Spin Chronicles Part 42: GNS Promenade

 Promenade: /ˌprɑm·əˈneɪd, -ˈnɑd/ to walk slowly along a street or path, usually where you can be seen by many people, for relaxation and pleasure. (Definition of promenade from the Cambridge Academic Content Dictionary.

In this post I am taking my Readers for a walk along the GNS construction that we have discussed in a number of previous posts Part 36-Part 41.

It will be a lazy promenade. Walking is important for philosophers. From Five philosophers on the joys of walking

"For Rousseau, the great benefit of walking is that you can move at your own time, doing as much or as little as you choose. You can see the country you’re traveling through, turn off to the right or left if you fancy, examine anything which interests you."

"Santayana points out that plants cannot move, whilst animals can.(...) Moving around allows animals to experience more of the world, to imagine how it might be elsewhere."

You can see the country you’re traveling through,
turn off to the right or left if you fancy,
examine anything which interests you

While walking we may recognize shapes and forms that are somewhat familiar to us, but there is always something new that enriches our knowledge and understanding of the reality. During the promenade we may notice other people, who look and dress differently. We were discussing the GNS construction. So let's see how it is being treated by mathematicians in an advanced textbook. The particular textbook I am inviting you to is "Fundamentals of the Theory of Operator Algebras, Volume 1", by Richard V. Kadison and John R. Ringrose, Academic Press 1983. It is a very nice book.

Operator algebras on promenade

It starts with Linear Spaces in Chapter 1, then, in Chapter 2, we have Basics of Hilbert Space and Linear Operators. Chapter 3 treats Banach Algebras. Chapter 4 discusses Elementary C*-Algebra Theory. In Ch. 4.5, p. 275,  we finally find the landscape that we are looking for "States and representations". On p. 277 we find, in Proposition 4.51,  something that looks familiar:

So, let's take a look.

Here the letter ρ is used for state. We used letter f. Our algebra A is denoted by a symbol like 𝒜, that i don't know how to reproduce using available fonts. Then we have a left ideal L_ρ, that we have denoted I_f in Part 40.

So, it is just a different notation, but nothing new for us. Of course the book has an adult C*-algebra, that is it works also for infinite-dimensional algebras, while we were playing with a finite-dimensional toy.

Let us walk a little bit further, on the next page we find Theorem 4.5.2 - the GNS construction. 

We see the term "cyclic representation". That means *-representation with a cyclic vector. We know what a cyclic vector vector is. We checked that "our" representation is cyclic in  Part 40. The book denotes the representation π_ρ, we called it ρ_f. The book calls the Hilbert space ℋ_ρ, we called it H_f. The book calls the cyclic vector xρ, we called it  Ω_f . Not a big deal. Finally, the last formula on the theorem is similar to our

(Ω, ρ(a)Ω) = f(a),

but the book has a different order under scalar product. That is because in the book the scalar product is assumed to be linear in the first argument and anti-linear in the second (the convention used by mathematicians), while our scalar product is anti-linear in the first argument and linear in the second (the convention used by physicists). Easy to remember.

So, thus far, nothing really new, just a different dress. Let us therefore move to the next picture at the exhibition. On p. 279 we meet Proposition 4.5.3:

This is something new. We need to stop here and examine it. So let us rewrite it using our notation. A is our finite-dimensional *-algebra. f is a state on A, ρ is a GNS cyclic representation of A on a Hilbert space H, with cyclic vector Ω, such that

f(a) = (Ω,ρ(a)Ω).                            (1)

Then, suppose we have some ρ',H',Ω' with the same properties. Then, the theorem states:

There is an isomorphism of Hilbert spaces U, from H to H', such that

UΩ = Ω',                                    (2)

and

ρ'(a) = Uρ(a)U^-1                        (3)

for all a in A.

The book uses U* instead of U^-1, but it has the same meaning.

This is a new result for us, and it needs a proof. It says that the GNS construction is essentially unique, up to a unitary equivalence. So, let us see about the proof. I will follow the reasoning in the book, with comments.

We start with establishing the equality that will be used to construct U. The equality is

||ρ(a)Ω|| = ||ρ'(a)Ω'||   for all a in A.        (4)

On the left we have the norm in H, on the right we have the norm in H'. To see that the equality indeed holds we use (1), which by assumption holds for ρ,H,Ω and for ρ',H',Ω':

||ρ(a)Ω||² = (ρ(a)Ω,ρ(a)Ω) = (Ω,ρ(a)*ρ(a)Ω) = (Ω,ρ(a*)ρ(a)Ω) = (Ω,ρ(a*a)Ω) = f(a*a),
||ρ'(a)Ω'||² = (ρ'(a)Ω',ρ'(a)Ω') = (Ω',ρ'(a)*ρ'(a)Ω) = (Ω',ρ'(a*)ρ'(a)Ω') = (Ω',ρ'(a*a)Ω') = f(a*a).

So, we have (4). We will now define U, a map from H to H'. We choose a vector in H. Since Ω is cyclic for ρ in H, there is an element a in A such that our chosen vector is of the form  ρ(a)Ω. In general there will be more than one such an "a", so we will need to be careful. We define U acting on ρ(a)Ω to be, by definition ρ'(a)Ω'.

Uρ(a)Ω = ρ'(a)Ω.

Now, does it define a map from H to H'? What if ρ(a)Ω=ρ(b)Ω? Well, then we have
0=ρ(a)Ω-ρ(b)Ω=ρ(a-b)Ω.

But  then, using (4), 0= ||ρ(a-b)Ω||=||ρ'(a-b)Ω'||=||ρ'(a)Ω'-ρ'(b)Ω'||, so that
ρ'(a)Ω'=ρ'(b)Ω'.

Therefore U is well defined, and it evidently maps H onto H' (Why?). The map U is evidently linear (Why?), and it is an isometry (Why?). Moreover, we have UΩ = Ω' (Why?), as required. Finally for any a,b in A we have

Uρ(a)ρ(b)Ω = Uρ(ab)Ω = ρ'(ab)Ω' = ρ'(a)ρ'(b)Ω' =ρ'(a)Uρ(b)Ω,

and since vectors ρ(b)Ω span the whole H, we have

Uρ(a) = ρ'(a)U, or

Uρ(a)U^-1 = ρ'(a),

QED.

We are done. We will use this result in the next post, where we will compare the GNS construction with that of Part 41. We can have a real promenade now.

  
Pictures at an Exhibition is a piano suite in ten movements, plus a recurring and varied Promenade theme, written in 1874 by Russian composer Modest Mussorgsky. 1874 My dear généralissime, Hartmann is boiling as Boris boiled—sounds and ideas hung in the air, I am gulping and overeating, and can barely manage to scribble them on paper. I am writing the 4th No.—the transitions are good (on the 'promenade'). I want to work more quickly and steadily. My physiognomy can be seen in the interludes. So far I think it's well turned ..

P.S. 30-01-25 14:56 A nice promenade with the GNS construction can be found in Ch. 4.3.2 "C* to Hilbert Space: The GNS Construction", p. 72 of the Ph. D. Thesis by Li Liu "Non-local Quantum Systems with Infinite Entanglement".
From the summary: " In Chapter 4, I give explicit definitions that build upon the C*-model to describe quantum systems that can be applied in topics of quantum information theory such as embezzlement of entanglement."

P.S. 30-01-25 16:05 Reading "Stumbling Blocks Against Unification On Some Persistent Misconceptions in Physics" by Matej Pavsic, World Scientific 2020 . There in "Ch. 4. Transformations of Spinors", we read:

"According to the generally accepted view spinors are objects that transform under rotations differently than vectors. However, it is also known by not so small group of experts that spinors can be considered as particular elements of Clifford algebras. "

A smaller group of experts knows that spinors can be constructed by a GNS construction.