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 . 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 over a field , we mean that is a -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 is an algebra over together with an involution, which is a unary operator , , satisfying,
- Anti-linearity: .
for all and .
Here, I am using to denote the complex conjugate of . Examples of *-algebras include:
- is a *-algebra, with involution given by complex conjugation.
- The algebra of complex polynomials over indeterminates is a commutative *-algebra. Involution, is given by taking complex conjugates of the polynomial coefficients.
- For a topological space , the collection of continuous functions is a commutative *-algebra, where addition and multiplication are defined pointwise, and involution is pointwise complex conjugation, .
- For a (complex) Hilbert space , the space of bounded linear operators is a *-algebra. Involution is defined as the operator adjoint, so for all .
A homomorphism of *-algebras is a map from to preserving the algebra operations,
for all and . 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 is not required. For homomorphisms of non-unitial *-algebras, we simply drop the requirement that .
An element of a *-algebra is called self-adjoint or real, if . Using to denote the set of self-adjoint elements of , anti-linearity implies that is closed under taking real linear combinations, so is a vector subspace of over the real numbers. The product of two self-adjoint elements need not be self adjoint. Considering , the identity shows that the product is in if and only if and commute. Finding self-adjoint elements of an algebra is easy enough. For an element then and are both self-adjoint,
Also, the identity is automatically self-adjoint,
Every can be uniquely decomposed into its real and imaginary parts,
for self-adjoint , which are given by
Commutative *-algebras are equivalent to commutative real algebras (precisely, they are equivalent categories). For any commutative *-algebra , the subset of self-adjoint elements is a commutative real algebra. In the opposite direction, if is a commutative real algebra, then we can consider its complexification . Every element of can be written uniquely as for , which is written more simply as . Multiplication and involution are given by
which makes into a commutative *-algebra. The self-adjoint elements of are precisely for . Hence, gives an isomorphism of real algebras from to . 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 is isomorphic to the complexification of the real algebra .
States on *-algebras are defined as follows.
Definition 2 Let be a *-algebra. Then, a linear map is
- real, or self-adjoint, if for all .
- positive if for all .
- a state if it is positive and .
It can be seen that is self-adjoint if and only if is real for all . If is positive then it must also be self-adjoint. Noting that, for , the value is a nonnegative real number so, writing as a combination of squares,
shows that 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 on a commutative *-algebra , its restriction, , to is a state. Conversely, any state extends uniquely to a state on . By writing in terms if its real and imaginary parts (1) then, by linearity, we have . The linear map satisfies positivity,
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 be a state on *-algebra , and be self-adjoint. Then, there exists a probability measure on such that
for all polynomials .
We do not need to assume that is commutative here, as it only depends only the restriction of to the subalgebra generated by , which is always commutative. We also translate theorem 5 of the previous post to the language of *-algebras.
Theorem 4 Let be a state on *-algebra and be commuting self-adjoint elements of satisfying Carleman’s condition
Then, there exists a unique probability measure on such that
for all polynomials .
Again, only the restriction of to the subalgebra generated by 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 is positive, then defines a semi-inner product on . Hence, the Cauchy–Schwarz inequality applies,
An alternative method by which states on *-algebras lead to classical probability measures is via groups of unitary elements. An element of a *-algebra is called unitary if it satisfies
States on a *-algebra automatically imply probability distributions on the unit circle associated to each unitary element.
Theorem 5 Let be a state on *-algebra , and be unitary. Then, there is a unique probability measure on satisfying
for all .
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 and states defined on certain spaces of unitary algebra elements.
Theorem 6 Let be a state on *-algebra , and be unitary elements of satisfying . If is continuous, then there is a unique probability measure on satisfying
for all .
Theorems 5 and 6 are both instances of a much more general form of Bochner’s theorem. For a locally compact abelian group , we denote its dual group by . This is the set of continuous group homomorphisms , and is itself a locally compact abelian group under the multiplication rule and topology of uniform convergence on compact sets. The elements can also be considered as maps under the action . This identifies elements of with elements of the dual of which, by Pontryagin duality, is an isomorphism. That is, the natural map from to its double-dual is an isomorphism. So, really, we have a pair of locally compact abelian groups and a pairing , , with respect to which is the dual of and is the dual of . Important examples of dual pairs of locally compact abelian groups are:
- and are dual groups under the pairing .
- is self-dual under the pairing .
- For any , the quotient under addition is self-dual under the pairing .
- if and are pairs of dual groups then so is .
We now give the general statement of Bochner’s theorem.
Theorem 7 Let be a state on *-algebra and be a locally compact abelian group. If are unitary elements of satisfying , and is continuous, then there is a unique regular probability measure on satisfying
for all .
Regularity of the probability measure simply means that, for any Borel , is the supremum of over compact . 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 , which I will express multiplicatively, so the group composition of two elements will be written as . We construct the group algebra . Elements of can be defined as formal sums
where is zero outside of a finite subset of , so (5) can be thought of as a finite sum. Multiplication and involution are given by
This makes into a *-algebra, with being unitary elements and being the identity, and is commutative if is commutative (i.e., abelian). If is a locally compact abelian group, then elements of the dual can be extended to operate on all of by,
Any probability measure on defines a state on as,
so that (4) holds. Noting that , we see that is a state. Hence, Bochner’s theorem provides a one-to-one mapping between states on such that is continuous and regular probability measures on .
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 , we use to denote the space of continuous functions , with the algebra operations of addition and multiplication defined pointwise, and involution being pointwise complex conjugation. This expresses as a *-algebra. We start with compact spaces which, by convention, I take to be Hausdorff. For a compact space , the theorem can be stated as follows.
Theorem 8 Let be a compact space and be a state on . Then, there is a unique regular probability measure on satisfying
for all .
A few comments are in order. In most statements of the theorem, nonnegativity of of the state is required in the sense that whenever . However, this is implied by definition 2 for states on a *-algebra, since . Sometimes, continuity of is also required but, again, this is implied by our definition of a state. By Cauchy–Schwarz (3), and using the fact that ,
There is a more general form of the Riesz-Markov theorem which applies to locally compact spaces. This involves linear functions on the space of continuous functions which vanish at infinity. To be precise, this means that for each , there exists a compact such that on . If is not itself compact, then the constant function is not in . This results in 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 is undefined. So, we need to extend the definition of a state to include non-unitial *-algebras. For the algebra , we have the norm equal to the supremum of over which, in fact, gives the structure of a C*-algebra. In this case, the operator norm can be used, which I will denote by , and is the smallest nonnegative real number satisfying
for all , and is infinite if there is no such real number. States on C*-algebras can be defined using the condition in place of . 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 . This can always be embedded in the unitial *-algebra , elements of which can be uniquely expressed as for and . Multiplication and involution in are defined as,
This makes into a unitial *-algebra, with unit . In the case where for a locally compact space , it can be seen that is isomorphic to , where is the compactification of formed by adding a `point at infinity’. Next, consider a linear map which is self-adjoint and positive, so that and for all . For each real , there is a unique extension of to a self-adjoint linear map satisfying , given by,
We would like to be a state on , which requires it to be positive and also that . We ask, for what values of is this positive? To answer this, we compute
This is positive for all if and only if
for all . By the standard result for quadratics, this is equivalent to
This suggests defining a norm of the positive linear map by
So, can be extended to a state on if and only if . We would want equality to hold since, otherwise, the state on assigns nonzero probability to the `point at infinity’. We arrive at the following definition of states on non-unitial *-algebras.
Definition 9 Let be a (non-unitial) *-algebra. Then, a linear map is
- real, or self-adjoint, if for all .
- positive if it is self-adjoint and for all .
- a state if it is positive and .
Note that, here, positive maps are required to also be self-adjoint. In the unitial case, (2) was used to express as a combination of squares and imply that 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 does, in fact, contain a unit. The Cauchy–Schwarz inequality (3) can be applied,
Theorem 10 Let be a locally compact space and be a state. Then, there is a unique regular probability measure on such that
for all .
Lemma 11 Let be a locally compact space and be positive. Then, .
Proof: Consider satisfying . Then giving,
So, . Next, suppose that . Then, by what we have just shown, so,
Taking the supremum over all such shows that . To prove the reverse inequality, consider any with and apply the Cauchy–Schwarz inequality (3),
Now, for each , by the definition of , there exists a compact such that outside of . By standard properties of locally compact spaces, there exists continuous with compact support such that on . Then, . Assuming that is finite, so that is norm-continuous,
giving 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 , we define a character to be a *-homomorphism which does not vanish everywhere. The collection of all characters is denoted , and can be given the weak topology, which is the weakest topology making the maps continuous, for each . It can be seen that this makes into a locally compact space and, in the case that has a unit, a compact space.
In the case where for a locally compact space, then we have a natural map given by , where . This can be seen to be a homeomorphism, so the space of characters allows us to reconstruct the space , up to homeomorphism, from the abstract representation of as a *-algebra. In the opposite direction, for any (non-unitial) *-algebra , we have a natural map given by , where . The Gelfand–Naimark theorem states that this is an isomorphism precisely when 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 satisfying the usual norm inequalities,
for all and , together with the C*-norm identity,
In order to be a true norm, rather than just a semi-norm, we require for all and, to be a C*-algebra, 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 be a (non-unitial) commutative C*-algebra and be a state. Then, there exists a unique regular probability measure on such that
for all .
This theorem can be used even in the case where is not a C*-algebra, by first constructing a C* completion. Let us suppose that is a (non-unitial) *-algebra and that is a state. As previously noted, this defines a semi-inner product on , which can only fail to be a true inner product as it need not be positive definite. Also, every defines a linear operator on given by left-multiplication, . We define to be the operator norm,
This need not be finite but, if it is, then we will say that is bounded. Repeated applications of the Cauchy–Schwarz inequality can be used to show that is increasing in and, if the algebra is commutative,
If every is bounded, then will satisfy all of the requirements of a C*-norm other than, possibly, positive definiteness and completeness. Furthermore, the state satisfies
so is continuous with respect to this norm. We can take the completion with respect to this norm, which will be a C*-algebra, and has a unique continuous extension to .
We use to denote the space of continuous characters . By definition, these satisfy for some . In fact, we can take . To see this, use the homomorphism property to obtain
Taking the limit as goes to infinity shows that a character is continuous if and only if
for all . Every has a unique continuous extension to and, conversely, characters on a C*-algebra can be shown to be continuous, so every character on restricts to a continuous character on . This identifies with the space of characters on .
With this notation, theorem 12 can be applied to the C*-algebra . 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 be a state on (non-unitial) commutative *-algebra such that each is bounded. Then, there exists a unique regular probability measure on satisfying
for all .