The never ending paper

by Orr Shalit

My paper On operator algebras associated with monomial ideals, written jointly with Evgenios Kakariadis, has recently appeared in Journal of Mathematical Analysis and Applications. They gave me a link to share (the link will work for the next several weeks): click here for an official version of the paper.

The paper is a very long paper, so it has a very long introduction too. To help to get into the heart of editors and referees, we wrote, at some point, a shorter cover letter which attempts to briefly explain what the main achievements are. See below the fold for that.

But first, a rant!

This paper took forever to publish.

In January, 2015, we submitted our paper to a top journal (and I announced arxiv submission in a previous blog post). About a month later it was rejected. We then submitted it to another journal, one tier lower. Two years (!) later it was returned to us with no referee report (!!), just an explanation that they did not manage to get it refereed, and maybe it is in our best interest to try somewhere else. We then sent the paper to several other pretty good journals, and it was rejected, at times after the editor received several mixed “quick opinions”, or after a referee said the work is good but “lacks applications”, or after a full referee report that was positive about some aspects of the paper but overall had some criticism about the length and “disjointedness” of the paper. In the end we sent it to JMAA and we are very pleased with the treatment it got there.

The bottom line: the paper was accepted almost four years after the first submission. In summary, a lot of time was wasted for referees, editors, and us.

I don’t want this rant to be long, but I just want to scream that the way things work is so screwed up! The pressure to publish in top journals, and the fact that this pressure affects everyone and therefore also increases the pressure on the journals and in particular the editors, leads to a big waste of time that has nothing to do with mathematics. Not to mention the sad fragmentation and specialization of mathematics.

You can say: “Well, your paper was weak. You are to blame for wasting the editor’s and referees’ time. If you sent it to a low tier journal right away, it would have been accepted faster, and less referees would have wasted their precious time”.

To this I can only answer: I think that our paper is very strong, and yes, we wanted this to be recognized. Dammit, it is stronger than papers that I accepted (as referee), to journals that rejected it. Oh, and BTW, referees who reject good papers are wasting their own time, as well as everyone else’s.

Along the way people asked: “Eighty pages? Why didn’t you split it into three papers?” Well, we finished about a year’s worth of hard work, and were very proud of the result. We felt that everything fitted beautifully together. Splitting the paper might have gotten us more publications faster, but we thought it would involve repetition and be a disservice to anyone trying to study the literature.

Maybe we could have played this better. But should this really be played like a game?

A little bit of constructive advice: Here are a few pieces of advice for authors/editors when dealing with long, mult-area papers that are hard to referee:

  1. At some point a wise colleague suggested that we include a cover letter that anticipates the objections that referees had (long, too many topics, why did you include this or that theorem), and answered them at the outset instead of answering them after rejection.
  2. At some point, we shared the information about where our paper has been considered earlier and also the names of editors. This led to editors communicating and possibly (I don’t know) using previous referee opinions reports that were good enough for one journal but not great enough for another (maybe).
  3. I made myself a rule of thumb that I will strive whenever possible to write papers that are less than 40 pages long. After all, we all wnat our ideas to get through, don’t we? It isn’t always possible, but in case it is possible at all, then there is no dilemma.
  4. Here is another piece of advice which I don’t know if I can follow (because I am working with junior people) but I know that I want to. One strategy with publishing is to always try great journals, and who knows, maybe we’ll get lucky. This works to some extent. In general, it is very hard to get excellent papers in the wrong field to top journals, the competition is very fierce. I would like to be able to say: I’ll submit only to a place where the chances of acceptance are high. I have used this strategy with very good (but not top) journals and I was very happy with the results.
  5. Overall, I think that the pressure we feel to publish in top journals is fake news. Of course everyone who looks at your CV and sees an annals paper will be automatically impressed, no matter how little they understand your field or care about it. But when you are evaluated for a position, tenure, or promotion, the really important things are the letters of evaluations. There need to be more than a handful of prominent mathematicians who will be able (and willing) to vouch for the merit of your work, and write something substantial about your achievement and their impact. If you have gotten to this stage, the journal names don’t matter that much any more, and the actual content of your work (and how well you communicated it) is what really matters. So most of the effort and also most of the worries should be aimed at making great research.
  6. On the other hand, don’t sell yourself too cheap. If you believe in your work, don’t give up.

Brief overview of the paper

Along with the paper, we submitted a cover letter that contained the following overview.

The paper concerns the study of operator algebras related to monomial ideals. We introduce several operator algebras arising from monomial ideals, and we study how these operator algebras relate to the initial data (that is, the actual monomials determining the ideal) and how these operator algebras relate to each other. This setting accommodates also algebras related to subshifts, from which we derive our motivation.

Our approach considers both C*-algebras and nonselfadjoint operator algebras that arise through the apparatus of C*-correspondences and subproduct systems. Results from C*-algebras feed in the examination of the nonselfadjoint ones, and vice versa.

Let us give a brief list of some connections.

Beginning with a monomial ideal we consider two nonselfadjoint algebras, namely:

  1. the nonselfadjoint algebra \mathcal{A}_X of the related subproduct system X, and
  2. the tensor algebra \mathcal{T}_E^+ of a certain C*-correspondence E.

Along with that there are several C*-algebras that can be associated to X and E, namely:

  1. the C*-algebras C^*(\mathcal{A}_X), C^*(\mathcal{A}_X)/\mathcal{K}, and
  2. the Toeplitz-Pimsner algebra \mathcal{T}_E and the Cuntz-Pimsner algebra \mathcal{O}_E.

When the monomial ideal comes from a subshift then C^*(\mathcal{A}_X)/\mathcal{K} is the Matsumoto algebra on the Fock space of allowable finite words.

In contrast to C*-algebras of C*-correspondences, the C*-algebras of subproduct systems are not well understood in general. However the C*-correspondence E that we introduce, is one of the novelties of the paper, and is useful to completely describe them (Theorem 6.1).

This also allows to further compute the C*-envelope of \mathcal{A}_X, by using that it sits inside the tensor algebra \mathcal{T}_E^+ of E (Theorem 7.6). The C*-envelope question is wide open for general subproduct systems, yet here we offer a C*-correspondence link to a definite answer.

The nonselfadjoint operator algebras also contribute to the study of the arising C*-algebras, as in Theorem 7.11. We note that in general the Cuntz-Pimsner algebra \mathcal{O}_E is not always C^*(\mathcal{A}_X)/\mathcal{K}. In fact we give necessary and sufficient conditions on the forbidden word set for that to be the case (Proposition 5.8 and Theorem 6.1).

This raises the question, what is then \mathcal{O}_E? We answer this in Proposition 10.2 where we show that \mathcal{O}_E is the graph algebra of the follower set graph of the subshift (a natural construction that gives a representation of the subshift).

The classification results (Corollary 8.12 and Theorem 9.2) are strikingly different: up to permutation of symbols for \mathcal{A}_X and and up to local piecewise conjugacy for \mathcal{T}_E^+. The preparatory work on the quantised dynamics (Section 4) allows us to easily illustrate the differences (see Example 9.8). As a further preparatory work we give general results on classification of general subproduct systems (Section 3) which are needed in Section 9 for the classification of the algebras \mathcal{A}_X. In particular we show that isometric isomorphisms (resp. bounded isomorphisms) are equivalent to isomorphism (resp. similarity) for general subproduct systems, which improves previous work in the area.

Rigidity for classes of C*-correspondences has been under considerable examination in the past years. Hence the classification results for the algebras \mathcal{T}_E^+ are follow-ups of the work of Davidson-Katsoulis and Davidson-Roydor. Part of our study addresses some subtle issues with their known results (in conjunction to Remarks 8.4 and 8.5), which called for some attention.

In the process we develop some alternative techniques that provide some additional insight and extnesions:

  • they apply in the study of relevant C*-algebras (Corollary 8.19), while,
  • they manage more examples beyond that of Davidson-Katsoulis (Application 11.1).