Spin Chronicles Part 45: positive domination

For some reasons I consider the concept of state on a *-algebra as important. I wrote a whole post on the subject, with a Theorem, its detailed proof, and with exercises for the Reader. I also wrote a long introduction. Then I have made one wrong click, and all was gone. Well, not all, the introduction has been preserved, since I have already managed to copy it to the blog. But introduction is just words. All the hard stuff, all formulas, are gone. So I decided to ask the I Ching oracle for advice. I Ching told me:

"Thunder and wind symbolize Duration,

The superior man stands firm

Without changing direction."

Fine. I stand firm. Will not change direction. But will provide a theorem, and skip the proof. The proof is lengthy, but not difficult. There will be no harm in skipping it. Maybe it will even be good. No distraction. So, first the beginning of the Introduction. The rest of the introduction will be at the end.

Introduction Part 1

 In this post, we will delve deeper into the concept of states on *-algebras. States are fundamental in physics, serving as a crucial link between the abstract formalism of algebra and the concrete realm of empirical measurement. They transform abstract algebraic elements into numbers, which we then interpret as experimental values. In this way, states act as a bridge between the invisible, abstract structures of theoretical physics and the tangible, measurable phenomena of the observable world. But this raises a profound and provocative question: Are states themselves "visible"? Or, more precisely, do they belong to the realm of the measurable, or do they reside in a liminal space between the abstract and the concrete?

This question is not merely technical; it is deeply philosophical. It forces us to confront the nature of representation, measurement, and the limits of human knowledge. States, in their role as mediators, challenge the boundary between the intelligible and the sensible, between what can be thought and what can be experienced. To ask whether states are "visible" is to grapple with the very nature of reality and our capacity to comprehend it. It is a question that mingles different levels of description—mathematical, physical, and metaphysical—in a way that is both illuminating and perilous. Such questions are logically treacherous, reminiscent of the paradoxical barber who shaves only those who do not shave themselves. They expose the fragility of our conceptual frameworks and the limitations of language itself.

The barber paradox

Let A be a finite-dimensional *-algebra with unit 1. Let f be a state on A, and let H be the Hilbert space of the GNS construction with cyclic vector Ω for a representation π (which previously we have denoted by ρ).

Definition. If f₁ is a positive functional on A, we say that f₁ is f-dominated if there exists a constant λ>0 such that

f₁(a*a) ≤ λ f(a*a) for all a in A.

We say that f₁ is f-absolutely continuous if f(a*a)=0 implies f₁(a*a) = 0.

It is evident that every f-dominated functional is f-absolutely continuous. (can you see it?) The converse is not immediately evident. But it will be seen from the following theorem that I adapt from the monograph by Naimark.

Here is the original theorem:

Let H be the Hilbert space of the GNS construction with cyclic vector Ω for a representation π (which previously we have denoted by ρ).

Theorem.  Let f₁ be a positive functional on A that is f-absolutely continuous. Then there exists a unique positive operator B in the commutant π(A)' of π(A) such that

f₁(a) = (Ω,π(a)BΩ).         (1)

Conversely, every such B determines, by (1) a positive functional  f₁ that is f-dominated.

Proof. Gone with the wind.

In a future post we will use this theorem to prove that a GNS representation is irreducible if and only if f is a pure state.

Introduction Part 2

Yet, it is precisely these kinds of questions that I find most compelling. They reveal the boundaries of our understanding and force us to confront the inadequacy of communicable language when grappling with the ineffable. By asking such questions, we quickly come to realize how constrained our knowledge is when confined to the tools of language and logic. Language, as a system of symbols, is inherently limited in its ability to capture the full depth of abstract thought and the complexity of reality. It is a filter through which we attempt to convey the inexpressible, but it inevitably falls short.

This tension between the communicable and the ineffable is at the heart of both science and philosophy. States on *-algebras, as abstract entities that give rise to measurable quantities, embody this tension. They are both a product of human thought and a reflection of an external reality that exists independently of our conceptual frameworks. In this sense, they invite us to consider the nature of existence itself: Is reality fundamentally mathematical, as some physicists and philosophers suggest? Or is mathematics merely a tool, a language we have invented to describe a reality that ultimately transcends it?

These questions are not merely academic; they have profound implications for how we understand the universe and our place within it. They challenge us to think beyond the limits of our current knowledge and to embrace the uncertainty and ambiguity that come with exploring the unknown. By engaging with such questions, we not only deepen our understanding of states on *-algebras but also confront the very nature of knowledge, reality, and the human condition. And in doing so, we may come to appreciate the beauty and mystery of a universe that is far more complex and enigmatic than our language and logic can ever fully capture.

P.S. 10-02-25 18:42 Glimpse into the future. Thanks to the inquires by Bjab we have already discussed the geometry of circles in the 2D plane. It was entertaining. And: "The more you get, the more you want." I decided to move to geometry of spheres (instead of geometry of points) in 3D space. Points are spheres of zero radius. Spin needs non-zero radius. You can't spin at a point. You need space for extending your hands. Electric current needs space to form a loop to create magnetic field. You can't understand spin restricting yourself to a zero-dimensional point. So, we will have to repeat our path, but at a higher level. Moving along a spiral. We will develop geometry of spheres much the same way as we have developed geometry of circles. Keeping the direction and moving forward. And we will see where it will lead us. But the promised Tomita-Takesaki will come first.

P.S. 11-02-25 7:06 Here is a translation from Russian to English of the translation from English to Russian of the beginning of one of the few books that, in the past,  made me to choose physics rather than electrical engineering as my profession.

Mitchell Wilson

Meeting at a Far Meridian

Translated by I. Gurova, N. Dekhtereva, N. Treneva

A man must be involved in the deeds and passions of his time, or else it may be thought that he never lived at all.

—Oliver Wendell Holmes

In the distant pre-war days, a man with Rennet’s specialty was usually allotted nothing more than a tiny room in some remote corner of the university—a room just large enough to fit a simple desk cluttered with hastily scribbled calculations and graphs from an experiment that might one day bring fame to its author. Among these papers, there were always three or four letters from other physicists in California, London, or Rome, left unanswered for long stretches of time because he rarely left his laboratory—a bare, cellar-like room with a few workbenches, a soapstone sink, and an incredibly complex apparatus he was working on at the time—a triumph of passion and ingenuity over a meager budget. The machine was so imposing that no outsider would have dared to ask, “Which side is the front, and which is the back?”

In those days, when the multinational population of the physics world numbered only a few thousand, and nearly everyone knew each other, if only by name, Rennet was just twenty-two years old. He had no idea that the beginning of his scientific career coincided with the end of an entire era—an era when science was still science and had not yet become politics, diplomacy, or war. In this respect, he was like a young heir to the throne whose first romantic escapade concluded on the night of the last ball before the revolution.

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P.S. 12-02-25 This is my reply to Anna's February 12, 2025 at 10:56 AM  comment.

First the last part: "Probably, the matter is in the concepts of 'completion' and 'dense space', which were not introduced in your story?". We are playing with a finite-dimensional case. I call it a "baby version". In finite-dimensional case every linear subspace of a vector space is automatically closed (in the standard topology of Rⁿ or Cⁿ). So it is automatically always "complete". We do not have to worry about about a subspace being only dense in another subspace, we do not have to worry about "completions". Reading "adult" stuff we can safely skip all these subtleties that are important only in infinite-dimensional cases. So the problem bothering Anna is somewhere else. Landsmann writes:

""First, assume that ρ is faithful in that ρ(a∗a) > 0 for all nonzero a ∈ A. It follows that (a, b) := ρ(a∗b) defines a positive definite sesquilinear form on A; the completion of A in the corresponding norm is a Hilbert space denoted by Hρ. By construction, it contains A as a dense subspace"."

Do we have faithful states? If so, what are they? From Part 40 we know a general form of a state on our algebra:

f(a) = =<f^½ ,a f^½ >,

where  f^½  is a positive element of A with norm 1: < f^½ , f^½ > = 1. When is it faithful? And when is it not faithful?

The simplest possible choice is to take f^½ =1 - the unit of the algebra. It is certainly positive (1=1*=1²), and, according to our definition of the scalar product, it has norm 1. For this choice we have 

f(a) = =<1,a>,

and

f(a*a) = =<1,a*a> = <a,a>.

This is certainly >0 for any nonzero a, since we already know that <,> is a Hilbert space scalar product on A. So, our state is faithful. Moreover, it is a cyclic vector for the representation L on A, since any a in A can be written as a = a1 = L(a)1. Thus A itself is, in this construction, the Hilbert space of the GNS construction, 1 is the cyclic vector, L is the representation.

Now another example. Using the notation from previous posts take f^½  = (1+n)/√2. Again we have a positive element of norm 1. Now

f(a) = <1+n,a(1+n)>/2.

Is it faithful? Take a=1-n. Then a*a=2(1-n), and (1-n)(1+n)=0. Thus f(a*a)=0. So, it is not faithful. The  vector f^½  is cyclic only in the subspace Af^½  of A, which is a two-dimensional left ideal of A, which we have investigated before.

In general our state is faithful if  the density matrix representing this state has no zero eigenvalues. Otherwise it is not a faithful state, and we have to take quotient in the GNS construction.