How does Algebraic QFT deal with renormalization? I'm reading David Wallace's essay on a critique of the algebraic approach to quantum field theory (AQFT). There he argued that AQFT failed to resolve the renormalization problem because it doesn't predict empirically testable assumptions, and there is no interacting algebraic quantum field theories in four dimensions.
However, I still don't think I understand how AQFT tackles the problem of infinity via renormalization. I learned that AQFT is based on the assumption that quantum fields can be defined on an arbitrarily open bounded spacetime region $\mathcal{O}$ of the Minkowski spacetime $\mathcal{M}$, under the correspondence $\mathcal{O}\rightarrow\mathfrak{A}(\mathcal{O})$, where $\mathfrak{A}(\mathcal{O})$ may be considered as the $C^*$-algebra of all bounded operators associated with  $\mathcal{O}$. In its totality, spacetime is covered by a "net" of local algebras
\begin{equation*}
\mathfrak{A}_{loc} = \bigcup_{\mathcal{O}}\mathfrak{A}(\mathcal{O})
\end{equation*}
How exactly is this relevant to renormalization? Is there a simple way I can see AQFT resolves the renormalization problem and removes the infinities in QFT?
(I found this video series helpful to understand some basic concepts in AQFT, which are also mentioned by @Valter Moretti)
 A: AQFT is an approach to QFT based on the notion of local net of $*$-algebras or $C^*$-algebras.  This is the intepretation of the A of AQFT.
It can be formulated in a generic (globally hyperbolic in general) spacetime. More precisely, nowadays, in the whole family of globally hyperbolic spacetimes simultaneously, in view of its functorial nature.
The use of abstract algebras instead of a given Hilbert space and operators in it permits one to deal with unitary inequivalent representations of the same algebra of observables. These naturally arise in view of the gravitational background and corresponding  unitarily inequivalent vacuum states.
It is worth stressing that fields are of quantum nature but gravity is classical here. Einstein equations, for instance, have to be interpreted in a semiclassical perspective $$\langle \Psi, T_{\mu\nu}(x) \Psi \rangle = k G_{\mu\nu}(x)\:.$$
In my view the main results of that approach (which can be traced back to the seminal paper by Haag and Kastler) concern a deep clarification of some physically important notions of QFT, like locality and covariance, in a very general setting.
Due to this generality, the approach is suitable to attack some intriguing physical phenomena as the Hawking radiation.
The approach is sufficiently flexible that it can be implemented in perturbative procedures of QFT  where renormalisation pops out.
I can say that I still consider myself a member of the international community of AQFT though, in the last 10 years, I passed  to deal with also different issues always at the basis of general quantum theory.
A relatively recent introductory book on the subject is    this https://link.springer.com/book/10.1007/978-3-319-21353-8,
(where I wrote, with I. Khavkine, an introductory chapter on Hadamard states and basic topics on renormalisation of Wick polynomials https://link.springer.com/chapter/10.1007/978-3-319-21353-8_5 exploiting techniques of microlocal analysis.)
Regarding renormalisation, the crucial fact is that in curved spacetime no natural use of Fourier transform is allowable due to the lack of translational invariance. Therefore  the general renormalisation machinery has to be improved. The basic idea, already known by Bogolyubov and explicitly developed in flat spacetime by Epstein and Glaser, is that the issues of renormalisation rely on the ill-definition of several distributions because they  are constructed through "forbidden" procedures. The use of thse procedures is however forced by basic requirements of the theory (the fact that interactions are local). Typically, a mathematically forbidden procedure is the  product of distributions.
These procedures can be partially tamed, but this generates ambiguous final results. The ambiguities can be fixed by hand to some extent by assuming three major hypotheses:
(a) a suitable (ultraviolet in the tangent space) scaling behaviour of these candidate distributions;
(b) a specific request of general covariance  and locality described within a suitable functorial setting (which definitely extends Haag and Kastler's view because now it considers all spacetimes simultaneously differently from H-K original approach);
(c) the use of a special family of states, said of Hadamard type, which in some sense extend the notion of Minkowski vacuum in curved spacetime.
When imposing these restrictions, the renormalisation ambiguities just amount to finite-order polynomials in the (covariant derivatives of) curvatures at the considered point of the spacetime. The  constant coefficients boil down to the familiar renormalisation constants when restricting to Minkowski spacetime. In flat spacetime the standard renormalisation procedure is substantially in agreement with the algebraic approach: the use of cutoffs is one of the possible (and correct) ways to deal with finite counterterms. I cannot see any substantial difference.
(see the nice book by G. Scharf "Finite Quantum Electrodynamics
The Causal Approach" https://link.springer.com/book/10.1007/978-3-642-57750-5).
One of the interesting results arising by general  locality (treated within a functorial perspective) is that renormalisation constants do not depend on the given spacetime nor on the considered state.
Recently, the AQFT setting has been profitably used to treat the problem of quantum measurement in local QFT (see the recent papers by my colleagues C.J. Fewster and R. Verch on the subject).
My personal view specifically  on the renormalisation within the AQFT approach is that, in fact, the problem appears to be clarified in its essence when looking at it from that perspective. For instance, the appearance of infinities is just a symptom and not a physical cause of the problem, as it instead may seem in the traditional formulation due to the  use of a special mathematical approach (very powerful in flat spacetime but useless as soon as we pass to curved spacetime). This is a great advantage, but, in my honest opinion,  it is not yet the solution of the physical problems.
(An international conference on some of these subjects and other related ideas  in Europe this summer https://indico.tpi.uni-jena.de/event/289/)
Personally I cannot share David Wallace's view on that rigid opposition between CQFT and AQFT.  It is clear to everybody who deals with fundamental physics that Science is a product of the human mind.  Therefore it is fallible and every apparently definite solution should be taken with the right distance. The ideological  opposition David Wallace speaks of does not exist in practice and it shouldn't be provoked.  In my honest opinion, his final statement "If the goal of philosophy of physics is to understand the deep structure of reality via our best extant physics, then to be lured away from the Standard Model by algebraic quantum field theory is sheer madness" is closer to Religion than to Science.
PS. The section "5 Understanding renormalization in AQFT" in David Wallace's essay proves that he has just a very superficial knowledge of what renormalisation is in AQFT. A pervasive confusion between AQFT and Constructive QFT takes place in his essay. Statements like "There is no known physically realistic interacting algebraic quantum field theory in four dimensions" refers to Constructive QFT, not AQFT, where the status is exactly the one of standard QFT. Since the AQFT perturbative theory includes standard QFT as special case (see the book I quoted above), treated in the Epstein-Glaser formalism in Minkowski spacetime. This formalism is nothing but the approach Bogolyubov uses in his books.  I am really astonished. He does not know almost anything on a technical  subject and he comes out with definte philosophical declarations on it, as the one I pointed out above.
