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. Continue reading “Completions of *-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}

Continue reading “The GNS Representation”

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”

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}}.

Continue reading “Algebraic Probability (continued)”