After the previous posts motivating the idea of studying probability spaces by looking at states on algebras, I will now make a start on the theory. The idea is that an abstract algebra can represent the collection of bounded, and complex-valued, random variables, with a state on this algebra taking the place of the probability measure. By allowing the algebra to be noncommutative, we also incorporate quantum probability.
I will take very small first steps in this post, considering only the basic definition of a *-algebra and positive maps. To effectively emulate classical probability theory in this context will involve additional technical requirements. However, that is not the aim here. We take a bare-bones approach, to get a feeling for the underlying constructs, and start with the definition of a *-algebra. I use to denote the complex conjugate of a complex number .
Definition 1 An algebra over field is a -vector space together with a binary product satisfying
for all and .
A *-algebra is an algebra over with a unary involution, satisfying
for all and .
An algebra is called unitial if there exists such that
for all . Then, is called the unit or identity of .
In contrast to my previous posts, I am not considering a *-algebra to contain a unit by default, and will refer to it as `unitial’ whenever the existence of a unit is required. An is called self-adjoint if and only if . It can be seen that the self-adjoint elements form a real-linear subspace of the algebra, which we denote by . For any , then and are both self-adjoint. Furthermore, every can be uniquely decomposed as
for . Using , this is easily solved to obtain and . If the algebra is unitial, then the identity must be self-adjoint, as can be seen from
A sub-*-algebra of is a subset which is closed under the algebra operations. That is , , and are all in , for any and . Any sub-*algebra is itself a *-algebra under these operations. An algebra is said to be commutative if the identity is satisfied.
Example 1 Let be a set and be the collection of functions . This is a commutative *-algebra under the pointwise operations
for and . The self-adjoint elements are the real-valued functions on .
Commutative *-algebras can often be represented as sub-*-algebras of the collection of complex valued functions from some set , although this does impose additional requirements. For example, any such algebra also satisfies whenever . In example 1, it can be seen that the positive elements of are precisely those that can be expressed in the form .
Noncommutative *-algebras arise as collections of linear operators on an inner product space.
Example 2 If is a vector space over a field , then the space of linear maps is a -algebra, where the algebra operations are defined by combining the linear maps in the usual way,
for all and . If is a complex vector space with inner product , let be the space of linear maps such that there exists an adjoint satisfying
for all . Then is a *-algebra. In particular, if is a Hilbert space, then the collection of all bounded linear maps is a *-algebra, with involution given by the operator adjoint.
In this example, if satisfies then
so that , as in example 1. Generally, we expect the property that implies for the cases that we will be interested in, although I will not impose this as a condition. Any element of of the form satisfies
so represents a positive linear operator.
Next, we define positive linear maps on a *-agebra.
Definition 2 Let be a *-algebra. Then, a linear map is,
- self-adjoint or real if for all .
- positive if it is self-adjoint and for all .
Example 3 Let be a finite measure space, and be the bounded measurable maps . Then, integration w.r.t. defines a positive linear map on ,
Example 4 Let be an inner product space, and be a sub-*-algebra of the space of linear maps as in example 2. Then, any defines a positive linear map on ,
Given a *-algebra and a positive linear map , we can define a semi-inner product by,
for all . This is only a semi-inner product, as it need not be positive definite. That is, it is possible that for some nonzero . The associated semi-norm is
I will refer to this as the semi-norm on . If you prefer to work with a true inner product, rather than a semi-inner product, it is always possible to quotient out by the space of for which . As acts on itself by left-multiplication, taking considered as a complex vector space shows that the construction of *-algebras in example 2 is quite general.
A left-ideal of a *-algebra is a subset, which is a subspace as a complex vector space, and which is closed under left-multiplication by elements of the algebra.
Lemma 3 Let be a positive linear map on *-algebra . Let denote the elements such that or, equivalently, . This is a left-ideal of .
Using to denote the quotient vector space, with quotient map , then the semi-inner product uniquely defines an inner product on by
and acts on by left-multiplication, .
Proof: If and then, by the triangle inequality,
So , showing that is a vector subspace. Also, for , Cauchy–Schwarz gives
so that is in which, therefore, is a left-ideal. Next, if then
and, similarly, the value of is unchanged by replacing with where . So, 3 is well-defined. Furthermore, if then , so that is a true inner product space. Finally, if then, as is a left-ideal, , so we can define the action of on by . ⬜
Now, for , consider the linear map on given by left-multiplication, (or, you can look at the action on , if preferred). We denote its operator norm by , so that,
Again, this may only be a semi-norm, as it is possible that for nonzero , and, alternatively, can be infinite. I will refer to as the seminorm and, sometimes, will drop the subscript and write simply , where it is unlikely to cause confusion. I will say that is bounded if it acts as a bounded operator, so that is finite. We show that this is a C*-seminorm. Whenever I say that a *-algebra acts on a semi-inner product space, I am requiring that this is in the sense of example 2 so that, in particular, (1) holds.
Lemma 4 Let be a *-algebra acting on semi-inner product space . Then, the operator norm on is an algebra semi-norm,
for all and . Furthermore, the C*-inequality,
holds for all , and .
Proof: The algebra seminorm properties are standard for any collection of operators on a normed space. For the C*-inequality, consider and . Cauchy–Schwarz gives
and the (5) follows. Next, cancelling from both sides of the following
gives . Using in place of gives the reverse inequality. ⬜
Once it is known that a semi-norm on a *-algebra satisfies the C*-inequality, then we get various identities for free. An element of a *-algebra is called normal if it commutes with its adjoint, . This includes, for example, all self-adjoint elements and all unitary elements which, by definition, satisfy .
Lemma 5 If is a finite seminorm on *-algebra satisfying the C*-inequality (5) then,
for all . Furthermore, for normal ,
for all and, more generally,
for all .
Proof: The C*-inequality gives
Cancelling gives . Replacing by gives the reverse inequality, so . Then,
giving the second equality. We now prove (7), starting with the case where is self-adjoint and is a power of 2. By what we have just shown,
Hence, by induction in ,
for all nonnegative integers . This proves (7) when is a power of 2, and we need to extend to the case where it is any positive integer. In that case, choose such that . Then,
Assuming that is nonzero, cancelling ,
gives the required equality. The case where is easily handled, since .
Now, if is normal then,
as required. ⬜
In particular, applying this to the action of on itself:
Proof: Let be the subalgebra of bounded elements of . By lemma 4, is finite for all , so is a *-subalgebra. As, again by lemma 4, the C*-inequality (5) holds, lemma 5 shows that the C*-identities (6) hold for bounded and (7,8) hold for bounded normal . It only remains to show that (6) holds for unbounded , but this is immediate from the C*-inequality (5) . ⬜