Here is a video recording of the talk.

BTW: You can see that someone in the audience asked me a question that I, embarrassingly, blacked out on: *do strongly commuting (unbounded) operators commute in the sense that there is some dense subspace on which the commutator is defined and equal to zero?* The answer is *yes* and is actually not hard to show with basic semigroup theory techniques. A little trickier is to show that strongly commuting operators have commuting spectral projections – which is an equivalent and perhaps more natural definition of “strong commutation” than the one I gave.

What is it that von Neumann was trying to do in that paper? I would like to understand this paper – the math as well as the physical consequences. It turns out that the only way I can really understand something is to try to explain it – hence this blog post. Luckily, this paper, which was originally written in German, was translated to English by Roderich Tumulka and published in the European Physical Journal H (and one can also find it on the arxiv). So following Tumulka’s translation I will now try to produce an annotated summary of the main part (Sections 0-2) of von Neumann’s paper.

With hindsight, after reading through the paper, I found that it very little to do with the stability questions that interested me, but it was a nice exercise.

**UPDATE: After publishing the post, I continued looking and somehow I reached the following commentary on von Neumann’s paper which was published by four mathematical/theoretical physicists including** **the translator (ironically, I wasn’t aware of it and did not think of looking for something like this until I finished wrestling with the paper by myself). I am sure that it will be more useful than this post for people who are interested in understanding von Neumann’s work:**

“Long time behavior of macroscopic quantum systems“, by Goldstein, Lebowitz, Tumulka dn Zanghi, The European Journal of Mathematics, 2010.

Below is my work-through the main parts of the paper. Sections (and subsections) as well as their numbering follow the sections in the paper.

**0.1** Here von Neumann describes the main goal of the paper: to “clarify the relations between the macroscopic and the microscopic point of view of complex systems”. The problem (dumbed down somewhat) is as follows: On the one hand, physics gives us laws which particles obey. In principle, if we have a system with a million particles (or a million degrees of freedom) we can write down all the corresponding equations of motion, solve them, and obtain a prediction to how our system should evolve. However, that’s not how large systems are analyzed. Large systems are analyzed using statistical mechanics and relatives, such a thermodynamics. For example, to describe the cooling of a cup of coffee, one doesn’t need to write down the position and momentum of every single particle within the cup; one uses “macroscopic variables” such as “volume”, “temperature”, etc., and then applies something like Newton’s law of cooling to obtain approximate results under certain assumptions.

von Neumann’s goal is to start from the basic principles of quantum mechanics and to deduce two results in statistical mechanics: one is the ergodic theorem and the other the H-theorem. What he means by this he does not explain at this point, so neither will I. I will just add that he sets up a mathematical formulation which takes into account both the basic formalism of quantum mechanics (how the system *really* evolves in time) as well as the practical and theoretical obstructions to observing the system (what we as macroscopic physicists actually see in the measurements) and proves theorems that confirm that in reality we should observe a system following the theorems of statistical mechanics.

**0.2 **This section contains the loose yet interesting connection to the modern notion of “stability”. The section starts by (very clearly IMO) explaining why using quantum mechanics for his goal is difficult: the phase space consisting of the position variables (the “coordinates”, in vN’s language) and the momentum variables are crucial for description of the statistical mechanics of Gibbs which he is trying to clarify, because on the one hand important quantities in that theory are expressed as functions of the variables, while on the other hand the Heisenberg uncertainty principle says that the values of these variables cannot be specified simultaneously (in a certain sense, not even approximately).

In the familiar formalism of quantum mechanics, the position and momentum variables and are represented by selfadjoint operators on given for by (multiplication by ) and . (The subscripts are there to specify the different degrees of freedom that arise from either the number of particles, the dimensions of space in which the particles can move, or whatever else, and denotes the total number of degrees of freedom of the system. Below I will omit them whenever possible).

**Short reminder:** Recall that in the formalism of quantum mechanics, when the system is in a certain (pure) state described by a unit vector , one cannot predict with certainty the value of an “observable” , which is represented by an operator denoted by the same letter, but rather one can predict its expectation (the mean of the value over many repeated measurements of the same observable in the same state):

Being able to predict only the average expected value is rather poor, but that’s life. However, we can do much better since we can also find the expected value of every *function* of , and therefore we have in principle the entire distribution of measurable values of the observable . In particular, one can calculate the *variance* of by

The value measures the uncertainty in measurement, and is referred to by vN as the “probable error” and also as ” as the “the spread”.

**Note:** vN writes the uncertainty principle as but in the notation I just introduced we would write . vN will soon introduce two somewhat vague quantities and that are not directly defined as spreads (although they are related).

Now, after this short reminder, back to the paper. The uncertainty principle follows from the noncommutativity of the variables: we have , and a formal computation shows that for any state. And indeed, vN knows that if and were commuting operators, then he would be able to compute any function in these variables (using an extension of his spectral theorem to a family of commuting selfadjoint operators, an extension that was certainly clear to him). So the problem is that we have operators that we cannot compute functions of, and this closely tied with the fact that these operators represent observables that cannot be specified simultaneously.

On the other hand, vN notes that all this doesn’t stop physicists from measuring position and momentum simultaneously (in footnote 6 there is an explanation of this can be done). There is no contradiction here, because the inaccuracy of macroscopic measuring devices is large enough that it doesn’t allow us to contradict the uncertainty principle. Still, one wants to be able to view this simultaneous-measurement-of-incompatible-variables within the formalism of quantum mechanics.

vN’s interpretation of this is quite neat: the variables that physicists are measuring simultaneously are really a pair of commuting operators, and . These operators are commuting selfadjoint operators , and one can compute functions of them and specify/measure their values simultaneously. Only one problem: they are not equal to the original operators of interest . The hope is that and are close in some sense to the original operators of interest and . In what sense? We’ll get back to that in a minute.

First, let’s point out that a general question arises: if we have given two operators that are *nearly commuting*, in the sense that (which is very small because is a tiny quantity), can we find *truly commuting* operators that are *near* to , in the sense that are small in some sense (small means in the order of magnitude of ). It is at this point where vN’s paper touches the problem of stability. In my recent paper with Malte Gerhold, we showed that if and are replaced by their infinite ampliations, then one can find commuting such that . This recovers a result of Haagerup and Rordam with a somewhat improved constant; in that same paper of Haagerup and Rordam they give an argument (which they attribute to Ryszard Nest) showing that the finite multiplicity of cannot be approximated in norm by commuting operators *at all*, in the sense that there are no bounded selfadjoint operators such that and commute.

vN’s notion of nearness is weaker than being close in norm. vN says that for simplicity, he will assume that the operators posses an orthonormal basis of joint eigenvectors. vN introduces numerical quantities that satisfy , and he requires that the spreads satisfy and when measured in every one of the eigenstates . This means

and

To see what this means, we note that we may as well replace whatever and we had with the diagonal operators where ; and where . With this notation, the requirement is that

and are small for all

This would certainly follow if we could arrange for and to be small in operator norm or in the Hilbert-Schmidt norm . However, the converse does not hold: it is possible that is small while both the Hilbert-Schmidt and the operator norms are not small (for an example, consider the case where is any orthonormal basis and the rank one orthogonal projection onto the span of . Then as , while ).

In the final two paragraphs of this section, vN summarizes the assumptions to be made in the rest of the paper: for every such that , one can find an orthonormal basis such that and . Here, play the role of (bounds on) in the above discussion. This is really all that is needed, and one doesn’t actually have to think too much about the operators . In footnote 9 vN claims that he has obtained this with with , but he does does not include a proof. The translator remarks that in the book that appeared three years later vN repeats this claim with ; this is some indication that perhaps vN’s first unpublished proof was in error, and we make a note to self to check this claim of vN at some point.

**0.3** In this subsection vN gives a recap of the basic formulation of quantum mechanics, where states of a system are represented by wave functions, stating that the expected value of a (function of an) observable in a pure state is given by . He also extends this principle to a statistical mixture of pure states, that is if the system is in mixed state where the states is with probability (for ), then the expected value of the observable is

where , and for any vector we let denote the rank one operator .

**0.4** In this section vN treats two operator theoretic formulations of ergodic theorem, deducing conditions under which they are satisfied, and finishes by arguing that the results actually do not imply the required ergodic theorem, because the results do not “mention the role of the macroscopic”. Let me try to briefly explain what vN does here (spoiler: it is just a motivational discussion).

The (quasi-)ergodic theorem that he wants to treat is the statement that “the time [a system’s point in phase space] spends in any region of [its energy surface] is proportional to the measure of that region”. vN gives two toy arguments for the validity of this statement, leaving the full justification for later in the paper (recall that we are still just in the introduction).

First, he considers the energy operator with eigenvalues corresponding to the eigenvectors . Then if the system is an initial state , then at time the solution is given by the solution to the initial value problem consisting of the Schrodinger equation together with the initial condition . The solution is

Since is a conserved quantity (conservation of energy), we have that is a constant, and moreover is constant in time for all , and assuming simple spectrum one can conclude that is constant for all , so that if we assume that , then the “energy surface” consists of all the states of the form where . The ergodic theorem then becomes the statmement that the sequence of points is uniformly distributed in the (space of sequences corresponding to the) energy surface. By a theorem of Kronecker, this happens if and only if the numbers are linearly independent over .

vN quickly then says that we have asked for too much, since the important quantity is actually the “statistical operator” , and not the vector . The th entry of is seen to be

Averaging this over one gets off the diagonal and on the diagonal. In order for the time average to be equal to the same diagonal operator, one requires for . Von Neumann then says that still is not what we really need because the state is still pure, and the considerations do not take into account the fact that we are observing macroscopic phenomena, and because the result would then contradict a known counter example to the ergodic theorem.

**0.5** + **0.6** Here vN roughly describes what will be carried out in the paper. There will be two types of conditions: one the size of phase cells (a condition which here is stated very roughly) and on the energy levels: the condition on the energy levels of the system is that all energy levels are distinct, and also that all differences are distinct. von Neumann says that under these two conditions he will prove the ergodic theorem and the so-called H-theorem. The classical H-theorem, roughly, states that the entropy of a system tends to increase. What vN says that we will actually prove is that the time average of the entropy of a system in a state is close to the entropy of the “micro-canonical ensemble” that is, the entropy of the system in an average state. These statements will be made more precise later on.

**0.7** Just a hat tip to previous work.

**1.1** Here the basic assumptions are laid down: all macroscopic measurements that can be made, can be made simultaneously. Thus, there is an orthonormal system that consists of the eigenvectors of all macroscopically measurable quantities. vN doesn’t specify the eigenvalues, because he hasn’t specified a particular observable – different observables will give rise to different eigenvalues for these eigenvectors.

Now, vN says that in practice we will not be able to know exactly in which state we are, so he groups the eigenvectors into subgroups where the are indistinguishable, in mathematical terms this means that they have the same eigenvalues w.r.t. to all macroscopic measurements.

For every , one can form the statistical operators where

Note that “corresponds to the pth one among the alternatives concerning the properties of the system that can be distinguished by macroscopic measurements”, and therefore according to von Neumann it corresponds to the “phase cells” in the Gibbsian statistical mechanics. The number is the number of real microscopic states in the cell — states that cannot be distinguished by measurement, and therefore quantifies the “coarseness of the macroscopic perspective”.

By the spectral theorem, every macroscopic observable can be represented as a weighted sum .

**1.2** vN now introduces the energy operator (aka Hamiltonian) . Note that vN wants the true Hamiltonian, not a macroscopic approximation, so here we need to use eigenvectors that are not the . **It seems that the are not the from the previous section (I mean Section 0, the introduction).** To account for the fact that we are not able to precisely measure the energy levels, the eigenvectors are regrouped into subgroups corresponding to eigenvalues in a way that “all the with the same are close to each other and only those with different can be macroscopically distinguished”.

**Note:** I have to say that it is not clear to me why this makes sense, that is, suppose the energy levels are the integers , then if want to group them as , then it is strange that can be readily distinguished from but not from ; one would expect some fuzziness rather than the separation into clear cut distinct groups. However, recall that this only four years (!) after Heisenberg and Schrodinger’s papers, and here already von Neumann is trying to obtain the fundamental results of statistical mechanics as corollaries of quantum mechanics, so we shouldn’t be surprised that he is making some assumptions and checking out what works.

**Note:** BTW, here the is an index, and has nothing to do with the observable that I discussed when recalling the formalism of quantum mechanics.

To formulate that the energy being in one of the groups is something that can be measured macroscopically, vN notes that when one applies the indicator function of to via the functional calculus, one obtains

and that this then should be a sum of a subset of the projections (which can be measured macroscopically):

and by taking the trace we get where . The family consists of mutually orthogonal projections that sum up to the identity, and they just give a different way to enumerate the projections , and we group them into subgroups that have a physical meaning. This gives rise to a new enumeration of the orthonormal basis elements ; we now write them as to keep track of which sum they belong to. The operator is a mixture of the states with equal weights, and also a weighted mixture of .

The operators correspond to the energy levels that can be measured by macroscopical means. So as far as the observer is concerned, corresponds to the “energy surface”, that is, a portion of phase space consisting of all states in which the energy is “constant”, in the sense that it remains in the macroscopically indistinguishable group . The energy surface decomposes into “phase cells” which are corresponds to different possibilities within the energy surface that can be distinguished via macroscopic measurements.

**1.3** Now given a certain pure state , von Neumann defines the “statistical operator”

which is a mixture of the mixed states with weights — according to the funny rules of quantum mechanics, this weight is the probability that a measurement the energy in this state will give a value in the group . This statistical operator is supposed to correspond to the “microcanonical ensemble” from statistical mechanics. von Neumann write: “Of course, this definition is really justified only afterwards by its success”. OK johnny.

von Neumann finishes by defining the entropies that he will analyze in the proof of the H-theorem:

This section is dense with calculations, so I will change the order a bit and start by telling you (spoiler) where vN is going. He is going to consider an initial state and study how it evolves in time according to Schrodinger’s equation ; the state at time will be denoted as (this is not the derivative w.r.t. time).

Then, he will work to find an estimate for the quantity , and he aims to show that it is bounded below by and above by some quantity (which in the paper is actually a complicated expression, not denoted by in the paper) that can be controlled under suitable assumptions. Then he considers the time averages , which should be bounded by . After some subsections he shows that **under various assumptions** can be shown to be **small** as goes to infinity, and that constitutes a proof for the H-theorem.

Similarly, he define expectation values corresponding to an observable , and shows that the time averages can be shown (under the same set of assumptions needed for the H-theorem) to be small for large , and this will constitute a proof of the ergodic theorem.

**2.1** Let be the orthonormal basis of eigenvalues of the Hamiltonian (energy operator) as described in Section 1, which is therefore given by

Note: here vN writes for what he earlier denoted as .

Therefore, if we write for the initial state , then the solution of Schrodinger’s equation is given by

Now vN introduces the notation

and

Note that depends on the time variable , and is independent of time (and therefore also well defined). In an important footnote (no. 33), vN notes that the time independence of implies that the statistical operator – “the microcanonical ensemble” – is also constant in time.

Also

and

vN rewrite the entropies in terms of these quantities:

A one page long series of calculations then gives

This inequality (vN refers to it as an *ansatz*, which is a fancy word for a certain expression of a mathematical entity that might be useful) will be used to prove the H-theorem.

**2.2** In this subsection vN seeks an *ansatz* for proving the ergodic theorem. Given a macroscopically observable quantity (maybe we should say a macroscopically measurable observable, or macroscopically observable observable…) we will want to eventually compare the time averages of the expected value of in the evolving state and the expected value of in the “microcanonical ensemble” . Thus, leaving the time averages for later, we shall require

and

Since is a macroscopically observable quantity, we assume that it has the form

A calculation gives (using the definition )

and

An rather straightforward estimation using the definitions and Cauchy-Schwarz then gives

where is the weighted “microcanonical average” of .

**Note:** In the above equation (equation (69) in the translation of the paper) one finds written instead of , however if one follows the computations one sees that this is a typo.

**2.3** Now vN considers the time averages of the quantities, which he denotes . At first one might think that this would mean , but looking further ahead one sees that he is actually interested in the limit as goes to infinity, that is, in the time average over *all* time.

We have from the previous two subsections

and

The goal is to show that the right hand side is *small*. It is worth pausing to note that we have here something interesting and quite unusual: von Neumann is only interested in proving that *we will observe* the theorems, hence the right hand side does not need to go to zero, but rather it should just be small (and if it is small enough, then we will observe it to be zero, hence he theorems are proved in the sense that it is proved that we shall observe these theorems to be true!). Another point to note is that vN is writing down the proof before the assumptions are stated, that is, he seems to be proceeding just as he has exploratively, and invites us to stumble upon the conditions required for the proof as we encounter them. This last part is not so unusual, there was a time in mathematics when it was popular to write proofs like that.

The expression we want to estimate are similar, and both contain the expression (note that of all the variables, only depends on time). Using the form von Neumann computes an explicit expression for and finds that

We actually need the more complicated term . Squaring the above, and averaging over time (meaning – as we commented above – averaging over an interval and then taking the limit as ) we get that every complex exponential with vanishes. If we make the two assumptions that all the differences for **all** values of are **(a)** nonzero and **(b)** distinct, then we can estimate

where

and

and now because we have that this is all bounded by the quantity

**(**)**

So now the goal becomes to bound and .

**2.4** In this subsection it turns out that the goal of showing that the quantity (**) is small, is **not** going to be always achieved. However, vN will begin an effort to prove that the quantity (**) is ** usually** small, by proving that its average value is small. Average over what? OK, let’s now start from the beginning of section 2.4 and follow von Neumann.

We assume that we have as defined in Section 2.1, and assume that all the differences for **all** values of are **(a)** nonzero and **(b)** distinct (in footnote 35 vN claims that these assumptions can be weakened somewhat). Recall also the definitions from Section 1.2, and in particular recall the operators for and that satisfy

and where . Now, all the objects will be held fixed (we may also think of as fixed), and we will play with the orthogonal decomposition . That is, we will consider all orthonormal families for the range of , and let

Now the quantities and that we need to estimate for controlling (**) depend on the choice of basis . vN explains why for some choices of basis the quantities will not, in fact, be small, for example when coincides with the basis (and is fixed). In this case vN argues (from pages 220 bottom, to 221 top) that

and so if the are large this will be large. von Neumann explains this thusly: *“The unfavorable result in this case arises, of course, from the fact that this choice of does not represent well their physical meaning: here, the have the same eigenfunctions as and thus commute with — which we expected not to be the case!”* If I understand correctly, what he means is that choosing a measurement basis as the eigenbasis of the Hamiltonian, we are not putting ourselves in the place of a macroscopic observer, so we should not expect to observe the phenomena of statistical mechanics.

**Note to self:** For me this is counter-intuitive, starting from the point of departure which was that I wanted to read this paper to understand how von Neumann uses approximation of non-commuting operators with commuting ones in analysis. In fact it seems to me that von Neumann * requires* the measurement operators to be far off from the true Hamiltonian in order to observe statistical mechanical phenomena.

vN then gives quick and dirty, but wrong (according to him) estimates for and as a warm up for the precise but cruder estimates to be presented in the next section.

**2.5** Here is the punchline of the proof. Everything done below will be under the assumption that (i.e., there are many phase cells for every energy level) and (i.e., the size of every phase cell is large, contains many true states) that in the text are thrown in at various stages.

Using averaging w.r.t. the Haar measure on the unitary group (calculations that are relegated to a lengthy appendix), he finds that the averages of and over all orthonormal bases (as in the previous subsection) are bounded by the quantities and , respectively.

Using these estimates, the average value of the quantity (**) can be bounded from above as follows

and this is approximately bounded by the simpler expression

where is the harmonic mean of the . von Neumann asks: *when is this expression small?*

By some simple computations and after introducing the arithmetic mean , von Neumann shows that for (**) to be small we will need

show that “the phase cells must be large compared to their number on the energy surface”.

This, together with the previous assumptions, is the condition under which the H-theorem and the ergodic theorem are “proved”. von Neumann closes the section by remarking that maybe this harsh condition is necessary for the theorems to hold, but that it is also possible that these conditions “merely arose from the imperfection of our methods”. He says that it is possible that if would suffices to assume that is large compared to unity, not to (which is also assumed large).

**Comment:** This last assumption is written mistakenly in the translation as instead of (I don’t know if in the original too).

In this section von Neumann gives an additional discussion of the physical meaning of the assumptions and the theorems, also discussing cases when they fail. There is no new mathematical content. For the physical significance of these results, one should take a look at the commentary that I linked to above.

Perhaps it is worth emphasizing that the appendix contains a lot of mathematical content, and is crucial to the work. I plan to revisit this appendix at some point. Surely I don’t want to give an impression, that these are not rote computations, rather they seem rather ingenious, especially given the time they were made. However, since my goal was to try to clarify *what* von Neumann was doing, and *why*, rather then *how*, I will not try to work through the calculations here.

Surprise: in a recent preprint, “Tensor algebras of subproduct systems and noncommutative function theory“, Michael Hartz and I show that no: not every tensor algebra is an algebra of continuous nc functions. We give a condition for when a tensor algebra is completely isometric to the algebra of continuous nc functions on its space of finite dimensional representations – this happens precisely when the homogeneous ideal corresponding to the subproduct system satisfies a certain perfect Nullstellensatz. We give an example showing that this Nullstellensatz (which always holds in the case of finitely many variables) might fail in the case of infinitely many variables. We also provide classes of examples for which the the Nullstellensatz always holds, e.g. monomial ideals.

Finally, we do end up solving the isomorphism problem for tensor algebras of subproduct systems with Hilbert space fibers of (possibly) infinite dimension: two such tensor algebras are isometrically isomorphic if and only if the subproduct systems are isomorphic (and a similar result for completely bounded isomorphism).

]]>**References for Lecture 1:**

In the first lectures I want describe ** classical dilation theory** as was developed from the 50s and on first by Sz.-Nagy, then by Sz.-Nagy and Foias, and also by many others. The central reference for the classical theory is

My goal in this series is to describe the development of the ideas of dilation theory from the very start and up to their current form (let’s see how far I get); that’s not the same thing as describing the ** history** of the subject. There have been many contributions that I ignore, and I will not say anything in the lectures about the related independent approaches developed by de Branges or Livsic and their collaborators. The above reference (“Harmonic Analysis of Operators on Hilbert Space”) is wonderful in that not only is it a very thorough presentation of the theory, but also it contains detailed “Notes” and “Additional results” sections, as well as a very extensive bibliography. This book might be the best place to start if you are really interested in this topic, and the “additional results” sections contain extensions and generalizations up-to-date as of 2010.

To quickly get an idea of how an application of the functional calculus (a corollary of the unitary dilation) to invariant subspaces works, look at this paper of Bercovici where he explains Brown, Chevru and Pearcy’s result that every contraction with spectrum containing the unit circle has a nontrivial invariant subspace. The invariant subspace problem was also solved for * operators of class *, which are operators that have a “minimal function”, that is, they are annihilated by some function (this is an extension of the notion of

Another related book that I used when preparing my lectures was Nikolskii’s book “Treatise on the Shift Operator” where the model theory (viewing operators as corestrictions of the shift) and its application are discussed in a detailed yet very elegant way (this book also contains extensive notes).

Finally, since it is rather brutal to send you to read three books, let me also mention my survey paper on the subject, **“Dilation theory: a guided tour”**. The first four sections (20 pages) contain the material of my first two lectures.

**Notes on Lecture 1:**

At the end of the lecture mentioned an important application of the unitary dilation: von Neumann’s inequality. Then I gave an important application of that: the functional calculus. Since I was running out of time I think I messed up a bit so here are a couple of corrections:

- The good definition of the disc algebra is simply the set of all continuous functions on the closed disc that are analytic on the interior of the disc (or the analytic functions on the open disc that extends to continuous functions on the closed disc). This is a nice function analytic definition, not the clumsy “closure of polynomials in the sup norm” which is how I defined it in class (the latter is a theorem, not a definition).
- The theorem is that for every contraction , there exists a contractive and unital homomorphism that maps to and the identity function to . This homomorphism is well defined thanks to von Neumann’s inequality, and one denotes .

One last note: I did not have time to mention that when an operator is pure, in the sense that in the strong operator topology, then the minimal isometric dilation is a shift (and the minimal unitary dilation is a bilateral shift). I plan to start with this my second lecture.

**References for lecture 2:**

Completely positive maps were introduced by Stinespring in the paper “Positive functions on C*-algebras“, where Stinespring’s dilation theorem was proved, as well as the theorem that we used, that a positive map on C*-algebra is completely positive. The best reference that I know of for studying about completely positive maps is Paulsen’s book “Completely Bounded Maps and Operator Algebras“, in particular see Chapter 3.

**Notes on lectures 2:** Contrary to what I said, the fact (which we didn’t use) that a positive map into a commutative C*-algebra is completely positive does not appear in Stinespring’s original paper. Stinespring proves the related result that a positive ** functional** is completely positive. The extension to the case when the range is a commutative C*-algebra appears (as a direct consequence of the scalar case) as Theorem 3.9 in Paulsen’s book.

**References for lecture 3:**

The book “Harmonic Analysis of Operators on Hilbert Space” contains material (and all references) on dilations of commuting contractions and failures of von Neumann’s inequality as well. It contains the commutant lifting theorem, but for a streamlined presentation of the application of commutant lifting to interpolation with bounded analytic functions **(which I planned to do but ended up skipping in lecture 3)** see Section 4 in my survey (this approach is originally due to Sarason in his paper “Generalized interpolation in “, though Sarason didn’t use the general framework of dilation theory but rather did the required commutant lifting in the setting of the problem).

Parrott’s example of commuting contractions that is from this paper of Parrott, where it is also shown that this example satisfies von Neumann’s inequality (for scalar valued polynomials). A simpler explanation of Parrott’s example appears on page 909 of this paper by Halmos. For a quick overview (with references) of von Neumann’s inequality holding or failing for three commuting contractions on a finite dimensional space, see this paper by Knese, which I blogged about enthusiastically at the time).

Arveson’s extension theorem is from his paper “Subalgebras of C*-algebras“. The proof I gave is an adaptation of the proof that appears in Paulsen’s book. Arveson’s dilation theorem is from his paper “Subalgebras of C*-algebras II“.

**References for lecture 4:** A very nice reference for obtaining the C*-envelope and the Shilov boundary via the unique extension property and maximal dilation is Arveson’s “Notes on the unique extension property“, an unpublished not that he published following Dritschel and McCullough’s proof. Arveson later managed to take these ideas to prove the existence of sufficiently many boundary representations in the separable case “The Noncommutative Choquet Boundary“. Davidson and Kennedy later proved the general case with the proof I discussed at the end of the talk: “The Choquet boundary of an operator system“.

Lecture 5 was given as a presentation, rather than a chalk talk. Here are the slides.

]]>Every time IWOTA is held at a different location – this year it was held in Krakow, Poland. This was the first time I traveled abroad since January 2020 and it was also the first conference in a couple of years for many of the participants and that was maybe the main (unofficial) theme of the conference: so nice to get back together, so nice get back to normal (honestly, I feel somewhat corrupt to speak of flying to conferences in a different continent, sleeping in hotels and eating out every day for a week as “normal” … about this we shall need to talk another day). It was nice to meet and shake hands with people whose papers I have read and admired, and exchange a few words. There was also a scientific program – check it here (clicking the speaker name leads to a pdf file with title and abstract).

I was invited to speak in two special sessions: the first one on “Functional Calculus and Spectral Constraints” and in it I spoke about “A von Neumann type inequality for commuting row contractions”. I spoke about my joint work with Hartz and Richter, which has been recently published in Math Z. In the comments and questions part of the talk I learned about a very interesting conjecture or Matsaev’s, which has been apparently solved in the negative by Drury a little more than a decade ago. The conjecture is that for every every polynomial and every contraction , it holds that

where is the unilateral shift on . It seemed to me that some of the participants in the session who knew about this paper didn’t completely believe the counter example because it requires numerical verification. Drury’s paper has a link to code that one can download and check. This problem and its solution is definitely something I’d like to learn more about, and maybe this will be next undergraduate project that I will offer. In any case, a friend told me that it is still an open question whether with some constant .

The second session I spoke in was “Operator Space Techniques in Operator Algebras”, and in it I spoke about “Dilations of unitary tuples and their surprising applications”, highlighting my recent work with Malte Gerhold. I also wanted to advertise my survey on dilation theory (because I recently recalled how good it is) but I ran out of time so I forgot to do the little advertisement in the end.

I usually actively look for talks that would excite and inspire me, to note to myself directions that I should follow up on after the conference. Let me tell you about the three talks that did it for me this time.

The first one was also the first talk in the conference, Nikhil Srivastava’s talk on numerical and algorithmic aspects of diagonalizing a matrix. I find it quite ** comforting** that supposedly completely solved problems like the diagonalization of a matrix are actually not completely settled once you look at them in practice. There are incredible software packages that one can use that actually usually work excellently; however, the speaker pointed out that that he is after algorithm algorithms that

The second talk that I want to mention was the talk by Ion Nechita, which highlighted connections between stuff I work on (free spectrahedra and matrix convexity) and stuff I don’t work on: quantum information. I was already aware of Nechita’s work but this talk really inspired me to look more into the interesting connections and applications of the mathematics I do to quantum information.

The last talk I want to highlight is the talk by Mirte van der Eyden, whose talk title was “Halos and the undecidability of tensor stable positive maps”, where she presented this paper with the same title. The talk was captivating both because of the very interesting content as well as for that fact that it was also the best ** talk**.

The problem treated is that of existence of essential tensor stable positive maps. A map is said to be ** tensor stable** if the maps , , , etc. are all positive. If a map is completely positive, then it is tensor stable. Also, if is a completely positive map followed by a transpose, then it is also tensor stable. A map that is of the previous two kinds is said to be a

The authors treat this problem in two interesting and non-standard ways. The first attack is literally based on nonstandard analysis. They work over the hypercomplex numbers (warning: on wikipedia these are called surcomplex numbers) and show that in that setting there do exist essential tensor stable maps (they can actually construct such a map). The “halo” in the title refers to the image in the hypercomplex world, where the tensor stable maps are surrounded by a halo consisting of the hypercomplex tensor stable maps.

The second approach to the problem is the following: they tried to show that the problem of deciding whether a given map is tensor stable is undecidable. Why would this be helpful? Well, deciding whether a map is completely positive, or a composition of a completely positive map and a transposition, is decidable. Thus, if they prove that the problem of deciding whether a map is tensor stable is an undecidable problem then there has to be an essential one! That’s quite clever. Unfortunately they can only show that a related problem is undecidable, but this is still very interesting.

Krakow is not far from the small town Staszow in Poland (which my grandmother would call Polania) which is where my father’s mother was born and raised. My grandmother immigrated to Israel before the war, but her family stayed behind and her parents and some of her siblings were murdered in the holocaust (two brothers fled the Nazis, and two of sisters survived the death camp, I think Treblinka). I have a friend who is a mathematician that works and lives in Krakow, and (knowing about my family’s origins) on the last day of the conference he very kindly suggested to drive me around the area, and since another friends of ours (a German) was already planning to go to Auschwitz, we decided to go there. And so we went to Auschwitz, a Pole, a German and a Jew. This was a very special experience that I will not forget.

What is there to say? By the end of the day I forgot that I was at a conference that ended that same morning. I don’t know what is the lesson to be learned and what I am supposed to do now. Just one thing is very clear: we should always remember and continue to remind.

]]>It was a nice surprise to discover that our dilation techniques have applications also in the realm of unbounded operators. It was also a nice surprise to find myself writing a 9 page paper, I forgot that this is even possible!

I will be speaking about this result in the upcoming IWOTA and then in the IMU meeting.

]]>The workshop NCAT 2022 that was held in honour of Paul Muhly at the Technion two weeks ago ended, to my great relief as organizer, with no disasters, thank god In fact it was a delightful event, with some excellent talks, some new collaborations starting up, a lot of words of kindness and friendship, old friendships and partnerships rekindled, I am very happy to have taken a part in the organization, together with my friends Baruch Solel and Adam Dor-On. Needless to say, all of the actual physical organization was done by the current and former CMS coordinators, the excellent Karla Konik and Yael Stern.

Check out the website to see the program, abstracts, and the slides of some of the talks. As we say in Hebrew: it was good, and it is good that it was.

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