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.
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.
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.
