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”

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)

Continue reading “Homomorphisms of *-Probability Spaces”

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.

Continue reading “States on *-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}.

Continue reading “*-Algebras”

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”