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 1Analgebraover field is a -vector space together with a binary product satisfyingfor all and .

A *-algebra is an algebra over with a unary involution, satisfying

for all and .

An algebra is called

unitialif 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 1Let 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 2If 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

(1)

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 2Let be a *-algebra. Then, a linear map is,

self-adjointorrealif for all .positiveif it is self-adjoint and for all .

Example 3Let be a finite measure space, and be the bounded measurable maps . Then, integration w.r.t. defines a positive linear map on ,

Example 4Let 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,

(2) |

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 3Let 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

(3)

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,

(4) |

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 4Let 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,

(5)

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 5If is a finite seminorm on *-algebra satisfying the C*-inequality (5) then,

(6) for all . Furthermore, for normal ,

(7) for all and, more generally,

(8)

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,

Finally,

as required. ⬜

In particular, applying this to the action of on itself:

Corollary 6If is a positive linear map on *-algebra , then the semi-norm is an algebra semi-norm satisfying the C*-identities (6), and satisfying (7,8) for bounded normal .

*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) . ⬜