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# Operator Theory and Analysis of Infinite Networks: Theory and Applications

Communicated by *Notices* Associate Editor Emily Olson

The quote that spoke to me the most (and the quote that I believe best summarizes the the content of this book) appears on page ix of the preface and states, “The literature on Hilbert space and linear operators frequently breaks into a dichotomoy: axiomatic vs. applied. In this book, we aim at linking the two sides: After introducing a set of axioms and using them to prove some theorems, we provide examples with explicit computations. For any application, there may be a host of messy choices for inner product, and often, only one of them is right (despite the presence of some axiomatic isomorphism).” In particular, as a mathematician who works in a subfield of operator algebras that deals with many different metrics that induce the same topology, the last sentence of this quote is something similar to what I tell my analysis/topology/functional analysis students all the time. And as an assistant professor at a liberal arts college who advises many undergraduate research projects, the promise of explicit computations in the examples of this book opened up the possibility of finding new research projects for my students. I found the authors to have fulfilled their promise as I read more and more of the text.

I have to admit that I was a little skeptical about the claim the authors had of “linking the two sides” of axiomatic and applied, and I wasn’t sure there was a place for this book in the wealth of math textbooks we have. However, in the introduction the authors have sections devoted to “What This Book is About” (starting on page xlii) and “What This Book Isn’t About” (starting on page li) to describe how this linking is accomplished. They also provide a detailed summary of each chapter. In particular, the section “What This Book Isn’t About” focuses on defending this textbook’s novel approach to these subjects and how it approaches well-known topics in a new and enlightening way. For example, in regard to spectral theory on page lii, the authors clarify, “Our approach differs from the extensive literature on spectral graph theory (see [Chu96] for an excellent introduction and an extensive list of further references) due to the fact that we eschew the basis for our investigations. We primarily study as an operator on and with respect to the energy inner product. The corresponding spectral theory is radically different from the spectral theory of in And in relation to operator algebras and an application of “infinite graphs to the study of quasi-periodicity in solid-state physics” on page liii, the authors state, “While periods and quasi-periods in graphs play a role in our current results, they enter our picture in quite different ways, for example via spectra and metrics that we compute from energy forms and associated Laplace operators” and that “There does not seem to be a direct comparison between our results and those of Guido .”*et al*.” In both of these cases and for the other topics they discuss in this section, they provide many references to support their bold claims.

Of course, I shouldn’t have been skeptical of this book having a place in the literature since the authors bring their notable expertise to the text. I can confidently say that Palle E T Jorgensen is known by everyone in the field of operator algebras and in many related fields due to his innumerous important contributions to these fields. The second author, Erin P J Pearse, has established himself as an expert in fractal geometry and holds a patent related to his work in data science. I can’t think of a better pair of authors for a text that links the axiomatic and applied sides of this particular subject.

Regarding the audience for this book, I believe it has something for everyone: from undergraduate to research mathematician (there are even conjectures scattered throughout the text, and I make some comments about these below). And as I said in the intro, I, as a professor in a liberal arts college whose area is in analysis, will use it as a reference for finding connections between the analysis presented in this text and my work to develop new undergraduate research projects for my students. While this text wouldn’t be sufficient for a course because it has no exercises, it could be used for an independent study or a reading seminar.

I will now focus on some highlights from the text. Of course, there are more highlights than the following, but for the sake of brevity, I thought it best to focus on the following.

For the first highlight, I am going to take an unorthodox approach and begin by focusing on Chapter 16, near the end of the text, and the Appendices, before making my way to Chapter 1. As a mathematician who works in operator algebras, when I first glanced at the text, a section entitled “The GNS Construction” caught my eye. However, there was something that initially confused me about this title. This confusion stemmed from the fact that the GNS construction is a well-known construction that feels more foundational, so I would suspect something like this would find a home in the Appendices. But as I read this section, I quickly realized why this content on the GNS appears in the text proper. The following statement from this chapter defends why this content is not reserved for the Appendices (note that the Appendices are amazing, so much so that I spend time on them in this review next, but they do serve a different purpose than the text proper). “We provide the following loose parallel [of the GNS construction with the Schoenberg-von Neumann theorem and also of Aronszajn’s theorem] for the interested reader to ponder further.” And the authors couldn’t be more correct about my desire to ponder this parallel further. To make it even clearer how important these parallels are, “The statement of the GNS construction is more similar in flavor to that of Aronszajn’s thoerem, but its proof is more similar to the Schoenberg-von Neumann theorem.” They then proceed to provide not only tables that display this important distinction, but also enlightening sketches of the proofs of the GNS construction (Theorem 16.4) and the Schoenberg-von Neumann theorem (Theorem A.17). I provide the aforementioned tables here to give a glimpse into why I felt the excitement that I felt.

**Schoenberg-von Neumann:**

**Aronszajn:**

**GNS Construction:**

The tables outline with incredible clarity the parallels between these constructions and I, for one, am grateful to the authors for providing them. A few pages before these tables, there is another table that caught my eye as a mathematician who works in some of the “noncommutative areas” of mathematics such as: noncommutative geometry, noncommutative metric geometry, and noncommutative topology. The next table in section 16.1 of Chapter 16 does an excellent job of displaying some noncommutative analogs to their commutative/classical counterparts, or in the authors’ terminology, some quantum analogs to their probabilistic/classical counterparts.

Continuing in this strange route of presenting my highlights of this text, I move on to the Appendices. I believe that some of the best proofs or descriptions of classical results can be found in the appendices of texts. One that particularly comes to mind is an appendix in John B. Conway’s *A Course in Functional Analysis*, which is Appendix C: The Dual of which contains a nice presentation of the Riesz-Markov-Kakutani representation theorem. In a similar way, the authors do not disappoint with their appendices. Appendix A mostly lists classical definitions in functional analysis, but toward the end, gifts us with an illuminating sketch of the proof of the aforementioned Schoenberg-von Neuamann theorem (Theorem A.17) along with a proof of a powerful uniqueness corollary (Theorem A.18). Appendix B provides some standard definitions as well as some detailed counterexamples that clarify some important concepts that appear in the text such as: a Hermitian operator that fails to be essentially self-adjoint (Example B.14) and two self-adjoint operators whose product is not essentially self-adjoint (Example B.18). Moreover, they provide a generalization of the Krein construction, while also offering a “more streamlined proof” a claim they make with which I fully agree. The diagram and table of Appendix C is also a great companion for the text proper since it, for instance, provides a summary of some of the main properties of the Laplacian operator in the various contexts in which it appears in the text. ,

Next, we travel to the beginning of the text proper, where I would like to highlight some aspects of Chapters 1, 5, and 6. Indeed, Chapters 5 and 6 caught my eye since they focus on the Laplacian. I find that the Laplacian is a difficult idea for students to grasp (I am including myself in the group of students that struggled with the Laplacian). I was pleasantly surprised with the presentation of this operator. It first appears in Chapter 1, where the authors do an excellent job of setting the stage for the gentle pace of the rest of the text. They immediately opt to prefer explanation over overwhelming detail but promise detail later and make good on that promise. This is made clear with statements like “We won’t worry about the domain of or **T** until Chapter 5.” I should note they don’t exclude readers that are more familiar with the material since after this statement they address a familiar approach so as not to confuse mathematicians who know the subject material. Another aspect I found impressive was the ability to utilize electrical resistance networks to motivate the definitions and results. The truly impressive part was the fact that while I have no knowledge about resistance networks, through their explanations I have a firmer grasp on the mathematical definitions (as an added bonus, I may now be able to actually understand my brother, who is an electrical engineer, when he talks about his job).

The amount of new research this text allows for is vast. The text not only contains a chapter devoted to Future Directions (Chapter 17), but also conjectures and open problems appear throughout the text. My favorite one of these appears in Chapter 3 (Conjecture 3.48), where they provide a detailed description of the conjecture and its importance as well as produce a “Nonproof” “in the hope that it will inspire the reader to find a correct proof.” I was very excited when I saw this and was delighted by the effort of the authors to provide their “Nonproof”!

Of course, the whole text is a pleasure to go through, but I believe the above highlights the novelty that this text furnishes to the mathematics community. In summary, this is a well done and thought out example-driven text. It’s a great resource for students and professors alike, and I know that I will be using it as a resource for research with my students.

## Credits

Photo of Konrad Aguilar is courtesy of Konrad Aguilar.