We previously defined noncommutative probability spaces as a *-algebra together with a nondegenerate state satisfying a completeness property. Justification for the stated definition was twofold. First, an argument similar to the construction of *measurable* random variables on classical probability spaces was used, by taking all possible limits for which an expectation can reasonably be defined. Second, I stated various natural mathematical properties of this construction, including the existence of completions and their functorial property, which allows us to pass from *pre*probability spaces, and homomorphisms between these, to the NC probability spaces which they generate. However, the statements were given without proof, so the purpose of the current post is to establish these results. Specifically, I will give proofs of each of the theorems stated in the post on noncommutative probability spaces, with the exception of the two theorems relating commutative *-probability spaces to their classical counterpart (theorems 2 and 10), which will be looked at in a later post.

Recall that a state on a *-algebra was defined as a positive linear map satisfying a normalisation condition. This can be expressed as for unitial algebras or, more generally, as using the following norm,

(1) |

We actually described three kinds of spaces. First, *-probability spaces, or *pre*probability spaces, which are nothing more than a pair consisting of a state on *-algebra . Next, W*-probability spaces are intended as a generalisation of classical probability spaces, and require the algebra to satisfy a weak completeness property. Finally, C*-algebras lie somewhere between the two definitions, requiring the weaker norm-completeness property. The idea is that if we start with a *-algebra, then this can be completed to give either a C* or a W*-probability space in an essentially unique way.

The method used to construct NC probability spaces here will be as algebras of operators on a Hilbert space together with a pure state. The *-algebra of bounded linear operators on a Hilbert space will be denoted by , and a *-subalgebra of will be referred to as a *-algebra on . An element will be called *cyclic* for the *-algebra if is a dense subspace of . The operator norm of will be denoted as , and the and seminorms with respect to a state are denoted by and respectively.

Lemma 1Let be a *-algebra on Hilbert space and have norm 1 and be cyclic for . Then,

(2)

is a nondegenerate state. Furthermore, the norm on coincides with the operator norm and the norm is given by .

*Proof:* Linearity of is immediate. For ,

This shows that is positive and that the -norm is

Also, if then,

shows that . Next, as is cyclic, there exists such that . In particular, so, dividing by if necessary, we suppose without loss of generality that . Then,

shows that , so that is a state.

Now, for ,

shows that . For the opposite inequality, for any , the fact that is cyclic means that there exists with . Then,

and, hence, as required.

Finally if satisfies then and, hence, , showing that is nondegenerate. ⬜

#### C*-completions

Each of the results in the post on noncommutative probability spaces was stated firstly for W*-probability spaces and, then, a C* version was also given. Here, I do the same, and give separate proofs for the results corresponding to W* and to C* spaces. I start by looking at C*-probability spaces, as this is simpler. The proofs of the W* versions of the results will be provided after that and will follow along very similar lines, but with some additional complications.

C*-probability spaces can be constructed by taking a pure state on the operator norm closure of a *-algebra on a Hilbert space.

Lemma 2Let be a *-algebra on Hilbert space . Then, its operator norm closure is a *-algebra. Furthermore, if has norm 1 and is cyclic for , then is a C*-probability space, where

(3)

*Proof:* That is a *-algebra follows from the fact that the *-algebra operations are norm-continuous. Explicitly, elements can be expressed as norm-limits of sequences . Then, and are, respectively, limits of the sequences and , so are in . Similarly, for , is the limit of , so is in which, therefore, is a *-algebra.

Lemma 1 says that is a bounded nondegenerate *-probability space, so it only remains to show that is -complete. By construction, it is a closed subspace of the complete space , so is complete under the operator norm. By lemma 1, this coincides with the norm. ⬜

Recall that the C*-completion of a bounded *-probability space consists of a C*-probability space together with a (state preserving) homomorphism with -dense image. C*-completions can be constructed with the aid of the GNS representation which, by definition, is a Hilbert space together with a *-homomorphism and an element which is cyclic for and satisfies .

Lemma 3Suppose that is a bounded *-probability space with GNS representation . Let be the operator norm closure of and define the state by (3).

Then, is a C*-probability space and defines a homomorphism from to which, furthermore, is a C*-completion.

*Proof:* Lemma (2) says that is a C*-probability space. Furthermore is, by definition, a *-homomorphism from to . To show that it also defines a homomorphism of *-probability spaces, we just need to show that it is state preserving. However, by the definition of the GNS representation,

as required. Finally, lemma 1 says that the norm coincides with the operator norm on with respect to which is dense, by construction. So, defines a C*-completion. ⬜

A homomorphism between *-probability spaces and gives an isomorphism on the underlying *-algebras if and only if it is one-to-one and onto. A sufficient condition to be one-to-one is that is nondegenerate and, for C* spaces, a sufficient condition to be onto is that is dense. So, a homomorphism between C*-probability spaces with dense image is necessarily an isomorphism.

Lemma 4A homomorphism of C*-probability spaces is an isomorphism if and only if is -dense in .

*Proof:* First, if is an isomorphism then is certainly dense. Conversely, suppose that is dense. The -seminorms on C*-algebras are complete norms and, by lemma 5 of the post on homomorphisms of *-probability spaces, is an isometry. Hence, is one-to-one and is a complete and dense subset of , so as required. ⬜

We now prove theorem 14 of the post on NC probability spaces, stating the functorial property of C*-completions. We first note, by lemma 4 of the post on homomorphisms of *-probability spaces, that C*-completions are always isometries.

Theorem 5Let be an -continuous homomorphism between bounded *-probability spaces and . Let

be C*-completions. Then, there is a unique homomorphism such that . Furthermore, is an isomorphism iff is -dense in .

*Proof:* Lemma 5 of the post on homomorphisms of *-probability spaces states that any homomorphism of *-probability spaces where the domain is a C*-probability space is automatically -isometric. In particular, by uniqueness of continuous extensions, is uniquely defined by its restriction to on which it is uniquely defined by . Hence, is unique.

We still need to show existence of satisfying the requirements of the theorem. As are all -isometries, for ,

(4) |

In particular, if then and, by nondegeneracy of , . So, we can define a *-homomorphism by

(5) |

As are state-preserving, the same is true of and, by (4), is an -isomtery. As is -complete, by continuous linear extensions uniquely extends to a continuous linear map . By -continuity of the *-algebra operations and of the state, it follows that is a *-homomorphism preserving the state as required.

Finally, and are -isometries with dense image, so that has dense image if and only if has dense image which, by lemma 4, is equivalent to being an isomorphism. ⬜

Theorem 13 of the NC probability post, stating the existence and uniqueness of C*-completions, follows easily.

Theorem 6Let be a bounded *-probability space. Then, it has a C*-completion, which is unique up to isomorphism. That is, for any two C*-completions

there exists a unique isomorphism such that .

*Proof:* Lemma 3 gives existence of the C*-completion, so only uniqueness remains. Let be as in the statement of the theorem, and be the identity automorphism on . Theorem 5 states the existence of a unique isomorphism from to satisfying

as required. ⬜

Theorem 15 of the NC probability post relating the C*-completion to the GNS construction also follows quickly.

Theorem 7Let be a bounded *-probability space, and be its GNS representation. Let be the operator norm closure of and define byThen,

- is a C*-probability space.
- is the C*-completion of .
- is a C*-probability space if and only if is an isomorphism between and .

*Proof:* The first two statements are already shown in lemma 3 above. Also, if is an isomorphism from to then is isomorphic to , so is a C*-probability space. It only remains to show that, if is a C*-probability space then is an isomorphism. However, is -dense in by definition of the C*-completion, so is an isomorphism by lemma 4. ⬜

It remains to establish theorems 11 and 12 of the NC probability post, which relate C*-probability spaces to states on C*-algebras. When we have a state specified on a C*-algebra, then there are two (semi)norms that can be applied. Firstly, there is the C*-algebra norm and, secondly, there is the seminorm defined by the state. It can be shown that, when the state is nondegenerate, they are the same and, more generally, the C*-algebra norm will be the stronger of the two. Using to denote the C*-algebra norm, this means that .

Lemma 8Let be a positive functional on C*-algebra . Then, the seminorm is weaker than the C*-norm and, if is nondegenerate, the and C*-norms coincide.

*Proof:* If then, for some self-adjoint (see lemma 5 of the *-algebra post). Multiplying on the right by and on the left by , and applying ,

for all , giving .

Now suppose that is nondegenerate, so that the -seminorm is positive definite. As we have shown that , it follows that is finite. Hence, we can let be its -completion, which is a C*-algebra. Then, the natural embedding is a one-to-one *-homomorphism, so is an isometry (see, eg, Blackadar II.2.2.9) under the respective C*-norms. That is, as required. ⬜

The previous result can be applied to prove theorem 11 of the NC probability post.

Theorem 9The pair is a C*-probability space if and only if is a C*-algebra and is a nondegenerate state.

*Proof:* First, suppose that is a C*-probability space. As the -seminorm satisfies the C*-properties and, by definition, is a complete norm, then is a C*-algebra. Conversely, suppose that is a C*-algebra and is a nondegenerate state. Lemma 8 says that the and C*-norms coincide, so is -complete. ⬜

Similarly, theorem 12 of the NC probability post can now be established, that even degenerate states on C*-algebras give rise to C*-probability spaces, so long as we quotient out by the *-ideal . The state on the quotient algebra is that given by the state under the quotient map .

Theorem 10Let be a C*-algebra and be a state. Then, is a C*-probability space.

*Proof:* As is, by definition, -closed and lemma 8 says that the C*-norm is stronger than , we see that is closed under the C*-norm. It is then standard (Blackadar II.5.1.1) that is a C*-algebra with respect to the norm

Furthermore, as is a nondegenerate state on (lemma 13 of the post on states), theorem 9 says that is a C*-probability space. ⬜

Finally, I show a basic result which is especially important for the definition of states used in these notes. For a *-algebra , a state is required to satisfy the positivity condition together with a normalisation criterion to ensure that the `total probability’ of the space is one. In the case where is unitial then the normalisation is simply . For C*-algebra with norm , the operator norm of the linear functional is defined by

(6) |

For nonunitial C*-algebras, the normalisation condition is commonly used in the definition of states and, for unitial C*-algebras, it is standard that positive linear functionals satisfy and, so, the normalisation condition given in terms of the operator norm is consistent with the condition for unitial algebras. In these notes, however, I did not want to assume that the algebra is unitial. Nor did I want to restrict up-front to C*-algebras. So, instead, I made use of of the norm (1) and applied the normalisation condition . For unitial C*-algebras, , and all three normalisation conditions are equivalent. To be sure that we are being consistent with the standard definition of states on nonunitial C*-algebras, we should show that the norms (1) and (6) coincide. This is a rather subtle point, and was not explicitly stated stated previously in these notes, so I give a proof here. The statement is known as Kadison’s inequality (Blackadar II.6.9.14).

Lemma 11Let be a C*-algebra and be a positive linear map. Then, .

*Proof:* We first show that . Choose with . Then, as , there exists a self-adjoint with . Using

gives , from which we obtain,

Hence, , giving

As this holds for all , we obtain .

We now show the reverse inequality, . Without loss of generality, suppose that is finite. I make use of the standard fact that every C*-algebra has an approximate identity (Blackadar II.4.1.3). That is is a net of self-adjoint elements with and for all . Applying Cauchy–Schwarz,

The second inequality here is using . Now, as and is bounded in the operator norm, we can take limits on the left hand side to obtain

and as required. ⬜

If and

#### W*-completions

Having established all of the required results for C*-probability spaces above, I now move on to W*-probability spaces. This very closely mirrors the C* case, and we will prove each of the W* versions on those results in the same order. However, there are additional complications here. Rather than just the norm, we now have to deal with the weak, strong, ultraweak and ultrastrong operator topologies, which can be used interchangeably in many situations. For this reason, we will require some additional helper lemmas, starting with the following.

A *-algebra on a Hilbert space will be said to act *nondegenerately* if for all implies that . Equivalently, the linear span of is dense in . We use to denote the closures of in the weak, strong, ultraweak and ultrastrong topologies respectively. The unit ball of is denoted by

where denotes the operator norm. The unit ball of the closure of below is denoted by .

Lemma 12Let be a *-algebra on Hilbert space . Then,

(7) Furthermore, if we let denote the closure of under the topologies above, then

- is a *-algebra.
- if acts nondegenerately on , then contains the identity operator, so is unitial.

*Proof:* Identities (7) are often stated as part of the bicommutant theorem (Blackadar I.9.1.2), although I will not make use of bicommutants here. Corollaries 16 and 20 of the post on normal maps say that and . As the ultraweak topology is stronger than the weak,

We make use of the Kaplansky density theorem, so that

The equality here is because the strong and ultrastrong topologies coincide on the unit ball of . By scaling, this gives , so (7) follows.

We now show that is a *-algebra. The fact that it is a linear subspace of closed under involution follows from the fact that the linear combination is jointly weakly continuous and the adjoint map is weakly continuous. It remains to show that is closed under products. Although the map is not jointly continuous, it is weakly continuous individually in each of and . By continuity in , for each fixed , for all . Then, by continuity in , for all .

Finally, suppose that acts nondegenerately. As is a C*-algebra, it has an approximate identity . This is a net with for all . Let be the set of such that . As is uniformly bounded, is a closed linear subspace of . Furthermore,

for all and , so . Hence, if acts nondegenerately then , so strongly showing that . ⬜

Recall that in the definition of a W*-probability space , we required the unit ball to be weakly complete. Actually, there are various equivalent requirements which could alternatively have been used, such as strongly complete, or weakly compact. We list these in the lemma below. For the third statement, I use for the Hilbert space completion, and for the continuous linear extension of the action of on , so that . This is a *-homomorphism, so that the image is a *-subalgebra of . In particular, when is a *-algebra on a Hilbert space, then we can take and , so that the third statement below is equivalent to being closed under any of the stated topologies. Where I list a sequence of words separated by a forwards slash `/’, this is intended to mean that the statement holds equivalently if either of these words are used. So, the lemma actually consists of ten equivalent statements. Recall that by a *-algebra representation, I am refering to a *-algebra acting on semi-inner product space in a manner consistent with the *-algebra operations so, in particular, includes the cases of a *-algebra on a Hilbert space and also bounded *-probability spaces considered as *-algebras acting on themselves by left multiplication.

Lemma 13Let be a bounded *-algebra representation. The following are equivalent.

- is weakly/ultraweakly compact.
- is weakly/strongly/ultraweakly/ultrastrongly complete.
- is weakly/strongly/ultraweakly/ultrastrongly closed in .

*Proof:* The first statement is equivalent under the weak and ultraweak topologies, since they coincide on the unit ball. Furthermore, as any compact subset of a vector space is automatically complete, this implies the second statement for the weak and ultraweak topologies.

Next, suppose that the second statement holds for the weak or, equivalently, the ultraweak topology. We show that it is strongly or, equivalently, ultrastrongly complete. Consider a strongly Cauchy net . As the weak topology is weaker than the strong, it is also weakly Cauchy and, by the assumption, has a weak limit . Then, for , weak convergence gives

and, so, strongly as required.

Now, supposing that is strongly complete, we prove the third statement. By (7) this is an equivalent statement under each of the mentioned topologies. Setting then,

The first equality here is using the Kaplansky density theorem, and the second uses the fact that is an isometry. So, to show that is strongly closed, it is sufficient to show that . Given , choose a net such that strongly. Then, and hence, strongly. So, by assumption, there exists a strong limit . Then, shows that as required.

Finally, assuming the third statement, we show that is ultraweakly compact. Theorem 21 of the post on normal maps expresses as the dual of a Banach space , with the weak-* topology corresponding to ultraweak convergence. Hence, is ultraweakly compact by the Banach–Alaoglu theorem. Then, as the ultraweak topology on given by its action on is the same as that given by the action on , this shows that is ultraweakly compact. ⬜

Lemma 1 above showed that the norm coincides with the operator norm for an algebra acting nondegenerately on a Hilbert space. We extend this to cover the remaining operator topologies.

Lemma 14Assume the hypotheses of lemma 1. Then,

- the ultraweak and ultrastrong topologies on defined with respect to the action on coincide with the respective topologies defined with respect to .
- the weak, strong, ultraweak and ultrastrong topologies on defined with respect to the action on coincide with the respective topologies defined with respect to .

*Proof:* The weak and ultraweak topologies coincide on , as do the strong and ultrastrong, so the second statement follows from the first. To prove the first statement, let be the semi-inner product space with set of elements equal to and inner product . By definition, the ultraweak and ultrastrong topologies defined by are equal to the respective topologies given by the left-multiplication action of on . By lemma 1, the linear map taking to is an isometry and, by assumption, has dense image in . So, is a Hilbert space completion. The result now follows from lemma 18 of the post on normal maps. ⬜

W*-probability spaces will be constructed by taking a pure state on the closure of a *-algebra on a Hilbert space, using a W* version of lemma 2.

Lemma 15Let be a *-algebra on Hilbert space and have norm 1 and be cyclic for . Then, is a W*-probability space, where is the weak closure of and is defined by (3).

*Proof:* Lemma 12 says that is a *-algebra, and lemma 1 says that is a nondegenerate state. As, by lemma 1, the norm coincides with the operator norm, the unit ball is the same whether defined with respect to or the operator norm. By lemma 13, is weakly complete under the weak topology on and, by lemma 14, the weak topology coincides with the one defined by . So, is a W*-probability space. ⬜

Whereas the C*-completion was defined via an dense homomorphism, for W*-completions we require the image of the unit ball to be a weakly dense subset of the unit ball in the codomain. However, there are several equivalent conditions which could equally be used. As above, the forwards slash `/’ means that either of the words are equivalent, so the lemma below actually consists of ten equivalent statements.

Lemma 16Let be a homomorphism of bounded *-probability spaces. The following are equivalent,

- is ultraweakly/ultrastrongly dense in .
- is weakly/strongly/ultraweakly/ultrastrongly dense in .
- is a weakly/strongly/ultraweakly/ultrastrongly dense subset of .

In case that the above equivalent conditions hold, then is normal.

*Proof:* Using , lemma 22 of the post on normal maps says that

(8) |

By scaling, a linear subspace is all of if and only if . Each of the the first two statements, using each of the stated topologies, are equivalent to the identical expressions in (8) being equal to , so these are all equivalent statements. Next, suppose that the first statement holds. Then, by lemma 10 of the post on normal maps, is normal. In particular, by lemma 2 of the same post, is an -isometry, so

(9) |

which, by the second statement, is a dense subset of under each of the operator topologies.

Finally, suppose that the third statement holds for any of the operator topologies mentioned. As , is -bounded and, by lemma 2 of the post on homomorphisms, is an -isometry. Hence, (9) holds, and the second statement follows. ⬜

Recall that the W*-completion of a bounded *-probability space is a homomorphism to a W*-probability space, such that is a weakly dense subset of . Alternatively, any of the equivalent statements of lemma 16 can be used for . We extend lemma 3 above to generate W*-completions.

Lemma 17Suppose that is a bounded *-probability space with GNS representation . Let be the weak closure of and define the state by (3).

Then, is a W*-probability space and defines a homomorphism from to which, furthermore, is a W*-completion.

*Proof:* Lemma (15) says that is a W*-probability space. Furthermore is, by definition, a state preserving *-homomorphism from to . Finally, lemma 14 says that the ultraweak topology on defined by coincides with the ultraweak topology defined by the action on , with respect to which is dense by lemma 12. So, defines a W*-completion. ⬜

In lemma 4 above, we showed that a homomorphism of C*-probability spaces is an isomorphism if and only if it has -dense image. I now give a W* version of this, where any of the equivalent statements of lemma 16 can be used.

Lemma 18A normal homomorphism of W*-probability spaces is an isomorphism if and only if is a weakly dense subset of .

*Proof:* First, if is an isomorphism then is certainly dense. Conversely, suppose that is dense. As is weakly continuous, and is weakly complete, it follows that is a complete and dense subset of and, hence, is all of . So, by scaling, . Finally, is one-to-one as is nondegenerate (by lemma 1 of the homomorphism post, if then and, by nondegeneracy, ). ⬜

I now prove theorem 7 from the NC probability post, showing the functorial property of W*-completions. This mirrors the proof of theorem 5 above.

Theorem 19Let be a normal homomorphism between bounded *-probability spaces and . Let

be W*-completions. Then, there is a unique normal homomorphism such that . Furthermore, is an isomorphism iff is weakly dense in .

*Proof:* As is ultraweakly continuous, it is uniquely determined by its values on the ultraweakly dense (lemma 16). In turn, it is uniquely detemined on by . So, is uniquely determined.

We need to show existence of . As and are isometries, (4) holds. So, whenever and, as in the proof of theorem 5, we can define the *-homomorphism by (5), which will be al -isometry.

As and are normal, they are weakly continuous on -bounded sets and, hence, so is . For any , maps into , which is weakly complete. As is weakly dense in by assumption, extends uniquely to a weakly continuous map from into . As , this uniquely defines a map

satisfying , and which is weakly continuous on . By weak continuity of the algebra operations, this is a normal *-homomorphism. Then, as preserves the state, and the states are ultraweakly continuous, it follows that preserves the state and is a normal homomorphism of *-probability spaces.

Finally, we show that is an isomorphism if and only if is weakly dense in . If it is an isomorphism, then and, as is weakly dense in , it follows that is weakly dense in . Hence, is weakly dense in .

Conversely, suppose that is weakly dense in . Then, is weakly dense in and, hence, so is . By lemma 18, is an isomorphism. ⬜

Theorem 6 of the NC probability post, stating the existence and uniqueness of W*-completions, can now be proven quite easily. This follows along similar lines to theorem 6 above for the C*-probability case.

Theorem 20Let be a bounded *-probability space. Then, it has a W*-completion, which is unique up to isomorphism. That is, for any two W*-completions

there exists a unique isomorphism such that .

*Proof:* Lemma 17 gives existence of the W*-completion, so only uniqueness remains. Let be as in the statement of the theorem, and be the identity automorphism on . Theorem 19 states the existence of a unique normal isomorphism from to satisfying

as required. Noting that an isomorphism of bounded *-probability spaces is necessarily normal, this is the unique isomorphism satisfying the required property. ⬜

Next, theorem 8 of the NC probability post follow quickly from the results above. This expresses W*-completions and W*-probability spaces in terms of the GNS representation, and mirrors the proof of the C* case, given in theorem 7 above.

Theorem 21Let be a bounded *-probability space, and be its GNS representation. Let be the weak closure of and define byThen,

- is a W*-probability space.
- is the W*-completion of .
- is a W*-probability space if and only if is an isomorphism between and .

*Proof:* The first two statements are already shown in lemma 17 above. Also, if is an isomorphism from to then is isomorphic to , so is a W*-probability space. It only remains to show that, if is a W*-probability space then is an isomorphism. However, is ultraweakly dense in by definition of the W*-completion, so is an isomorphism by lemma 18. ⬜

We move on to establish theorems 3 and 4 of the NC probability post, which express W*-probability spaces in terms of normal states on von Neumann algebras. I start with a W* version of lemma 8. Recall that a von Neumann algebra on a Hilbert space is a weakly closed and unitial *-subalgebra of the bounded linear operators on the space. Hence, the operator topologies can be defined with respect to the action on this space. In addition, a state on the algebra, also defines operator topologies, so we need to relate these two sets of topologies. For a *-algebra acting on a semi-inner product space , I will use -weak (-strong, etc) to denote the weak (strong, etc) topology defined by this action. Similarly for a state , I use -weak (-strong, etc) to denote the topologies defined by the state or, equivalently, by the action of the algebra on itself by left multiplication with the semi-inner product.

Lemma 22Let be a von Neumann algebra on Hilbert space and be a normal state. Then, the -ultraweak (resp., -ultrastrong) topology is weaker than the -ultraweak (resp., -ultrastrong). If the state is nondegenerate, the ultraweak and ultrastrong topologies defined by coincide with those defined by the action on .

*Proof:* Let have the inner product , and acts on this by left-multiplication. Define the homomorphism of *-algebra representations to be the identity on . By assumption, is normal, so is weakly continuous on the unit ball (defined by the operator norm of the action of ). Consider and a net tending weakly to zero (w.r.t. the action on ). Then, is a norm-bounded net tending weakly to zero and, hence, , showing that is normal and, hence, is ultraweakly and ultrastrongly continuous. This implies that the ultraweak (resp., ultrastrong) topology defined w.r.t. the action on is stronger than with the action on .

Now suppose that is nondegenerate. By lemma 8, is an isometry, so gives a continuous bijective map from under the -weak topology, which is compact by lemma 13, onto under the -weak topology, which is Hausdorff by nondegeneracy. Hence, it gives a homeomorphism of the unit ball, so is normal and is ultraweakly and ultrastrongly continuous. This implies that the ultraweak (resp., ultrastrong) topology defined w.r.t. the action on is stronger than w.r.t. the action on . ⬜

We can now prove theorem 3 of the post on NC probability spaces, stating that W*-probability spaces consist of nondegenerate normal states on a von Neumann algebra. Compare with the proof of theorem 9 above. By definition, a von Neumann algebra can be represented by an action on a Hilbert space, with respect to which it forms a weakly closed *-subalgebra of the collection of bounded operators. By saying that the state is normal, we mean that it is normal with respect to this action.

Theorem 23The pair is a W*-probability space if and only if is a von Neumann algebra with respect to which is a nondegenerate normal state.

*Proof:* First, suppose that is a W*-probability space. By theorem 21, is a *-isomorphism between and the weakly closed which, by lemma 12, is a unitial *-subalgebra. This represents as a von Neumann algebra on , with respect to which is normal and nondegerate.

Conversely, suppose that is a von Neumann algebra on Hilbert space and that is a nondegenerate normal state. The norm and the ultraweak topology defined by coincides with that defined by the action on (lemmas 8 and 22). Hence, the unit ball is ultraweakly complete (lemma 13), so is a W*-probability space. ⬜

I conclude with a proof of theorem 4 of the post on NC probability spaces. Compare with theorem 10 above for the C* case.

Theorem 24Let be a von Neumann algebra and be a normal state. Then, is a W*-probability space.

*Proof:* As is just the -ultraweak closure of the point , and the -ultraweak topology is stronger than the -ultraweak (lemma 22), it follows that is -ultraweakly closed,

Suppose that is a von Neumann algebra on Hilbert space . It is standard that the quotient by an ultraweakly closed *-ideal is again a von Neumann algebra. Specifically, every such ideal is generated by a central projection (Blackadar III.1.13). That is, , where is a projection which commutes with all elements of the algebra. Then, is *-isomorphic to the ultraweakly closed complementary ideal and, hence, is a von Neumann algebra acting on the closed subspace . The *-homomorphism given by multiplication by has kernel , so generates the isomorphism . The state is given by the restriction of to and, hence, is also normal. So, is a normal state on , and the result follows from theorem 23. ⬜