# Noncommutative Probability Spaces

In classical probability theory, we start with a sample space ${\Omega}$, a collection ${\mathcal F}$ of events, which is a sigma-algebra on ${\Omega}$, and a probability measure ${{\mathbb P}}$ on ${(\Omega,\mathcal F)}$. The triple ${(\Omega,\mathcal F,{\mathbb P})}$ is a probability space, and the collection ${L^\infty(\Omega,\mathcal F,{\mathbb P})}$ of bounded complex-valued random variables on the probability space forms a commutative algebra under pointwise addition and products. The measure ${{\mathbb P}}$ defines an expectation, or integral with respect to ${{\mathbb P}}$, which is a linear map

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle L^\infty(\Omega,\mathcal F,{\mathbb P})\rightarrow{\mathbb C},\smallskip\\ &\displaystyle X\mapsto{\mathbb E}[X]=\int X(\omega)d{\mathbb P}(\omega). \end{array}$

In this post I provide definitions of probability spaces from the algebraic viewpoint. Statements of some of their first properties will be given in order to justify and clarify the definitions, although any proofs will be left until later posts. In the algebraic setting, we begin with a *-algebra ${\mathcal A}$, which takes the place of the collection of bounded random variables from the classical theory. It is not necessary for the algebra to be represented as a space of functions from an underlying sample space. Since the individual points ${\omega\in\Omega}$ constituting the sample space are not required in the theory, this is a pointless approach. By allowing multiplication of random variables’ in ${\mathcal A}$ to be noncommutative, we incorporate probability spaces which have no counterpart in the classical setting, such as are used in quantum theory. The second and final ingredient is a state on the algebra, taking the place of the classical expectation operator. This is a linear map ${p\colon\mathcal A\rightarrow{\mathbb C}}$ satisfying the positivity constraint ${p(a^*a)\ge1}$ and, when ${\mathcal A}$ is unitial, the normalisation condition ${p(1)=1}$. Algebraic, or noncommutative probability spaces are completely described by a pair ${(\mathcal A,p)}$ consisting of a *-algebra ${\mathcal A}$ and a state ${p}$. Noncommutative examples include the *-algebra of bounded linear operators on a Hilbert space with pure state ${p(a)=\langle\xi,a\xi\rangle}$ for a fixed unit vector ${\xi}$. Continue reading “Noncommutative Probability Spaces”

# The GNS Representation

As is well known, the space of bounded linear operators on any Hilbert space forms a *-algebra, and (pure) states on this algebra are defined by unit vectors. Considering a Hilbert space ${\mathcal H}$, the space of bounded linear operators ${\mathcal H\rightarrow\mathcal H}$ is denoted as ${B(\mathcal H)}$. This forms an algebra under the usual pointwise addition and scalar multiplication operators, and involution of the algebra is given by the operator adjoint,

$\displaystyle \langle x,a^*y\rangle=\langle ax,y\rangle$

for any ${a\in B(\mathcal H)}$ and all ${x,y\in\mathcal H}$. A unit vector ${\xi\in\mathcal H}$ defines a state ${p\colon B(\mathcal H)\rightarrow{\mathbb C}}$ by ${p(a)=\langle\xi,a\xi\rangle}$.

The Gelfand-Naimark–Segal (GNS) representation allows us to go in the opposite direction and, starting from a state on an abstract *-algebra, realises this as a pure state on a *-subalgebra of ${B(\mathcal H)}$ for some Hilbert space ${\mathcal H}$.

Consider a *-algebra ${\mathcal A}$ and positive linear map ${p\colon\mathcal A\rightarrow{\mathbb C}}$. Recall that this defines a semi-inner product on the *-algebra ${\mathcal A}$, given by ${\langle x,y\rangle=p(x^*y)}$. The associated seminorm is denoted by ${\lVert x\rVert_2=\sqrt{p(x^*x)}}$, which we refer to as the ${L^2}$-seminorm. Also, every ${a\in\mathcal A}$ defines a linear operator on ${\mathcal A}$ by left-multiplication, ${x\mapsto ax}$. We use ${\lVert a\rVert_\infty}$ to denote its operator norm, and refer to this as the ${L^\infty}$-seminorm. An element ${a\in\mathcal A}$ is bounded if ${\lVert a\rVert_\infty}$ is finite, and we say that ${(\mathcal A,p)}$ is bounded if every ${a\in\mathcal A}$ is bounded.

Theorem 1 Let ${(\mathcal A,p)}$ be a bounded *-probability space. Then, there exists a triple ${(\mathcal H,\pi,\xi)}$ where,

• ${\mathcal H}$ is a Hilbert space.
• ${\pi\colon\mathcal A\rightarrow B(\mathcal H)}$ is a *-homomorphism.
• ${\xi\in\mathcal H}$ satisfies ${p(a)=\langle\xi,\pi(a)\xi\rangle}$ for all ${a\in\mathcal A}$.
• ${\xi}$ is cyclic for ${\mathcal A}$, so that ${\{\pi(a)\xi\colon a\in\mathcal A\}}$ is dense in ${\mathcal H}$.

Furthermore, this representation is unique up to isomorphism: if ${(\mathcal H^\prime,\pi^\prime,\xi^\prime)}$ is any other such triple, then there exists a unique invertible linear isometry of Hilbert spaces ${L\colon\mathcal H\rightarrow\mathcal H^\prime}$ such that

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \pi^\prime(a)=L\pi(a)L^{-1},\smallskip\\ &\displaystyle \xi^\prime=L\xi. \end{array}$

# Normal Maps

Given two *-probability spaces ${(\mathcal A,p)}$ and ${(\mathcal A^\prime,p^\prime)}$, we want to consider maps ${\varphi\colon\mathcal A\rightarrow\mathcal A^\prime}$. For example, we can look homomorphisms, which preserve the *-algebra operations, and can also consider restricting to state-preserving maps satisfying ${p^\prime(\varphi(a))=p(a)}$. In algebraic probability theory, however, it is often necessary to include a continuity condition, leading to the idea of normal maps, which I look at in this post. In fact, as we will see, all *-homomorphisms between commutative probability spaces which preserve the state are normal, so this concept is most important in the noncommutative setting.

In contrast to the previous few posts on algebraic probability, the current post is a bit of a gear-change. We are still concerned with with the basic concepts of *-algebras and states. However, the main theorem stated below, which reduces to the Radon-Nikodym theorem in the commutative case, is deeper and much more difficult to prove than the relatively simple results with which I have been concerned with so far. Continue reading “Normal Maps”

# Operator Topologies

We previously defined the notion of positive linear maps and states on *-algebras, and noted that there always exists seminorms defining the ${L^2}$ and ${L^\infty}$ topologies. However, for applications to noncommutative probability theory, these are often not the most convenient modes of convergence to be using. Instead, the weak, strong, ultraweak and ultrastrong operator topologies can be used. This, rather technical post, is intended to introduce these concepts and prove their first properties.

Weak convergence on a *-probability space ${(\mathcal A,p)}$ is straightforward to define. A net ${a_\alpha\in\mathcal A}$ tends weakly to the limit ${a}$ if and only if ${p(xa_\alpha y)\rightarrow p(xay)}$ for all ${x,y\in\mathcal A}$. Continue reading “Operator Topologies”

# Homomorphisms of *-Probability Spaces

I previously introduced the concept of a *-probability space as a pair ${(\mathcal A,p)}$ consisting of a state ${p}$ on a *-algebra ${\mathcal A}$. As we noted, this concept is rather too simplistic to properly capture a noncommutative generalisation of classical probability spaces, and I will later give conditions for ${(\mathcal A,p)}$ to be considered as a true probability space. For now, I continue the investigation of these preprobability spaces, and will look at homomorphisms in this post.

A *-homomorphism between *-algebras ${\mathcal A}$ and ${\mathcal A^\prime}$ is a map ${\varphi\colon\mathcal A\rightarrow\mathcal A^\prime}$ preserving the algebra operations,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle \varphi(\lambda a+\mu b)=\lambda\varphi(a)+\mu\varphi(b),\smallskip\\ &\displaystyle \varphi(ab)=\varphi(a)\varphi(b),\smallskip\\ &\displaystyle \varphi(a^*)=\varphi(a)^*, \end{array}$

for all ${a,b\in\mathcal A}$ and ${\lambda,\mu\in{\mathbb C}}$. The term *-homomorphism’ is used to distinguish it from the concept of simple algebra homomorphisms which need not preserve the involution (the third identity above). Next, I will say that ${\varphi}$ is a homomorphism of *-probability spaces ${(\mathcal A,p)}$ and ${(\mathcal A^\prime,p^\prime)}$ if it is a *-homomorphism from ${\mathcal A}$ to ${\mathcal A^\prime}$ which preserves the state,

$\displaystyle p^\prime(\varphi(a))=p(a),$

for all ${a\in\mathcal A}$.

Now, recall that for any *-probability space ${(\mathcal A,p)}$, we define a semi-inner product ${\langle x,y\rangle=p(x^*y)}$ on ${\mathcal A}$ and the associated ${L^2(p)}$ seminorm, ${\lVert x\rVert_2=\sqrt{p(x^*x)}}$. Homomorphisms of *-probability spaces are clearly ${L^2}$-isometries,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle \langle\varphi(x),\varphi(y)\rangle&\displaystyle=p^\prime\left(\varphi(x)^*\varphi(y)\right)=p^\prime\left(\varphi(x^*y)\right)\smallskip\\ &\displaystyle=p(x^*y)=\langle x,y\rangle. \end{array}$

For each ${a\in\mathcal A}$, the ${L^\infty(p)}$ seminorm ${\lVert a\rVert_\infty}$ is defined as the operator norm of the left-multiplication map ${x\mapsto ax}$ on ${\mathcal A}$, considered as a vector space with the ${L^2}$ seminorm. Homomorphisms of *-probability spaces do not need to be ${L^\infty}$-isometric.

Lemma 1 If ${\varphi\colon(\mathcal A,p)\rightarrow(\mathcal A^\prime,p^\prime)}$ is a homomorphism of *-probability spaces then, for any ${a\in\mathcal A}$,

 $\displaystyle \lVert\varphi(a)\rVert_\infty\ge\lVert a\rVert_\infty.$ (1)

# States on *-Algebras

So far, we have been considering positive linear maps on a *-algebra. Taking things a step further, we want to consider positive maps which are normalized so as to correspond to expectations under a probability measure. That is, we require ${p(1)=1}$, although this is only defined for unitial algebras. I use the definitions and notation of the previous post on *-algebras.

Definition 1 A state on a unitial *-algebra ${\mathcal A}$ is a positive linear map ${p\colon\mathcal A\rightarrow{\mathbb C}}$ satisfying ${p(1)=1}$.

Examples 3 and 4 of the previous post can be extended to give states.

Example 1 Let ${(X,\mathcal E,\mu)}$ be a probability space, and ${\mathcal A}$ be the bounded measurable maps ${X\rightarrow{\mathbb C}}$. Then, integration w.r.t. ${\mu}$ defines a state on ${\mathcal A}$,

$\displaystyle p(f)=\int f d\mu.$

Example 2 Let ${V}$ be an inner product space, and ${\mathcal A}$ be a *-algebra of the space of linear maps ${a\colon V\rightarrow V}$ as in example 2 of the previous post, and including the identity map ${I}$. Then, any ${\xi\in V}$ with ${\lVert\xi\rVert=1}$ defines a state on ${\mathcal A}$,

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

# *-Algebras

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 ${\bar\lambda}$ to denote the complex conjugate of a complex number ${\lambda}$.

Definition 1 An algebra ${\mathcal A}$ over field ${K}$ is a ${K}$-vector space together with a binary product ${(a,b)\mapsto ab}$ satisfying

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle a(bc)=(ab)c,\smallskip\\ &\displaystyle \lambda(ab)=(\lambda a)b=a(\lambda b)\smallskip\\ &\displaystyle a(b+c)=ab+ac,\smallskip\\ &\displaystyle (a+b)c=ac+bc, \end{array}$

for all ${a,b,c\in\mathcal A}$ and ${\lambda\in K}$.

A *-algebra ${\mathcal A}$ is an algebra over ${{\mathbb C}}$ with a unary involution, ${a\mapsto a^*}$ satisfying

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle (\lambda a+\mu b)^*=\bar\lambda a^*+\bar\mu b^*,\smallskip\\ &\displaystyle (ab)^*=b^*a^*\smallskip\\ &\displaystyle a^{**}=a. \end{array}$

for all ${a,b,c\in\mathcal A}$ and ${\lambda,\mu\in{\mathbb C}}$.

An algebra is called unitial if there exists ${1\in\mathcal A}$ such that

$\displaystyle 1a=a1=a$

for all ${a\in\mathcal A}$. Then, ${1}$ is called the unit or identity of ${\mathcal A}$.

# Algebraic Probability: Quantum Theory

We continue the investigation of representing probability spaces as states on algebras. Whereas, previously, I focused attention on the commutative case and on classical probabilities, in the current post I will look at non-commutative quantum probability.

Quantum theory is concerned with computing probabilities of outcomes of measurements of a physical system, as conducted by an observer. The standard approach is to start with a Hilbert space ${\mathcal H}$, which is used to represent the states of the system. This is a vector space over the complex numbers, together with an inner product ${\langle\cdot,\cdot\rangle}$. By definition, this is linear in one argument and anti-linear in the other,

$\displaystyle \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle\langle\phi,\lambda\psi+\mu\chi\rangle=\lambda\langle\phi,\psi\rangle+\mu\langle\phi,\chi\rangle,\smallskip\\ &\displaystyle\langle\lambda\phi+\mu\psi,\chi\rangle=\bar\lambda\langle\phi,\chi\rangle+\bar\mu\langle\psi,\chi\rangle,\smallskip\\ &\displaystyle\langle\psi,\phi\rangle=\overline{\langle\phi,\psi\rangle}, \end{array}$

for ${\phi,\psi,\chi\in\mathcal H}$ and ${\lambda,\mu\in{\mathbb C}}$. Positive definiteness is required, so that ${\langle\psi,\psi\rangle > 0}$ for ${\psi\not=0}$. I am using the physicists’ convention, where the inner product is linear in the second argument and anti-linear in the first. Furthermore, physicists often use the bra–ket notation ${\langle\phi\vert\psi\rangle}$, which can be split up into the bra’ ${\langle\phi\vert}$ and ket’ ${\vert\psi\rangle}$ considered as elements of the dual space of ${\mathcal H}$ and of ${\mathcal H}$ respectively. For a linear operator ${A\colon\mathcal H\rightarrow\mathcal H}$, the expression ${\langle\phi,A\psi\rangle}$ is often expressed as ${\langle\phi\vert A\vert\psi\rangle}$ in the physicists’ language. By the Hilbert space definition, ${\mathcal H}$ is complete with respect to the norm ${\lVert\psi\rVert=\sqrt{\langle\psi,\psi\rangle}}$. Continue reading “Algebraic Probability: Quantum Theory”

# Algebraic Probability (continued)

Continuing on from the previous post, I look at cases where the abstract concept of states on algebras correspond to classical probability measures. Up until now, we have considered commutative real algebras but, before going further, it will help to look instead at algebras over the complex numbers ${{\mathbb C}}$. In the commutative case, we will see that this is equivalent to using real algebras, but can be more convenient, and in the non-commutative case it is essential. When using complex algebras, we will require the existence of an involution, which can be thought of as a generalisation of complex conjugation.

Recall that, by an algebra ${\mathcal A}$ over a field ${K}$, we mean that ${\mathcal A}$ is a ${K}$-vector space together with a binary product operation satisfying associativity, distributivity over addition, compatibility with scalars, and which has a multiplicative identity.

Definition 1 A *-algebra ${\mathcal A}$ is an algebra over ${{\mathbb C}}$ together with an involution, which is a unary operator ${\mathcal A\rightarrow\mathcal A}$, ${a\mapsto a^*}$, satisfying,

1. Anti-linearity: ${(\lambda a+\mu b)^*=\bar\lambda a^*+\bar\mu b^*}$.
2. ${(ab)^*=b^*a^*}$.
3. ${a^{**}=a}$

for all ${a,b\in\mathcal A}$ and ${\lambda,\mu\in{\mathbb C}}$.

# Algebraic Probability

The aim of this post is to motivate the idea of representing probability spaces as states on a commutative algebra. We will consider how this abstract construction relates directly to classical probabilities.

In the standard axiomatization of probability theory, due to Kolmogorov, the central construct is a probability space ${(\Omega,\mathcal F,{\mathbb P})}$. This consists of a state space ${\Omega}$, an event space ${\mathcal F}$, which is a sigma-algebra of subsets of ${\Omega}$, and a probability measure ${{\mathbb P}}$. The measure ${{\mathbb P}}$ is defined as a map ${{\mathbb P}\colon\mathcal F\rightarrow{\mathbb R}^+}$ satisfying countable additivity and normalised as ${{\mathbb P}(\Omega)=1}$.

A measure space allows us to define integrals of real-valued measurable functions or, in the language of probability, expectations of random variables. We construct the set ${L^\infty(\Omega,\mathcal F)}$ of all bounded measurable functions ${X\colon\Omega\rightarrow{\mathbb R}}$. This is a real vector space and, as it is closed under multiplication, is an algebra. Expectation, by definition, is the unique linear map ${L^\infty\rightarrow{\mathbb R}}$, ${X\mapsto{\mathbb E}[X]}$ satisfying ${{\mathbb E}[1_A]={\mathbb P}(A)}$ for ${A\in\mathcal F}$ and monotone convergence: if ${X_n\in L^\infty}$ is a nonnegative sequence increasing to a bounded limit ${X}$, then ${{\mathbb E}[X_n]}$ tends to ${{\mathbb E}[X]}$.

In the opposite direction, any nonnegative linear map ${p\colon L^\infty(\Omega,\mathcal F)\rightarrow{\mathbb R}}$ satisfying monotone convergence and ${p(1)=1}$ defines a probability measure by ${{\mathbb P}(A)=p(1_A)}$. This is the unique measure with respect to which expectation agrees with the linear map, ${{\mathbb E}=p}$. So, probability measures are in one-to-one correspondence with such linear maps, and they can be viewed as one and the same thing. The Kolmogorov definition of a probability space can be thought of as representing the expectation on the subset of ${L^\infty}$ consisting of indicator functions ${1_A}$. In practice, it is often more convenient to start with a different subset of ${L^\infty}$. For example, probability measures on ${{\mathbb R}^+}$ can be defined via their Laplace transform, ${\mathcal L_{{\mathbb P}}(a)=\int e^{-ax}d{\mathbb P}(x)}$, which represents the expectation on exponential functions ${x\mapsto e^{-ax}}$. Generalising to complex-valued random variables, probability measures on ${{\mathbb R}}$ are often represented by their characteristic function ${\varphi(a)=\int e^{iax}d{\mathbb P}(x)}$, which is just the expectation of the complex exponentials ${x\mapsto e^{iax}}$. In fact, by the monotone class theorem, we can uniquely represent probability measures on ${(\Omega,\mathcal F)}$ by the expectations on any subset ${\mathcal K\subseteq L^\infty}$ which is closed under taking products and generates the sigma-algebra ${\mathcal F}$. Continue reading “Algebraic Probability”