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I am reading a circle of papers which use arguments based on Fredholm determinants of nuclear operators to compute numerical quantities associated to real-analytic and holomorphic dynamical systems. In order to understand these papers I am learning about Grothendieck's theorems on traces and determinants of nuclear operators. (My reference for these results is the book Traces and Determinants of Linear Operators by Goh'berg, Goldberg and Krupnik, which I am finding easier to read than the original papers of Grothendieck.) The operators are typically defined on classical Banach spaces of holomorphic functions, sometimes of several variables.

In order for Grothendieck's results on the existence of traces and Fredholm determinants to be applied it seems that the Banach space $\mathfrak{X}$ on which the operators are to be studied must satisfy the approximation property: for every compact subset $K$ of $\mathfrak{X}$ and every $\varepsilon>0$ we must be able to find a finite-rank operator $F$ such that $\|x-Fx\|<\varepsilon$ for all $x \in K$. I was not previously familiar with the approximation property (my background is mainly in ergodic theory and dynamical systems) and am trying to better understand the scope of this particular hypothesis.

Unfortunately the approximation property does not seem to be very explicitly treated in the papers I am reading and I have not yet discovered any useful references. I wonder if anyone can point me to a reference for the following:

Let $D \subset \mathbb{C}^d$ be nonempty, open and bounded. Is it known whether or not the Banach space of bounded holomorphic functions $D \to \mathbb{C}$ with continuous extensions to $\overline{D}$, equipped with the uniform norm, has the approximation property?

Some qualifications:

  • I would be happy with a reference which states that the problem is open, as well as with a definite answer either way.
  • I'm interested both in cases where $D$ is connected, and in cases where it is not connected. Any information about either would be welcome.
  • I understand that the approximation property for $H^\infty(D)$ is unknown when $D\subset \mathbb{C}$ is the unit disc. Can anyone give me a reference stating that this problem is open?
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    $\begingroup$ I'm fairly sure that the algebra in your main question is known to have the AP if D is a polydisc (=Cartesian product of copies of open unit disc) but I don't recall where I've seen this stated. One place that might mention this, at least for $d=1$, is Wojtaszcyk's book Banach spaces for analysts, but I don't have a copy to hand right now. I also suspect that book would mention that "does $H^\infty({\mathbb D})$ have the AP?" is an open problem $\endgroup$ – Yemon Choi Aug 7 '17 at 18:22
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    $\begingroup$ For the polydisc case, I guess something like a Fejer kernel argument (plus max modulus principle) should give a hands-on proof of the AP, but this is very much off the top of my head $\endgroup$ – Yemon Choi Aug 7 '17 at 18:24
  • $\begingroup$ Thanks. In general I don't expect to know anything about $D$, and there's no reason for it to be a polydisc. Is it unknown for more general domains, do you know? $\endgroup$ – Ian Morris Aug 7 '17 at 19:08
  • $\begingroup$ I'm afraid I don't know off the top of my head. I had though that people like Aron or Dineen would have looked at this, but right now this guess is not leading anywhere... I'll stop for now, to avoid clogging the comments, unless I find something clear and citeable $\endgroup$ – Yemon Choi Aug 8 '17 at 2:49
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A classical reference for the case of the unit disk:

Bourgain, J. and Reinov, O. On the approximation properties for the space $H^\infty$. Math. Nachr. 122 (1985), 19–27. (Link)

As recently as 2013, the question is still mentioned as open in:

Brudnyi, A. Banach-valued holomorphic functions on the maximal ideal space of $H^\infty$. Invent. Math. 193 (2013), no. 1, 187–227. (Link)

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