Completions of *-Probability Spaces

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 preprobability 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 ${p}$ on a *-algebra was defined as a positive linear map satisfying a normalisation condition. This can be expressed as ${p(1)=1}$ for unitial algebras or, more generally, as ${\lVert p\rVert=1}$ using the following norm,

 $\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle \lVert p\rVert &\displaystyle = \sup\left\{\lvert p(a)\rvert^2\colon a\in\mathcal A, p(a^*a)\le1\right\}\smallskip\\ &\displaystyle = \inf\left\{K\in{\mathbb R}^+\colon\lvert p(a)\rvert^2\le Kp(a^*a)\ \forall a\in\mathcal A\right\}. \end{array}$ (1)

We actually described three kinds of spaces. First, *-probability spaces, or preprobability spaces, which are nothing more than a pair ${(\mathcal A,p)}$ consisting of a state ${p}$ on *-algebra ${\mathcal A}$. 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 ${\mathcal H}$ will be denoted by ${B(\mathcal H)}$, and a *-subalgebra of ${B(\mathcal H)}$ will be referred to as a *-algebra on ${\mathcal H}$. An element ${\xi\in\mathcal H}$ will be called cyclic for the *-algebra ${\mathcal A\subseteq B(\mathcal H)}$ if ${\mathcal A\xi=\{a\xi\colon a\in\mathcal A\}}$ is a dense subspace of ${\mathcal H}$. The operator norm of ${a\in\mathcal A}$ will be denoted as ${\lVert a\rVert}$, and the ${L^2}$ and ${L^\infty}$ seminorms with respect to a state ${p}$ are denoted by ${\lVert a\rVert_2}$ and ${\lVert a\rVert_\infty}$ respectively.

Lemma 1 Let ${\mathcal A}$ be a *-algebra on Hilbert space ${\mathcal H}$ and ${\xi\in\mathcal H}$ have norm 1 and be cyclic for ${\mathcal A}$. Then,

 $\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle p\colon\mathcal A\rightarrow{\mathbb C},\smallskip\\ &\displaystyle p(a)=\langle \xi,a\xi\rangle, \end{array}$ (2)

is a nondegenerate state. Furthermore, the ${L^\infty(p)}$ norm on ${\mathcal A}$ coincides with the operator norm and the ${L^2(p)}$ norm is given by ${\lVert a\rVert_2=\lVert a\xi\rVert}$.

Proof: Linearity of ${p}$ is immediate. For ${a\in\mathcal A}$,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle p(a^*)=\langle\xi,a^*\xi\rangle=\langle a\xi,\xi\rangle=\overline{p(a)},\smallskip\\ &\displaystyle p(a^*a)=\langle\xi,a^*a\xi\rangle=\langle a\xi,a\xi\rangle\ge0. \end{array}$

This shows that ${p}$ is positive and that the ${L^2(p)}$-norm is

$\displaystyle \lVert a\rVert_2=\sqrt{p(a^*a)}=\sqrt{\langle a\xi,a\xi\rangle}=\lVert a\xi\rVert.$

Also, if ${\lVert a\rVert_2\le1}$ then,

$\displaystyle \lvert p(a)\rvert^2=\lvert\langle\xi,a\xi\rangle\rvert^2\le\lVert a\xi\rVert^2\le1$

shows that ${\lVert p\rVert\le1}$. Next, as ${\xi}$ is cyclic, there exists ${a_n\in\mathcal A}$ such that ${a_n\xi\rightarrow\xi}$. In particular, ${\lVert a_n\rVert_2\rightarrow1}$ so, dividing by ${\lVert a_n\rVert_2}$ if necessary, we suppose without loss of generality that ${\lVert a_n\rVert_2=1}$. Then,

$\displaystyle p(a_n)=\langle\xi,a_n\xi\rangle\rightarrow\langle\xi,\xi\rangle=1$

shows that ${\lVert p\rVert\ge1}$, so that ${p}$ is a state.

Now, for ${a,b\in\mathcal A}$,

$\displaystyle \lVert ab\rVert_2=\lVert ab\xi\rVert\le\lVert a\rVert \lVert b\xi\rVert=\lVert a\rVert\lVert b\rVert_2$

shows that ${\lVert a\rVert_\infty\le\lVert a\rVert}$. For the opposite inequality, for any ${x\in\mathcal H}$, the fact that ${\xi}$ is cyclic means that there exists ${b_n\in\mathcal A}$ with ${b_n\xi\rightarrow x}$. Then,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle \lVert ax\rVert &\displaystyle =\lim_n\lVert ab_n\xi\rVert=\lim_n\lVert ab_n\rVert_2\smallskip\\ &\displaystyle \le\lim_n\lVert a\rVert_\infty\lVert b_n\rVert_2\smallskip\\ &\displaystyle =\lim_n\lVert a\rVert_\infty\lVert b_n\xi\rVert=\lVert a\rVert_\infty\lVert x\rVert \end{array}$

and, hence, ${\lVert a\rVert\le\lVert a\rVert_\infty}$ as required.

Finally if ${a\in\mathcal A}$ satisfies ${\lVert a\rVert_\infty=0}$ then ${\lVert a\rVert=0}$ and, hence, ${a=0}$, showing that ${p}$ 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 2 Let ${\mathcal A}$ be a *-algebra on Hilbert space ${\mathcal H}$. Then, its operator norm closure ${\bar{\mathcal A}}$ is a *-algebra. Furthermore, if ${\xi\in\mathcal H}$ has norm 1 and is cyclic for ${\mathcal A}$, then ${(\bar{\mathcal A},\bar p)}$ is a C*-probability space, where

 $\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \bar p\colon\bar{\mathcal A}\rightarrow{\mathbb C},\smallskip\\ &\displaystyle \bar p(a)=\langle \xi,a\xi\rangle. \end{array}$ (3)

Proof: That ${\bar{\mathcal A}}$ is a *-algebra follows from the fact that the *-algebra operations are norm-continuous. Explicitly, elements ${a,b\in\bar{\mathcal A}}$ can be expressed as norm-limits of sequences ${a_n,b_n\in\bar{\mathcal A}}$. Then, ${ab}$ and ${a^*}$ are, respectively, limits of the sequences ${a_nb_n}$ and ${a_n^*}$, so are in ${\bar{\mathcal A}}$. Similarly, for ${\lambda,\mu\in{\mathbb C}}$, ${\lambda a+\mu b}$ is the limit of ${\lambda a_n+\mu b_n}$, so is in ${\bar{\mathcal A}}$ which, therefore, is a *-algebra.

Lemma 1 says that ${(\bar{\mathcal A},\bar p)}$ is a bounded nondegenerate *-probability space, so it only remains to show that ${\bar{\mathcal A}}$ is ${L^\infty(p)}$-complete. By construction, it is a closed subspace of the complete space ${B(\mathcal H)}$, so is complete under the operator norm. By lemma 1, this coincides with the ${L^\infty(\bar p)}$ norm. ⬜

Recall that the C*-completion of a bounded *-probability space ${(\mathcal A,p)}$ consists of a C*-probability space ${(\bar{\mathcal A},\bar p)}$ together with a (state preserving) homomorphism ${\pi\colon(\mathcal A,p)\rightarrow(\bar{\mathcal A},\bar p)}$ with ${L^\infty}$-dense image. C*-completions can be constructed with the aid of the GNS representation ${(\mathcal H,\pi,\xi)}$ which, by definition, is a Hilbert space ${\mathcal H}$ together with a *-homomorphism ${\pi\colon\mathcal A\rightarrow B(\mathcal H)}$ and an element ${\xi\in\mathcal H}$ which is cyclic for ${\pi(\mathcal A)}$ and satisfies ${p(a)=\langle\xi,\pi(a)\xi\rangle}$.

Lemma 3 Suppose that ${(\mathcal A,p)}$ is a bounded *-probability space with GNS representation ${(\mathcal H,\pi,\xi)}$. Let ${\bar{\mathcal A}\subseteq B(\mathcal H)}$ be the operator norm closure of ${\pi(\mathcal A)}$ and define the state ${\bar p\colon\bar{\mathcal A}\rightarrow{\mathbb C}}$ by (3).

Then, ${(\bar{\mathcal A},\bar p)}$ is a C*-probability space and ${\pi}$ defines a homomorphism from ${(\mathcal A,p)}$ to ${(\bar{\mathcal A},\bar p)}$ which, furthermore, is a C*-completion.

Proof: Lemma (2) says that ${(\bar{\mathcal A},\bar p)}$ is a C*-probability space. Furthermore ${\pi}$ is, by definition, a *-homomorphism from ${\mathcal A}$ to ${\bar{\mathcal A}}$. 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,

$\displaystyle \bar p(\pi(a))=\langle\xi,\pi(a)\xi\rangle=p(a)$

as required. Finally, lemma 1 says that the ${L^\infty(\bar p)}$ norm coincides with the operator norm on ${\bar{\mathcal A}}$ with respect to which ${\pi(\mathcal A)}$ is dense, by construction. So, ${\pi}$ defines a C*-completion. ⬜

A homomorphism ${\varphi}$ between *-probability spaces ${(\mathcal A,p)}$ and ${(\mathcal A^\prime,p^\prime)}$ 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 ${p}$ is nondegenerate and, for C* spaces, a sufficient condition to be onto is that ${\varphi(\mathcal A)}$ is dense. So, a homomorphism between C*-probability spaces with dense image is necessarily an isomorphism.

Lemma 4 A homomorphism ${\varphi\colon(\mathcal A,p)\rightarrow(\mathcal A,p^\prime)}$ of C*-probability spaces is an isomorphism if and only if ${\varphi(\mathcal A)}$ is ${L^\infty}$-dense in ${\mathcal A^\prime}$.

Proof: First, if ${\varphi}$ is an isomorphism then ${\varphi(\mathcal A)=\mathcal A^\prime}$ is certainly dense. Conversely, suppose that ${\varphi(\mathcal A)}$ is dense. The ${L^\infty}$-seminorms on C*-algebras are complete norms and, by lemma 5 of the post on homomorphisms of *-probability spaces, ${\varphi}$ is an ${L^\infty}$ isometry. Hence, ${\varphi}$ is one-to-one and ${\varphi(\mathcal A)}$ is a complete and dense subset of ${\mathcal A^\prime}$, so ${\varphi(\mathcal A)=\mathcal A^\prime}$ 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 ${L^\infty}$ isometries.

Theorem 5 Let ${\varphi}$ be an ${L^\infty}$-continuous homomorphism between bounded *-probability spaces ${(\mathcal A,p)}$ and ${(\mathcal A^\prime,p^\prime)}$. Let

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \pi\colon(\mathcal A,p)\rightarrow(\bar{\mathcal A},\bar p),\smallskip\\ &\displaystyle \pi^\prime\colon(\mathcal A^\prime,p^\prime)\rightarrow(\bar{\mathcal A}^\prime,\bar p^\prime). \end{array}$

be C*-completions. Then, there is a unique homomorphism ${\bar\varphi\colon(\bar{\mathcal A},\bar p)\rightarrow(\bar{\mathcal A}^\prime,\bar p^\prime)}$ such that ${\bar\varphi\circ\pi=\pi^\prime\circ\varphi}$. Furthermore, ${\bar\varphi}$ is an isomorphism iff ${\varphi(\mathcal A)}$ is ${L^\infty}$-dense in ${\mathcal A^\prime}$.

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 ${L^\infty}$-isometric. In particular, by uniqueness of continuous extensions, ${\bar\varphi}$ is uniquely defined by its restriction to ${\pi(\mathcal A)}$ on which it is uniquely defined by ${\bar\varphi(\pi(a))=\pi^\prime(\varphi(a))}$. Hence, ${\bar\varphi}$ is unique.

We still need to show existence of ${\bar\varphi}$ satisfying the requirements of the theorem. As ${\varphi,\pi,\pi^\prime}$ are all ${L^\infty}$-isometries, for ${a\in\mathcal A}$,

 $\displaystyle \lVert \pi^\prime(\varphi(a))\rVert_\infty=\lVert a\rVert_\infty=\lVert\pi(a)\rVert_\infty.$ (4)

In particular, if ${\pi(a)=0}$ then ${\lVert \pi^\prime(\varphi(a))\rVert_\infty=0}$ and, by nondegeneracy of ${\bar p^\prime}$, ${\pi^\prime(\varphi(a))=0}$. So, we can define a *-homomorphism ${\varphi_0\colon\pi(\mathcal A)\rightarrow\bar{\mathcal A}^\prime}$ by

 $\displaystyle \varphi_0(\pi(a))=\pi^\prime(\varphi(a)).$ (5)

As ${\pi,\pi^\prime,\varphi}$ are state-preserving, the same is true of ${\varphi_0}$ and, by (4), is an ${L^\infty}$-isomtery. As ${\bar{\mathcal A}^\prime}$ is ${L^\infty}$-complete, by continuous linear extensions ${\varphi_0}$ uniquely extends to a continuous linear map ${\bar\varphi\colon\bar{\mathcal A}\rightarrow\bar{\mathcal A}^\prime}$. By ${L^\infty}$-continuity of the *-algebra operations and of the state, it follows that ${\bar\varphi}$ is a *-homomorphism preserving the state as required.

Finally, ${\pi}$ and ${\pi^\prime}$ are ${L^\infty}$-isometries with dense image, so that ${\varphi}$ has dense image if and only if ${\bar\varphi}$ has dense image which, by lemma 4, is equivalent to ${\bar\varphi}$ being an isomorphism. ⬜

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

Theorem 6 Let ${(\mathcal A,p)}$ be a bounded *-probability space. Then, it has a C*-completion, which is unique up to isomorphism. That is, for any two C*-completions

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \pi\colon(\mathcal A,p)\rightarrow(\bar{\mathcal A},\bar p),\smallskip\\ &\displaystyle \pi^\prime\colon(\mathcal A,p)\rightarrow(\mathcal A^\prime,p^\prime) \end{array}$

there exists a unique isomorphism ${\varphi\colon(\bar{\mathcal A},\bar p)\rightarrow(\mathcal A^\prime,p^\prime)}$ such that ${\pi^\prime=\varphi\circ\pi}$.

Proof: Lemma 3 gives existence of the C*-completion, so only uniqueness remains. Let ${\pi,\pi^\prime}$ be as in the statement of the theorem, and ${\iota}$ be the identity automorphism on ${(\mathcal A,p)}$. Theorem 5 states the existence of a unique isomorphism ${\varphi}$ from ${(\bar{\mathcal A},\bar p)}$ to ${(\mathcal A^\prime,p^\prime)}$ satisfying

$\displaystyle \pi^\prime=\pi^\prime\circ\iota=\varphi\circ\pi$

as required. ⬜

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

Theorem 7 Let ${(\mathcal A,p)}$ be a bounded *-probability space, and ${(\mathcal H,\pi,\xi)}$ be its GNS representation. Let ${\bar{\mathcal A}\subseteq B(\mathcal H)}$ be the operator norm closure of ${\pi(\mathcal A)}$ and define ${\bar p\colon\bar{\mathcal A}\rightarrow{\mathbb C}}$ by

$\displaystyle \bar p(a)=\langle\xi,a\xi\rangle.$

Then,

• ${(\bar{\mathcal A},\bar p)}$ is a C*-probability space.
• ${\pi\colon(\mathcal A,p)\rightarrow(\bar{\mathcal A},\bar p)}$ is the C*-completion of ${(\mathcal A,p)}$.
• ${(\mathcal A,p)}$ is a C*-probability space if and only if ${\pi}$ is an isomorphism between ${\mathcal A}$ and ${\bar{\mathcal A}}$.

Proof: The first two statements are already shown in lemma 3 above. Also, if ${\pi}$ is an isomorphism from ${\mathcal A}$ to ${\bar{\mathcal A}}$ then ${(\mathcal A,p)}$ is isomorphic to ${(\bar{\mathcal A},\bar p)}$, so is a C*-probability space. It only remains to show that, if ${(\mathcal A,p)}$ is a C*-probability space then ${\pi}$ is an isomorphism. However, ${\pi(\mathcal A)}$ is ${L^\infty}$-dense in ${\bar{\mathcal A}}$ by definition of the C*-completion, so ${\pi}$ 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 ${L^\infty}$ 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 ${\lVert a\rVert_*}$ to denote the C*-algebra norm, this means that ${\lVert a\rVert_\infty\le\lVert a\rVert_*}$.

Lemma 8 Let ${p\colon\mathcal A\rightarrow{\mathbb C}}$ be a positive functional on C*-algebra ${\mathcal A}$. Then, the ${L^\infty(p)}$ seminorm is weaker than the C*-norm and, if ${p}$ is nondegenerate, the ${L^\infty}$ and C*-norms coincide.

Proof: If ${K=\lVert a\rVert_*}$ then, ${K^2-a^*a=b^2}$ for some self-adjoint ${b\in{\mathbb C}\oplus\mathcal A}$ (see lemma 5 of the *-algebra post). Multiplying on the right by ${x\in\mathcal A}$ and on the left by ${x^*}$, and applying ${p}$,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle K^2\lVert x\rVert_2^2-\lVert ax\rVert_2^2 &\displaystyle =p(x^*(K^2-a^*a)x)\smallskip\\ &\displaystyle =p((bx)^*(bx))\ge0 \end{array}$

for all ${x\in\mathcal A}$, giving ${\lVert a\rVert_\infty\le K}$.

Now suppose that ${p}$ is nondegenerate, so that the ${L^\infty}$-seminorm is positive definite. As we have shown that ${\lVert a\rVert_\infty\le\lVert a\rVert_* < \infty}$, it follows that ${\lVert\cdot\rVert_\infty}$ is finite. Hence, we can let ${\bar{\mathcal A}}$ be its ${L^\infty}$-completion, which is a C*-algebra. Then, the natural embedding ${\mathcal A\hookrightarrow\bar{\mathcal A}}$ is a one-to-one *-homomorphism, so is an isometry (see, eg, Blackadar II.2.2.9) under the respective C*-norms. That is, ${\lVert a\rVert_*=\lVert a\rVert_\infty}$ as required. ⬜

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

Theorem 9 The pair ${(\mathcal A,p)}$ is a C*-probability space if and only if ${\mathcal A}$ is a C*-algebra and ${p}$ is a nondegenerate state.

Proof: First, suppose that ${(\mathcal A,p)}$ is a C*-probability space. As the ${L^\infty(p)}$-seminorm satisfies the C*-properties and, by definition, is a complete norm, then ${\mathcal A}$ is a C*-algebra. Conversely, suppose that ${\mathcal A}$ is a C*-algebra and ${p}$ is a nondegenerate state. Lemma 8 says that the ${L^\infty}$ and C*-norms coincide, so ${\mathcal A}$ is ${L^\infty}$-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 ${\mathcal N=\{a\in\mathcal A\colon\lVert a\rVert_\infty=0\}}$. The state ${p^\prime}$ on the quotient algebra ${\mathcal A/\mathcal N}$ is that given by the state ${p}$ under the quotient map ${\mathcal A\rightarrow\mathcal A/\mathcal N}$.

Theorem 10 Let ${\mathcal A}$ be a C*-algebra and ${p\colon\mathcal A\rightarrow{\mathbb C}}$ be a state. Then, ${(\mathcal A/\mathcal N,p^\prime)}$ is a C*-probability space.

Proof: As ${\mathcal N}$ is, by definition, ${L^\infty(p)}$-closed and lemma 8 says that the C*-norm is stronger than ${L^\infty}$, we see that ${\mathcal N}$ is closed under the C*-norm. It is then standard (Blackadar II.5.1.1) that ${\mathcal A/\mathcal N}$ is a C*-algebra with respect to the norm

$\displaystyle \lVert a+\mathcal N\rVert_*=\inf\left\{\lVert a+b\rVert_*\colon b\in\mathcal N\right\}.$

Furthermore, as ${p^\prime}$ is a nondegenerate state on ${\mathcal A/\mathcal N}$ (lemma 13 of the post on states), theorem 9 says that ${(\mathcal A/\mathcal N,p^\prime)}$ 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 ${\mathcal A}$, a state ${p\colon\mathcal A\rightarrow{\mathbb C}}$ is required to satisfy the positivity condition ${p(a^*a)\ge0}$ together with a normalisation criterion to ensure that the total probability’ of the space is one. In the case where ${\mathcal A}$ is unitial then the normalisation is simply ${p(1)=1}$. For C*-algebra ${\mathcal A}$ with norm ${\lVert\cdot\rVert}$, the operator norm of the linear functional ${p}$ is defined by

 $\displaystyle \lVert p\rVert_{\rm op}=\sup\left\{\vert p(a)\rvert\colon a\in\mathcal A,\lVert a\rVert\le1\right\}.$ (6)

For nonunitial C*-algebras, the normalisation condition ${\lVert p\rVert_{\rm op}=1}$ is commonly used in the definition of states and, for unitial C*-algebras, it is standard that positive linear functionals satisfy ${\lVert p\rVert_{\rm op}=p(1)}$ 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 ${\lVert p\rVert=1}$. For unitial C*-algebras, ${\lVert p\rVert_{\rm op}=p(1)=\lVert p\rVert}$, 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 ${\lVert p\rVert\le\lVert p\rVert_{\rm op}}$ is known as Kadison’s inequality (Blackadar II.6.9.14).

Lemma 11 Let ${\mathcal A}$ be a C*-algebra and ${p\colon\mathcal A\rightarrow{\mathbb C}}$ be a positive linear map. Then, ${\lVert p\rVert=\lVert p\rVert_{\rm op}}$.

Proof: We first show that ${\lVert p\rVert_{\rm op}\le\lVert p\rVert}$. Choose ${a\in\mathcal A}$ with ${\lVert a\rVert_*\le1}$. Then, as ${\lVert aa^*\rVert_*=\lVert a\rVert_*^2\le 1}$, there exists a self-adjoint ${b\in{\mathbb C}\oplus\mathcal A}$ with ${1-aa^*=b^2}$. Using

$\displaystyle a^*a-(a^*a)^2=a^*(1-aa^*)a=(ba)^*(ba)$

gives ${p(a^*a-(a^*a)^2)\ge0}$, from which we obtain,

$\displaystyle p(a^*a)^2\le\lVert p\rVert p((a^*a)^2)\le\lVert p\rVert p(a^*a).$

Hence, ${p(a^*a)\le\lVert p\rVert}$, giving

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

As this holds for all ${\lVert a\rVert_*\le1}$, we obtain ${\lVert p\rVert_{\rm op}\le\lVert p\rVert}$.

We now show the reverse inequality, ${\lVert p\rVert\le\lVert p\rVert_{\rm op}}$. Without loss of generality, suppose that ${\lVert p\rVert_{\rm op}}$ is finite. I make use of the standard fact that every C*-algebra has an approximate identity ${u_\alpha}$ (Blackadar II.4.1.3). That is ${u_\alpha\in\mathcal A}$ is a net of self-adjoint elements with ${\lVert u_\alpha\rVert_*\le1}$ and ${\lVert u_\alpha a-a\rVert_*\rightarrow0}$ for all ${a\in\mathcal A}$. Applying Cauchy–Schwarz,

$\displaystyle \lvert p(u_\alpha a)\rvert^2 \le p(u_\alpha^2)p(a^*a)\le\lVert p\rVert_{\rm op}p(a^*a).$

The second inequality here is using ${\lVert u_\alpha^2\rVert_*\le1}$. Now, as ${\lVert u_\alpha a-a\rVert_*\rightarrow0}$ and ${p}$ is bounded in the operator norm, we can take limits on the left hand side to obtain

$\displaystyle \lvert p(a)\rvert^2\le\lVert p\rVert_{\rm op}p(a^*a),$

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

If ${\lVert a\rVert_*\le1}$ and

$\displaystyle \lvert p(a)\rvert^2\le\lVert p\rVert p(a^*a)\le\lVert p\rVert \lVert p\rVert_{\rm op}$

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 ${L^\infty}$ 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 ${\mathcal A}$ on a Hilbert space ${\mathcal H}$ will be said to act nondegenerately if ${ax=0}$ for all ${a\in\mathcal A}$ implies that ${x=0}$. Equivalently, the linear span of ${\{ax\colon a\in\mathcal A,x\in\mathcal H\}}$ is dense in ${\mathcal H}$. We use ${\mathcal S^{\rm w},\mathcal S^{\rm s},\mathcal S^{\rm uw},\mathcal S^{\rm us}}$ to denote the closures of ${S\subseteq\mathcal A}$ in the weak, strong, ultraweak and ultrastrong topologies respectively. The unit ball of ${\mathcal A}$ is denoted by

$\displaystyle \mathcal A_1=\left\{a\in\mathcal A\colon a\in\mathcal A,\ \lVert a\rVert\le1\right\}$

where ${\lVert\cdot\rVert}$ denotes the operator norm. The unit ball of the closure of ${\mathcal A}$ below is denoted by ${\bar{\mathcal A}_1=(\bar{\mathcal A})_1}$.

Lemma 12 Let ${\mathcal A}$ be a *-algebra on Hilbert space ${\mathcal H}$. Then,

 $\displaystyle \mathcal A^{\rm w}=\mathcal A^{\rm s}=\mathcal A^{\rm uw}=\mathcal A^{\rm us}.$ (7)

Furthermore, if we let ${\bar{\mathcal A}}$ denote the closure of ${\mathcal A}$ under the topologies above, then

• ${\bar{\mathcal A}}$ is a *-algebra.
• if ${\mathcal A}$ acts nondegenerately on ${\mathcal H}$, then ${\bar{\mathcal A}}$ 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 ${\mathcal A^{\rm w}=\mathcal A^{\rm s}}$ and ${\mathcal A^{\rm uw}=\mathcal A^{\rm us}}$. As the ultraweak topology is stronger than the weak,

$\displaystyle \mathcal A^{\rm uw}=\mathcal A^{\rm us}\subseteq\mathcal A^{\rm w}=\mathcal A^{\rm s}.$

We make use of the Kaplansky density theorem, so that

$\displaystyle \mathcal A^{\rm s}\cap B(\mathcal H)_1\subseteq(\mathcal A\cap B(\mathcal H)_1)^{\rm s}=(\mathcal A\cap B(\mathcal H)_1)^{\rm us}\subseteq\mathcal A^{\rm us}.$

The equality here is because the strong and ultrastrong topologies coincide on the unit ball of ${B(\mathcal H)}$. By scaling, this gives ${\mathcal A^{\rm s}\subseteq\mathcal A^{\rm us}}$, so (7) follows.

We now show that ${\bar{\mathcal A}}$ is a *-algebra. The fact that it is a linear subspace of ${B(\mathcal H)}$ closed under involution follows from the fact that the linear combination ${(a,b)\mapsto\lambda a+\mu b}$ is jointly weakly continuous and the adjoint map ${a\mapsto a^*}$ is weakly continuous. It remains to show that ${\bar{\mathcal A}}$ is closed under products. Although the map ${(a,b)\mapsto ab}$ is not jointly continuous, it is weakly continuous individually in each of ${a}$ and ${b}$. By continuity in ${b}$, for each fixed ${a\in\mathcal A}$, ${ab\in\bar{\mathcal A}}$ for all ${b\in\bar{\mathcal A}}$. Then, by continuity in ${a}$, ${ab\in\bar{\mathcal A}}$ for all ${a,b\in\bar{\mathcal A}}$.

Finally, suppose that ${\mathcal A}$ acts nondegenerately. As ${\bar{\mathcal A}}$ is a C*-algebra, it has an approximate identity ${u_\alpha\in\bar{\mathcal A}_1}$. This is a net with ${\lVert u_\alpha b-b\rVert\rightarrow0}$ for all ${b\in\bar{\mathcal A}}$. Let ${V}$ be the set of ${x\in\mathcal H}$ such that ${u_\alpha x\rightarrow x}$. As ${u_\alpha}$ is uniformly bounded, ${V}$ is a closed linear subspace of ${\mathcal H}$. Furthermore,

$\displaystyle \lVert u_\alpha bx-bx\rVert\le\lVert u_\alpha b-b\rVert\lVert x\rVert\rightarrow0$

for all ${b\in\mathcal A}$ and ${x\in\mathcal H}$, so ${bx\in V}$. Hence, if ${\mathcal A}$ acts nondegenerately then ${V=\mathcal H}$, so ${u_\alpha\rightarrow I}$ strongly showing that ${I\in\bar{\mathcal A}}$. ⬜

Recall that in the definition of a W*-probability space ${(\mathcal A,p)}$, we required the unit ball ${\mathcal A_1}$ 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 ${\iota\colon V\rightarrow\mathcal H}$ for the Hilbert space completion, and ${\pi\colon\mathcal A\rightarrow B(\mathcal H)}$ for the continuous linear extension of the action of ${a\in\mathcal A}$ on ${\mathcal H}$, so that ${\pi(a)\iota x=\iota(ax)}$. This is a *-homomorphism, so that the image ${\pi(\mathcal A)}$ is a *-subalgebra of ${B(\mathcal H)}$. In particular, when ${\mathcal A}$ is a *-algebra on a Hilbert space, then we can take ${\mathcal H=V}$ and ${\pi(\mathcal A)=\mathcal A}$, so that the third statement below is equivalent to ${\mathcal A}$ 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 ${\mathcal A}$ acting on semi-inner product space ${V}$ 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 13 Let ${(\mathcal A,V)}$ be a bounded *-algebra representation. The following are equivalent.

1. ${\mathcal A_1}$ is weakly/ultraweakly compact.
2. ${\mathcal A_1}$ is weakly/strongly/ultraweakly/ultrastrongly complete.
3. ${\pi(\mathcal A)}$ is weakly/strongly/ultraweakly/ultrastrongly closed in ${B(\mathcal H)}$.

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 ${a_\alpha\in\mathcal A_1}$. As the weak topology is weaker than the strong, it is also weakly Cauchy and, by the assumption, has a weak limit ${a\in\mathcal A_1}$. Then, for ${x\in V}$, weak convergence gives

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle \lVert (a_\alpha-a)x\rVert^2&\displaystyle=\lim_\beta\langle(a_\alpha-a)x,(a_\alpha-a_\beta)x\rangle\smallskip\\ &\displaystyle\le\lVert(a_\alpha-a)x\rVert\lim_\beta\lVert(a_\alpha-a_\beta)x\rVert\smallskip\\ &\displaystyle\le 2\lVert x\rVert\lim_\beta\lVert(a_\alpha-a_\beta)x\rVert\rightarrow0 \end{array}$

and, so, ${a_\alpha\rightarrow a}$ strongly as required.

Now, supposing that ${\mathcal A_1}$ is strongly complete, we prove the third statement. By (7) this is an equivalent statement under each of the mentioned topologies. Setting ${\mathcal B=\pi(\mathcal A)}$ then,

$\displaystyle \mathcal B^{\rm s}\cap B(\mathcal H)_1=\left(\mathcal B\cap B(\mathcal H)_1\right)^{\rm s}=\pi(\mathcal A_1)^{\rm s}.$

The first equality here is using the Kaplansky density theorem, and the second uses the fact that ${\pi}$ is an ${L^\infty}$ isometry. So, to show that ${\mathcal B}$ is strongly closed, it is sufficient to show that ${\pi(\mathcal A_1)^{\rm s}=\pi(\mathcal A_1)}$. Given ${a\in\pi(\mathcal A_1)^{\rm s}}$, choose a net ${a_\alpha\in\mathcal A_1}$ such that ${\pi(a_\alpha)\rightarrow a}$ strongly. Then, ${\pi(a_\alpha-a_\beta)\rightarrow0}$ and hence, ${a_\alpha-a_\beta\rightarrow0}$ strongly. So, by assumption, there exists a strong limit ${a_\alpha\rightarrow b\in\mathcal A_1}$. Then, ${\pi(b)=\lim\pi(a_\alpha)=a}$ shows that ${a\in\pi(\mathcal A_1)}$ as required.

Finally, assuming the third statement, we show that ${\mathcal A_1}$ is ultraweakly compact. Theorem 21 of the post on normal maps expresses ${\pi(\mathcal A)}$ as the dual of a Banach space ${\mathcal A_*}$, with the weak-* topology corresponding to ultraweak convergence. Hence, ${\pi(\mathcal A_1)}$ is ultraweakly compact by the Banach–Alaoglu theorem. Then, as the ultraweak topology on ${\mathcal A_1}$ given by its action on ${V}$ is the same as that given by the action on ${\mathcal H}$, this shows that ${\mathcal A_1}$ is ultraweakly compact. ⬜

Lemma 1 above showed that the ${L^\infty}$ 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 14 Assume the hypotheses of lemma 1. Then,

1. the ultraweak and ultrastrong topologies on ${\mathcal A}$ defined with respect to the action on ${\mathcal H}$ coincide with the respective topologies defined with respect to ${p}$.
2. the weak, strong, ultraweak and ultrastrong topologies on ${\mathcal A_1}$ defined with respect to the action on ${\mathcal H}$ coincide with the respective topologies defined with respect to ${p}$.

Proof: The weak and ultraweak topologies coincide on ${\mathcal A_1}$, as do the strong and ultrastrong, so the second statement follows from the first. To prove the first statement, let ${V}$ be the semi-inner product space with set of elements equal to ${\mathcal A}$ and inner product ${\langle x,y\rangle=p(x^*y)}$. By definition, the ultraweak and ultrastrong topologies defined by ${p}$ are equal to the respective topologies given by the left-multiplication action of ${\mathcal A}$ on ${V}$. By lemma 1, the linear map ${V\rightarrow\mathcal H}$ taking ${x\in V}$ to ${x\xi}$ is an isometry and, by assumption, has dense image in ${\mathcal H}$. So, ${V\rightarrow\mathcal H}$ 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 15 Let ${\mathcal A}$ be a *-algebra on Hilbert space ${\mathcal H}$ and ${\xi\in\mathcal H}$ have norm 1 and be cyclic for ${\mathcal A}$. Then, ${(\bar{\mathcal A},\bar p)}$ is a W*-probability space, where ${\bar{\mathcal A}}$ is the weak closure of ${\mathcal A}$ and ${\bar p}$ is defined by (3).

Proof: Lemma 12 says that ${\bar{\mathcal A}}$ is a *-algebra, and lemma 1 says that ${\bar p}$ is a nondegenerate state. As, by lemma 1, the ${L^\infty(\bar p)}$ norm coincides with the operator norm, the unit ball ${\bar{\mathcal A}_1}$ is the same whether defined with respect to ${L^\infty}$ or the operator norm. By lemma 13, ${\bar{\mathcal A}_1}$ is weakly complete under the weak topology on ${B(\mathcal H)}$ and, by lemma 14, the weak topology coincides with the one defined by ${p}$. So, ${(\bar{\mathcal A},\bar p)}$ is a W*-probability space. ⬜

Whereas the C*-completion was defined via an ${L^\infty}$ 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 16 Let ${\varphi\colon(\mathcal A,p)\rightarrow(\mathcal A^\prime,p^\prime)}$ be a homomorphism of bounded *-probability spaces. The following are equivalent,

1. ${\varphi(\mathcal A)}$ is ultraweakly/ultrastrongly dense in ${\mathcal A^\prime}$.
2. ${\varphi(\mathcal A)\cap\mathcal A^\prime_1}$ is weakly/strongly/ultraweakly/ultrastrongly dense in ${\mathcal A^\prime_1}$.
3. ${\varphi(\mathcal A_1)}$ is a weakly/strongly/ultraweakly/ultrastrongly dense subset of ${\mathcal A^\prime_1}$.

In case that the above equivalent conditions hold, then ${\varphi}$ is normal.

Proof: Using ${\mathcal B=\varphi(\mathcal A)}$, lemma 22 of the post on normal maps says that

 $\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \mathcal B^{\rm uw}\cap\mathcal A^\prime_1 =\mathcal B^{\rm us}\cap\mathcal A^\prime_1 =\left(\mathcal B\cap\mathcal A^\prime_1\right)^{\rm w}\smallskip\\ &\displaystyle =\left(\mathcal B\cap\mathcal A^\prime_1\right)^{\rm s} =\left(\mathcal B\cap\mathcal A^\prime_1\right)^{\rm uw} =\left(\mathcal B\cap\mathcal A^\prime_1\right)^{\rm us}. \end{array}$ (8)

By scaling, a linear subspace ${S\subseteq\mathcal A^\prime}$ is all of ${\mathcal A^\prime}$ if and only if ${S\cap\mathcal A^\prime_1=\mathcal A^\prime_1}$. Each of the the first two statements, using each of the stated topologies, are equivalent to the identical expressions in (8) being equal to ${\mathcal A^\prime_1}$, so these are all equivalent statements. Next, suppose that the first statement holds. Then, by lemma 10 of the post on normal maps, ${\varphi}$ is normal. In particular, by lemma 2 of the same post, ${\varphi}$ is an ${L^\infty}$-isometry, so

 $\displaystyle \varphi(\mathcal A_1)=\varphi(\mathcal A)\cap \mathcal A^\prime_1$ (9)

which, by the second statement, is a dense subset of ${\mathcal A^\prime_1}$ under each of the operator topologies.

Finally, suppose that the third statement holds for any of the operator topologies mentioned. As ${\varphi(\mathcal A_1)\subseteq\mathcal A_1^\prime}$, ${\varphi}$ is ${L^\infty}$-bounded and, by lemma 2 of the post on homomorphisms, is an ${L^\infty}$-isometry. Hence, (9) holds, and the second statement follows. ⬜

Recall that the W*-completion of a bounded *-probability space ${(\mathcal A,p)}$ is a homomorphism ${\pi\colon(\mathcal A,p)\rightarrow(\bar{\mathcal A},\bar p)}$ to a W*-probability space, such that ${\pi(\mathcal A_1)}$ is a weakly dense subset of ${\bar{\mathcal A}_1}$. Alternatively, any of the equivalent statements of lemma 16 can be used for ${\pi}$. We extend lemma 3 above to generate W*-completions.

Lemma 17 Suppose that ${(\mathcal A,p)}$ is a bounded *-probability space with GNS representation ${(\mathcal H,\pi,\xi)}$. Let ${\bar{\mathcal A}\subseteq B(\mathcal H)}$ be the weak closure of ${\pi(\mathcal A)}$ and define the state ${\bar p\colon\bar{\mathcal A}\rightarrow{\mathbb C}}$ by (3).

Then, ${(\bar{\mathcal A},\bar p)}$ is a W*-probability space and ${\pi}$ defines a homomorphism from ${(\mathcal A,p)}$ to ${(\bar{\mathcal A},\bar p)}$ which, furthermore, is a W*-completion.

Proof: Lemma (15) says that ${(\bar{\mathcal A},\bar p)}$ is a W*-probability space. Furthermore ${\pi}$ is, by definition, a state preserving *-homomorphism from ${\mathcal A}$ to ${\bar{\mathcal A}}$. Finally, lemma 14 says that the ultraweak topology on ${\bar{\mathcal A}}$ defined by ${\bar p}$ coincides with the ultraweak topology defined by the action on ${\mathcal H}$, with respect to which ${\pi(\mathcal A)}$ is dense by lemma 12. So, ${\pi}$ 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 ${L^\infty}$-dense image. I now give a W* version of this, where any of the equivalent statements of lemma 16 can be used.

Lemma 18 A normal homomorphism ${\varphi\colon(\mathcal A,p)\rightarrow(\mathcal A,p^\prime)}$ of W*-probability spaces is an isomorphism if and only if ${\varphi(\mathcal A_1)}$ is a weakly dense subset of ${\mathcal A^\prime_1}$.

Proof: First, if ${\varphi}$ is an isomorphism then ${\varphi(\mathcal A_1)=\mathcal A^\prime_1}$ is certainly dense. Conversely, suppose that ${\varphi(\mathcal A_1)}$ is dense. As ${\varphi}$ is weakly continuous, and ${\mathcal A_1}$ is weakly complete, it follows that ${\varphi(\mathcal A_1)}$ is a complete and dense subset of ${\mathcal A^\prime_1}$ and, hence, is all of ${\mathcal A^\prime_1}$. So, by scaling, ${\varphi(\mathcal A)=\mathcal A^\prime}$. Finally, ${\varphi}$ is one-to-one as ${p}$ is nondegenerate (by lemma 1 of the homomorphism post, if ${\varphi(a)=0}$ then ${\lVert a\rVert_\infty=0}$ and, by nondegeneracy, ${a=0}$). ⬜

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 19 Let ${\varphi}$ be a normal homomorphism between bounded *-probability spaces ${(\mathcal A,p)}$ and ${(\mathcal A^\prime,p^\prime)}$. Let

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \pi\colon(\mathcal A,p)\rightarrow(\bar{\mathcal A},\bar p),\smallskip\\ &\displaystyle \pi^\prime\colon(\mathcal A^\prime,p^\prime)\rightarrow(\bar{\mathcal A}^\prime,\bar p^\prime). \end{array}$

be W*-completions. Then, there is a unique normal homomorphism ${\bar\varphi\colon(\bar{\mathcal A},\bar p)\rightarrow(\bar{\mathcal A}^\prime,\bar p^\prime)}$ such that ${\bar\varphi\circ\pi=\pi^\prime\circ\varphi}$. Furthermore, ${\bar\varphi}$ is an isomorphism iff ${\varphi(\mathcal A_1)}$ is weakly dense in ${\mathcal A^\prime_1}$.

Proof: As ${\bar\varphi}$ is ultraweakly continuous, it is uniquely determined by its values on the ultraweakly dense ${\pi(\mathcal A)\subseteq\bar{\mathcal A}}$ (lemma 16). In turn, it is uniquely detemined on ${\pi(\mathcal A)}$ by ${\bar\varphi(\pi(a))=\pi^\prime(\varphi(a))}$. So, ${\bar\varphi}$ is uniquely determined.

We need to show existence of ${\bar\varphi}$. As ${\pi}$ and ${\pi^\prime}$ are isometries, (4) holds. So, ${\pi^\prime(\varphi(a))=0}$ whenever ${\pi(a)=0}$ and, as in the proof of theorem 5, we can define the *-homomorphism ${\varphi_0\colon\pi(\mathcal A)\rightarrow\bar{\mathcal A}^\prime}$ by (5), which will be al ${L^\infty}$-isometry.

As ${\pi}$ and ${\varphi}$ are normal, they are weakly continuous on ${L^\infty}$-bounded sets and, hence, so is ${\varphi_0}$. For any ${r > 0}$, ${\varphi_0}$ maps ${\pi(\mathcal A)\cap\bar{\mathcal A}_r}$ into ${\bar{\mathcal A}_r}$, which is weakly complete. As ${\pi(\mathcal A_r)}$ is weakly dense in ${\bar{\mathcal A}_r}$ by assumption, ${\varphi_0}$ extends uniquely to a weakly continuous map from ${\bar{\mathcal A}_r}$ into ${\bar{\mathcal A}^\prime}$. As ${\bar{\mathcal A}=\bigcup_{r > 0}\bar{\mathcal A}_r}$, this uniquely defines a map

$\displaystyle \bar\varphi\colon\bar{\mathcal A}\rightarrow\bar{\mathcal A}^\prime$

satisfying ${\bar\varphi\circ\pi=\pi^\prime\circ\varphi}$, and which is weakly continuous on ${\bar{\mathcal A}_r}$. By weak continuity of the algebra operations, this is a normal *-homomorphism. Then, as ${\varphi_0}$ preserves the state, and the states ${\bar p,\bar p^\prime}$ are ultraweakly continuous, it follows that ${\bar\varphi}$ preserves the state and is a normal homomorphism of *-probability spaces.

Finally, we show that ${\bar\varphi}$ is an isomorphism if and only if ${\varphi(\mathcal A_1)}$ is weakly dense in ${\mathcal A^\prime_1}$. If it is an isomorphism, then ${\bar\varphi(\bar{\mathcal A}_1)=\bar{\mathcal A}^\prime_1}$ and, as ${\pi(\mathcal A_1)}$ is weakly dense in ${\bar{\mathcal A}_1}$, it follows that ${\pi^\prime(\varphi(\mathcal A_1))=\bar\varphi(\pi(\mathcal A_1))}$ is weakly dense in ${\bar{\mathcal A}^\prime_1}$. Hence, ${\varphi(\mathcal A_1)}$ is weakly dense in ${\mathcal A^\prime_1}$.

Conversely, suppose that ${\varphi(\mathcal A_1)}$ is weakly dense in ${\mathcal A^\prime}$. Then, ${\bar\varphi(\pi(\mathcal A_1))=\pi^\prime(\varphi(\mathcal A_1))}$ is weakly dense in ${\bar{\mathcal A}^\prime_1}$ and, hence, so is ${\bar\varphi(\bar{\mathcal A}_1)}$. By lemma 18, ${\bar\varphi}$ 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 20 Let ${(\mathcal A,p)}$ be a bounded *-probability space. Then, it has a W*-completion, which is unique up to isomorphism. That is, for any two W*-completions

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \pi\colon(\mathcal A,p)\rightarrow(\bar{\mathcal A},\bar p),\smallskip\\ &\displaystyle \pi^\prime\colon(\mathcal A,p)\rightarrow(\mathcal A^\prime,p^\prime) \end{array}$

there exists a unique isomorphism ${\varphi\colon(\bar{\mathcal A},\bar p)\rightarrow(\mathcal A^\prime,p^\prime)}$ such that ${\pi^\prime=\varphi\circ\pi}$.

Proof: Lemma 17 gives existence of the W*-completion, so only uniqueness remains. Let ${\pi,\pi^\prime}$ be as in the statement of the theorem, and ${\iota}$ be the identity automorphism on ${(\mathcal A,p)}$. Theorem 19 states the existence of a unique normal isomorphism ${\varphi}$ from ${(\bar{\mathcal A},\bar p)}$ to ${(\mathcal A^\prime,p^\prime)}$ satisfying

$\displaystyle \pi^\prime=\pi^\prime\circ\iota=\varphi\circ\pi$

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 21 Let ${(\mathcal A,p)}$ be a bounded *-probability space, and ${(\mathcal H,\pi,\xi)}$ be its GNS representation. Let ${\bar{\mathcal A}\subseteq B(\mathcal H)}$ be the weak closure of ${\pi(\mathcal A)}$ and define ${\bar p\colon\bar{\mathcal A}\rightarrow{\mathbb C}}$ by

$\displaystyle \bar p(a)=\langle\xi,a\xi\rangle.$

Then,

• ${(\bar{\mathcal A},\bar p)}$ is a W*-probability space.
• ${\pi\colon(\mathcal A,p)\rightarrow(\bar{\mathcal A},\bar p)}$ is the W*-completion of ${(\mathcal A,p)}$.
• ${(\mathcal A,p)}$ is a W*-probability space if and only if ${\pi}$ is an isomorphism between ${\mathcal A}$ and ${\bar{\mathcal A}}$.

Proof: The first two statements are already shown in lemma 17 above. Also, if ${\pi}$ is an isomorphism from ${\mathcal A}$ to ${\bar{\mathcal A}}$ then ${(\mathcal A,p)}$ is isomorphic to ${(\bar{\mathcal A},\bar p)}$, so is a W*-probability space. It only remains to show that, if ${(\mathcal A,p)}$ is a W*-probability space then ${\pi}$ is an isomorphism. However, ${\pi(\mathcal A_1)}$ is ultraweakly dense in ${\bar{\mathcal A}_1}$ by definition of the W*-completion, so ${\pi}$ 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 ${V}$, I will use ${V}$-weak (${V}$-strong, etc) to denote the weak (strong, etc) topology defined by this action. Similarly for a state ${p}$, I use ${p}$-weak (${p}$-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 ${L^2(p)}$ semi-inner product.

Lemma 22 Let ${\mathcal A}$ be a von Neumann algebra on Hilbert space ${\mathcal H}$ and ${p\colon\mathcal A\rightarrow{\mathbb C}}$ be a normal state. Then, the ${p}$-ultraweak (resp., ${p}$-ultrastrong) topology is weaker than the ${\mathcal H}$-ultraweak (resp., ${\mathcal H}$-ultrastrong). If the state is nondegenerate, the ultraweak and ultrastrong topologies defined by ${p}$ coincide with those defined by the action on ${\mathcal H}$.

Proof: Let ${V=\mathcal A}$ have the inner product ${\langle x,y\rangle=p(x^*y)}$, and ${\mathcal A}$ acts on this by left-multiplication. Define the homomorphism of *-algebra representations ${\varphi\colon(\mathcal A,\mathcal H)\rightarrow(\mathcal A,V)}$ to be the identity on ${\mathcal A}$. By assumption, ${p}$ is normal, so is weakly continuous on the unit ball (defined by the operator norm of the action of ${\mathcal H}$). Consider ${x,y\in\mathcal A}$ and a net ${a_\alpha\in\mathcal A_1}$ tending weakly to zero (w.r.t. the action on ${\mathcal H}$). Then, ${x^*a_\alpha y}$ is a norm-bounded net tending weakly to zero and, hence, ${\langle x,a_\alpha y\rangle=p(x^*a_\alpha y)\rightarrow0}$, showing that ${\varphi}$ is normal and, hence, is ultraweakly and ultrastrongly continuous. This implies that the ultraweak (resp., ultrastrong) topology defined w.r.t. the action on ${\mathcal H}$ is stronger than with the action on ${V}$.

Now suppose that ${p}$ is nondegenerate. By lemma 8, ${\varphi}$ is an isometry, so gives a continuous bijective map from ${\mathcal A_1}$ under the ${\mathcal H}$-weak topology, which is compact by lemma 13, onto ${\mathcal A_1}$ under the ${V}$-weak topology, which is Hausdorff by nondegeneracy. Hence, it gives a homeomorphism of the unit ball, so ${\varphi^{-1}}$ is normal and is ultraweakly and ultrastrongly continuous. This implies that the ultraweak (resp., ultrastrong) topology defined w.r.t. the action on ${V}$ is stronger than w.r.t. the action on ${\mathcal H}$. ⬜

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 23 The pair ${(\mathcal A,p)}$ is a W*-probability space if and only if ${\mathcal A}$ is a von Neumann algebra with respect to which ${p}$ is a nondegenerate normal state.

Proof: First, suppose that ${(\mathcal A,p)}$ is a W*-probability space. By theorem 21, ${\pi}$ is a *-isomorphism between ${\mathcal A}$ and the weakly closed ${\bar{\mathcal A}\subseteq B(\mathcal H)}$ which, by lemma 12, is a unitial *-subalgebra. This represents ${\mathcal A}$ as a von Neumann algebra on ${\mathcal H}$, with respect to which ${p(a)=\langle\xi,\pi(a)\xi\rangle}$ is normal and nondegerate.

Conversely, suppose that ${\mathcal A}$ is a von Neumann algebra on Hilbert space ${\mathcal H}$ and that ${p\colon\mathcal A\rightarrow{\mathbb C}}$ is a nondegenerate normal state. The ${L^\infty}$ norm and the ultraweak topology defined by ${p}$ coincides with that defined by the action on ${\mathcal H}$ (lemmas 8 and 22). Hence, the unit ball is ultraweakly complete (lemma 13), so ${(\mathcal A,p)}$ 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 24 Let ${\mathcal A}$ be a von Neumann algebra and ${p\colon\mathcal A\rightarrow{\mathbb C}}$ be a normal state. Then, ${(\mathcal A/\mathcal N,p^\prime)}$ is a W*-probability space.

Proof: As ${\mathcal N}$ is just the ${p}$-ultraweak closure of the point ${\{0\}}$, and the ${\mathcal H}$-ultraweak topology is stronger than the ${p}$-ultraweak (lemma 22), it follows that ${\mathcal N}$ is ${\mathcal H}$-ultraweakly closed,

Suppose that ${\mathcal A}$ is a von Neumann algebra on Hilbert space ${\mathcal H}$. It is standard that the quotient ${\mathcal A/\mathcal N}$ by an ultraweakly closed *-ideal is again a von Neumann algebra. Specifically, every such ideal is generated by a central projection ${P}$ (Blackadar III.1.13). That is, ${\mathcal N=\mathcal AP}$, where ${P=P^*P}$ is a projection which commutes with all elements of the algebra. Then, ${\mathcal A/\mathcal N}$ is *-isomorphic to the ultraweakly closed complementary ideal ${\mathcal N^\prime=\mathcal A(1-P)}$ and, hence, is a von Neumann algebra acting on the closed subspace ${(1-P)\mathcal H\subseteq\mathcal H}$. The *-homomorphism ${\mathcal A\rightarrow\mathcal N^\prime}$ given by multiplication by ${1-P}$ has kernel ${\mathcal N}$, so generates the isomorphism ${\mathcal A/\mathcal N\rightarrow\mathcal N^\prime}$. The state ${p^\prime}$ is given by the restriction of ${p}$ to ${\mathcal N^\prime}$ and, hence, is also normal. So, ${p^\prime}$ is a normal state on ${\mathcal A/\mathcal N}$, and the result follows from theorem 23. ⬜