### Measure theory is a must

###### [This post started out as an introduction to a post I was planning to write on convergence theorems for the Riemann integral. The introduction kind of got out hand, so I decided to post it separately. Since I have to get back to my real work, I will postpone writing that post on convergence theorems for the Riemann integral for another time, probably during the Passover break (but in any case before we need them for the course I am teaching this term, Calculus 2)].

Mathematicians love to argue about subjective opinions. One of the most tiresome and depressing subjects of debate is “What should an undergraduate math major curriculum contain?”

### K-spectral sets and the holomorphic functional calculus

In two previous posts I discussed the holomorphic functional calculus as part of a standard course in functional analysis (lectures notes 18 and 19). In this post I wish to discuss a slightly different approach, which relies also on the notion of K-spectral sets, and relies a little less on contour integration of Banach-space valued functions.

In my very personal opinion this approach is a little more natural then the standard one, and it would be even more natural if one was able to altogether remove the dependence on Banach-space valued integrals (unfortunately, right now I don’t know how to do this completely).

### Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties)

In this post we continue our discussion of the holomorphic functional calculus for elements of a Banach algebra (or operators). The beginning of this discussion can be found in Notes 18. Read the rest of this entry »

### Advanced Analysis, Notes 18: The holomorphic functional calculus I (motivation, definition, line integrals of holomorphic Banach-space valued functions)

This course, Advanced Analysis, contains some lectures which I have not written up as posts. For the topic of Banach algebras and C*-algebras the lectures I give in class follow pretty closely Arveson’s presentation from “A Short Course in Spectral Theory” (except that we do more examples in class). But there is one topic  - the holomorphic functional calculus -for which I decided to take a slightly different route, and for the students’ reference I am writing up my point of view.

Throughout this lecture we fix a unital Banach algebra $A$. By “unital Banach algebra” we mean that $A$ is a Banach algebra with normalised unit $1_A$.  For a complex number $t \in \mathbb{C}$ we write $t$ for $t \cdot 1_A$; in particular $1 = 1_A$.  The spectrum $\sigma(a)$ of an element $a \in A$ is the set

$\sigma(a) = \{t \in \mathbb{C} : a- t \textrm{ is not invertible in } A\}.$

The resolvent set of $a$, $\rho(a)$, is defined to be the complement of the spectrum,

$\rho(a) = \mathbb{C} \setminus \sigma(a)$.

### Thoughts following the Notices opinion article

The December issue of the Notices of the AMS has quite a thought provoking opinion article by Doron Zeilberger. In fact I already read this piece earlier in Opinions of Dr. Z, but re-finding it in the notices re-kindled a feeling that I get so often: we (mathematicians) are lost. More precisely: we have lost contact with the ground. Read the rest of this entry »

### Major advances in the operator amenability problem

Laurent Marcoux and Alexey Popov recently published a preprint, whose title speaks for itself :”Abelian, amenable operator algebras are similar to C*-algebras“. This complements another recent contribution, by Yemon Choi, Ilijas Farah and Narutaka Ozawa, “A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra“.

The open problem that these two papers address is whether every amenable Banach algebra, which is a subalgebra of $B(H)$, is similar to a (nuclear) C*-algebra. As the titles clearly indicate (good titling!), we now know that an abelian amenable operator algebra is similar to a C*-algebra, and on the other hand, that a non-separable, non-abelian operator algebra is not necessarily similar to a C*-algebra.

I recommend reading the introduction to the Marcoux-Popov paper (which is very friendly to non-experts too) to get a picture of this problem, its history, and an outline of the solution.

### Spectral sets and distinguished varieties in the symmetrized bidisc

In this post I will write about a new paper, “Spectral sets and distinguished varieties in the symmetrized bidisc“, that Sourav Pal and I posted on the arxiv, and give the background to understand what we do in that paper.

### Advanced analysis – this week’s lectures

The semester here at BGU began, and I am teaching Advanced Analysis again. For the students’ convenience, I am putting up links to lecture notes which are relevant to this week.

Introduction, parts one and two.

### Essential normality, essential norms and hyper rigidity

Matt Kennedy and I recently posted on the arxiv a our paper “Essential normality, essential norms and hyper rigidity“. This paper treats Arveson’s conjecture on essential normality (see the first open problem in this previous post). From the abstract:

Let $S = (S_1, \ldots, S_d)$ denote the compression of the $d$-shift to the complement of a homogeneous ideal $I$ of $\mathbb{C}[z_1, \ldots, z_d]$. Arveson conjectured that $S$ is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary.

Previous works on the conjecture verified it for certain classes of ideals, for example ideals generated by monomials, principal ideals, or ideals of “low dimension”. In this paper we find results that hold for all ideals, but – alas! – these are only partial results.

Denote by $Z = (Z_1, \ldots, Z_d)$ the image of $S$ in the Calkin algebra (here as in the above paragraph, $S$ is the compression of the $d$-shift to the complement of an ideal $I$ in $H^2_d$). Another way of stating Arveson’s conjecture is that the C*-algebra generated by $Z$ is commutative. This would have implied that the norm closed (non-selfadjoint) algebra generated by $Z$ is equal to the sup-norm closure of polynomials on the zero variety of the ideal $I$. One of our main results is that we are able to show that the non-selfadjoint algebra is indeed as the conjecture predicts, and this gives some evidence for the conjecture. This is also enough to obtain a von Neumann inequality on subvarieties of the ball, what would have been a consequence of the conjecture being true.

Another main objective is to connect between essential normality and the noncommutative Choquet boundary (see this and this previous posts). A main result here is  we have is that the tuple $S$ is essentially normal if and only if it is hyperrigid  (meaning in particular that all irreducible representations of $C^*(S)$ are boundary representations).

### Survey on the Drury-Arveson space: more-or-less ready for use

Several weeks ago I posted a link to the survey I wrote: “Operator theory and function theory on Drury-Arveson space and its quotients“. Now after several rounds of corrections and additions I think that it is more or less in final form.

This survey is written for Handbook in Operator Theory, ed. Daniel Alpay, to appear in the Springer References Works in Mathematics series.

I wish to thank Joav Orovitz, Guy Salomon, Matthew Kennedy and Joseph Ball for finding many mistakes, suggesting additional topics and references, and other improvements. Their help was truly invaluable.

Shana Tova to all.