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Quantum mechanics as we know it is the source of many puzzling questions. The simplest quantum mechanical system is the hydrogen atom, consisting of a negative charge, an electron, moving around a positive charge, a proton. This reminds electrodynamics, and accepting the fact that the electron is a bit of a slippery particle, whose position and speed are described by probability, rather than by exact formulae, the hydrogen atom can indeed be solved, by starting with electrodynamics, and making a long series of corrections, for the most coming from experiments, but sometimes coming as well from intuition, with the idea in mind that beautiful mathematics should correspond to true physics. The solution, as we presently know it, is something quite complicated.

Mathematically, the commonly accepted belief is that the good framework for the study of quantum mechanics is an infinite dimensional complex Hilbert space [math]H[/math], whose vectors can be thought of as being states of the system, and with the linear operators [math]T:H\to H[/math] corresponding to the observables. This is however to be taken with care, because in order to do “true physics”, things must be far sharper than that. Always remember indeed that the simplest object of quantum mechanics is the hydrogen atom, whose simplest states and observables are something quite complicated. Thus when talking about “states and observables”, we have a whole continuum of possible considerations and theories, ranging from true physics to very abstract mathematics.

For making things worse, even the existence and exact relevance of the Hilbert space [math]H[/math] is subject to debate. This is something more philosophical, related to the 2-body hydrogen problem evoked above, which has twisted the minds of many scientists, starting with Einstein and others. Can we get someday to a better quantum mechanics, by adding more variables to those available inside [math]H[/math]? No one really knows the answer here.

The present book is an introduction to the algebras [math]A\subset B(H)[/math] that the bounded linear operators [math]T:H\to H[/math] can form, once a Hilbert space [math]H[/math] is given. There has been an enormous amount of work on such algebras, starting with von Neumann in the 1930s, and we will insist here on the aspects which are beautiful. With the idea, or rather hope in mind, that beautiful mathematics should correspond to true physics.

So, what is beauty, in the operator algebra framework? In our opinion, the source of all possible beauty is an old result of von Neumann, related to the Spectral Theorem for normal operators, which states that any commutative von Neumann algebra [math]A\subset B(H)[/math] must be of the form [math]A=L^\infty(X)[/math], with [math]X[/math] being a measured space.

This is something subtle and interesting, which suggests doing several things with the von Neumann algebras [math]A\subset B(H)[/math]. Given such an algebra we can write the center as [math]Z(A)=L^\infty(X)[/math], we have then a decomposition of type [math]A=\int_XA_xdx[/math], and the problem is that of understanding the structure of the fibers, called “factors”. This is what von Neumann himself, and then Connes and others, did. Another idea, more speculative, following later work of Connes, and in parallel work of Voiculescu, is that of writing [math]A=L^\infty(X)[/math], with [math]X[/math] being an abstract “quantum measured space”, and then trying to understand the geometry and probabilistic theory of [math]X[/math]. Finally, yet another beautiful idea, due this time to Jones, is that of looking at the inclusions [math]A_0\subset A_1[/math] of von Neumann algebras, instead at the von Neumann algebras themselves, the point being that the “symmetries” of such an inclusion lead to interesting combinatorics.

All in all, many things that can be done with a von Neumann algebra [math]A\subset B(H)[/math], and explaining the basics, plus having a look at the above 4 directions of research, is already what a medium sized book can cover. And this book is written exactly with this idea in mind. We will talk about all the above, keeping things as simple as possible, and with everything being accessible with a minimal knowledge of undergraduate mathematics.

The book is organized in 4 parts, with Part I explaining the basics of operator theory, Part II explaining the basics of operator algebras, with a look into geometry and probability too, then Part III going into the structure of the von Neumann factors, and finally Part IV being an introduction to the subfactor theory of Jones.

This book contains, besides the basics of the operator algebra theory, some modern material as well, namely quantum group illustrations for pretty much everything, and I am grateful to Julien Bichon, Beno\^ it Collins, Steve Curran and the others, for our joint work. Many thanks go as well to my cats. Their views and opinions on mathematics, and knowledge of advanced functional analysis, have always been of great help.

\ Cergy, August 2024

Teo Banica \baselineskip=15.95pt \tableofcontents \baselineskip=14pt

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].