(x)}. This can be seen to be a homeomorphism, so the space of characters allows us to reconstruct the space {X}, up to homeomorphism, from the abstract representation of {C_0(X)} as a *-algebra. In the opposite direction, for any (non-unitial) *-algebra {\mathcal A}, we have a natural map {\mathcal A\rightarrow C_0(\widehat{\mathcal A})} given by {a\mapsto f_a}, where {f_a(\chi)=\chi(a)}. The Gelfand–Naimark theorem states that this is an isomorphism precisely when {\mathcal A} is a (non-unitial) commutative C*-algebra and, hence, gives an equivalence between locally compact spaces and commutative C*-algebras.

For a (non-unitial) *-algebra to be a C*-algebra, we require that it has a norm {\lVert\cdot\rVert} satisfying the usual norm inequalities,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\lVert\lambda a\rVert=\lvert\lambda\rvert\lVert a\rVert,\smallskip\\ &\displaystyle\lVert a+b\rVert\le\lVert a\rVert+\lVert b\rVert,\smallskip\\ &\displaystyle\Vert ab\rVert\le\lVert a\rVert\lVert b\rVert, \end{array}

for all {\lambda\in{\mathbb C}} and {a\in\mathcal A}, together with the C*-norm identity,

\displaystyle  \lVert a^*a\rVert=\lVert a\rVert^2.

In order to be a true norm, rather than just a semi-norm, we require {\lVert a\rVert > 0} for all {a\not=0} and, to be a C*-algebra, {\mathcal A} should be complete with respect to this norm. Combining the Gelfand–Naimark theorem with theorem 10 gives an equivalence between states on a C*-algebra and regular probability measures on its character space.

Theorem 12 Let {\mathcal A} be a (non-unitial) commutative C*-algebra and {p\colon\mathcal A\rightarrow{\mathbb C}} be a state. Then, there exists a unique regular probability measure {\mu} on {\widehat{\mathcal A}} such that

\displaystyle  p(a)=\int\chi(a)d\mu(\chi)

for all {a\in\mathcal A}.

This theorem can be used even in the case where {\mathcal A} is not a C*-algebra, by first constructing a C* completion. Let us suppose that {\mathcal A} is a (non-unitial) *-algebra and that {p\colon\mathcal A\rightarrow{\mathbb C}} is a state. As previously noted, this defines a semi-inner product {\langle x,y\rangle=p(x^*y)} on {\mathcal A}, which can only fail to be a true inner product as it need not be positive definite. Also, every {a\in\mathcal A} defines a linear operator on {\mathcal A} given by left-multiplication, {x\mapsto ax}. We define {\lVert a\rVert_p} to be the operator norm,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle\lVert a\rVert_p &\displaystyle= \inf\left\{K\in{\mathbb R}^+\colon p(x^*a^*ax)\le K^2p(x^*x){\rm\ for\ all\ }x\in\mathcal A\right\}\smallskip\\ &\displaystyle=\sup\left\{p(x^*a^*ax)^{1/2}\colon x\in\mathcal A, p(x^*x)\le1\right\}. \end{array}

This need not be finite but, if it is, then we will say that {a} is bounded. Repeated applications of the Cauchy–Schwarz inequality can be used to show that {p((a^*a)^n)^{\frac1{2n}}} is increasing in {n} and, if the algebra is commutative,

\displaystyle  \lVert a\rVert_p=\lim_{n\rightarrow\infty}p((a^*a)^n)^{\frac1{2n}}.

If every {a\in\mathcal A} is bounded, then {\lVert\cdot\rVert_p} will satisfy all of the requirements of a C*-norm other than, possibly, positive definiteness and completeness. Furthermore, the state satisfies

\displaystyle  \lvert p(a)\rvert\le\lVert a\rVert_p,

so is continuous with respect to this norm. We can take the completion {\mathcal A\rightarrow\overline{\mathcal A}} with respect to this norm, which will be a C*-algebra, and {p} has a unique continuous extension to {\overline{\mathcal A}}.

We use {\mathcal A_p} to denote the space of continuous characters {\chi\colon\mathcal A\rightarrow{\mathbb C}}. By definition, these satisfy {\lvert\chi(a)\rvert\le K\lVert a\rVert_p} for some {K\in{\mathbb R}^+}. In fact, we can take {K=1}. To see this, use the homomorphism property {\chi(a^n)=\chi(a)^n} to obtain

\displaystyle  \lvert\chi(a)\rvert=\lvert\chi(a^n)\rvert^{\frac1n}\le\left(K\lVert a^n\rVert_p\right)^{\frac1n}\le K^{\frac1n}\lVert a\rVert_p.

Taking the limit as {n} goes to infinity shows that a character is continuous if and only if

\displaystyle  \lvert\chi(a)\rvert\le\lVert a\rVert_p

for all {a\in\mathcal A}. Every {\chi\in\widehat{\mathcal A}_p} has a unique continuous extension to {\overline{\mathcal A}} and, conversely, characters on a C*-algebra can be shown to be continuous, so every character on {\overline{\mathcal A}} restricts to a continuous character on {\mathcal A}. This identifies {\widehat{\mathcal A}_p} with the space of characters on {\overline{\mathcal A}}.

With this notation, theorem 12 can be applied to the C*-algebra {\overline{\mathcal A}}. Subject to boundedness conditions, this gives a natural representation of a state on a commutative *-algebra as the expectation under a regular probability measure on an algebra of continuous functions on a locally compact space.

Theorem 13 Let {p} be a state on (non-unitial) commutative *-algebra {\mathcal A} such that each {a\in\mathcal A} is bounded. Then, there exists a unique regular probability measure {\mu} on {\widehat{\mathcal A}_p} satisfying

\displaystyle  p(a)=\int\chi(a)d\mu(\chi)

for all {a\in\mathcal A}.

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