A uniform algebra approach
to an approximation theorem of Sahutoğlu and Tikaradze

In [ST19] Sönmez Sahutoğlu and Akaki Tikaradze proved, on what they referred to as $L^{\infty}$-pseudoconvex domains, an approximation result that can be viewed as a several complex variables generalization of a weak form of an earlier approximation result in one complex variable due to Christopher Bishop¹¹1Christopher Bishop should not be confused with Errett Bishop after whom the antisymmetric decomposition, which will appear later in our paper, is named. [Bis89]. They used their approximation result to give a generalization to several complex variables of a theorem of Sheldon Axler, Željko Čučkovič, and Nagisetti Rao regarding commuting Toeplitz operators [AČR00]. The main purpose of the present paper is to give a simple proof of the approximation result of Sahutoğlu and Tikaradze, under a different hypothesis on the underlying domain, using methods from the theory of uniform algebras.

We introduce here some notation and terminology we will use. Throughout the paper, $\Omega$ will be an open set in $\mathbb{C}^{n}$ or the in the Riemann sphere. The boundary of $\Omega$ will be denoted by $b\Omega$. Following [ST19], given a holomorphic map $f:\Omega\rightarrow\mathbb{C}^{m}$ we will denote by $\Omega_{f,\lambda}$ the set of all nonisolated points of $f^{-1}(\lambda)$ and we set $\Omega_{f}=\bigcup_{\lambda\in\mathbb{C}^{m}}\Omega_{f,\lambda}$. For a compact space $X$, we denote by $C(X)$ the algebra of all continuous complex-valued functions on $X$. A uniform algebra on $X$ is a supremum norm closed subalgebra of $C(X)$ that contains the constant functions and separates the points of $X$. In particular, a uniform algebra is a commutative Banach algebra. We will denote the maximal ideal space of a commutative Banach algebra $A$ by $\mathfrak{M}_{A}$. Given $x\in A$ we will denote the Gelfand transform of $x$ as usual by $\widehat{x}$. If $A$ is a Banach algebra of continuous complex-valued functions on a subset of $\mathbb{C}^{n}$ and the complex coordinate functions $z_{1},\ldots,z_{n}$ belong to $A$, we will let $\pi_{A}:\mathfrak{M}_{A}\rightarrow\mathbb{C}^{n}$ denote the map given by $\pi_{A}(x)=\bigl{(}\widehat{z}_{1}(x),\ldots,\widehat{z}_{n}(x)\bigr{)}$. As usual ${H^{\infty}(\Omega)}$ will denote the algebra of bounded holomorphic functions on $\Omega$ equipped with the supremum norm. If $A$ is an algebra of bounded continuous complex-valued functions on $\Omega$ and $f_{1},\ldots,f_{m}$ are bounded continuous complex-valued functions on $\Omega$, we will denote by $A[f_{1},\ldots,f_{m}]$ the norm closed subalgebra of $L^{\infty}(\Omega)$ generated by $A$ and the functions $f_{1},\ldots,f_{m}$. This last notation, which is rather standard, differs from the notation in [ST19] in that in [ST19] the notation $A[f_{1},\ldots,f_{m}]$ is used to denote the algebra generated by $A$ and $f_{1},\ldots,f_{m}$ without taking closure.

In the terminology of Sahutoğlu and Tikaradze, an $L^{\infty}$-pseudoconvex domain is a pseudoconvex domain on which the $\overline{\partial}$ problem is solvable in $L^{\infty}$. (See [ST19] for the precise definition.) The approximation theorem of Sahutoğlu and Tikaradze referred to above is the following.

This theorem can be regarded as a partial extension to several variables of an approximation theorem of Christopher Bishop.

Sahutoğlu and Tikaradze’s proof of Theorem 1 was inspired by Bishop’s proof of Theorem 2, and like Bishop’s proof, it is rather long and complicated. A simpler proof of Bishop’s theorem was given by the second author of the present paper in [Izz93] using uniform algebra methods. Here we will use uniform algebra methods to give a simple proof of Theorem 1 with the hypothesis that $\Omega$ is an $L^{\infty}$-pseudoconvex domain replaced by the hypothesis that $\Omega$ is open when regarded as a subset of the maximal ideal space $\mathfrak{M}_{{H^{\infty}(\Omega)}}$ of ${H^{\infty}(\Omega)}$. (We regard $\Omega$ as a subset of $\mathfrak{M}_{{H^{\infty}(\Omega)}}$ by identifying each point $\lambda$ in $\Omega$ with the functional “evaluation at $\lambda$”.) We state the result explicitly here.

By exactly the same prove we will also establish the analogous assertion for the algebra $A(\Omega)$ of continuous complex-valued functions on $\overline{\Omega}$ that are holomorphic on $\Omega$.

We will see also that a similar argument in combination with a result in the second author’s paper [Izz03] yields yet another proof of Theorem 2.

The class of domains $\Omega$ for which $\Omega$ is open in $\mathfrak{M}_{{H^{\infty}(\Omega)}}$ (or $\mathfrak{M}_{A(\Omega)}$) is quite broad. To see this, note that for $A$ a Banach algebra of continuous complex-valued functions on $\Omega$ containing the functions $z_{1},\ldots,z_{n}$, the set $\Omega$ is open in $\mathfrak{M}_{A}$ whenever $\pi_{A}$ is injective over $\Omega$, since in that case $\Omega$ (regarded as a subset of $\mathfrak{M}_{A}$) coincides with $\pi_{A}^{-1}(\Omega)$. Furthermore this injectivity over $\Omega$ obviously holds whenever Gleason’s problem is solvable for $A$, i.e., whenever, for every point $a=(a_{1},\ldots,a_{n})\in\Omega$, the functions $z_{1}-a_{1},\ldots,z_{n}-a_{n}$ generate the ideal of functions in $A$ vanishing at $a$. Gleason’s problem has been extensively studied and is known to be solvable for ${H^{\infty}(\Omega)}$ and $A(\Omega)$ on many classes of domains. (See for instance, [AS79] for the case of strongly pseudoconvex domains, or for the particular case of the ball [Rud08].)

Theorems 1, 3, and 4 can be reformulated using the notion of essential set. For a uniform algebra $A$ on a compact space $X$, the essential set ${\mathscr{E}}$ for $A$ is the smallest closed subset of $X$ such that $A$ contains every continuous complex-valued function on $X$ that vanishes on ${\mathscr{E}}$. The existence of the essential set was proved by Herbert Bear [Bea59] (or see [Bro69]). Theorem 4 asserts that under the given hypotheses on $\Omega$ and $f$, the essential set for $A(\Omega)[\overline{f}_{1},\ldots,\overline{f}_{m}]$ is contained in $b\Omega\cup\overline{\Omega}_{f}$. The conclusion of Theorems 1 and 3 can be reformulated as the assertion that the essential set for ${H^{\infty}(\Omega)}[\overline{f}_{1},\ldots,\overline{f}_{m}]$ regarded as a uniform algebra on its maximal ideal space is contained in $(\mathfrak{M}_{{H^{\infty}(\Omega)}}\setminus\Omega)\cup\overline{\Omega}_{f}$.

One reason for interest in the above theorems stems from an application to Toeplitz operators given by Sahutoğlu and Tikaradze. Let $L^{2}_{a}(\Omega)$ denote the Bergman space, i.e., the space of square integrable, holomorphic functions on $\Omega$, and let $P:L^{2}(\Omega)\rightarrow L^{2}_{a}(\Omega)$ denote the Bergman projection, i.e., the orthogonal projection of $L^{2}(\Omega)$ onto $L^{2}_{a}(\Omega)$. For $\phi\in L^{\infty}(\Omega)$ the Toeplitz operator $T_{\phi}:L^{2}_{a}(\Omega)\rightarrow L^{2}_{a}(\Omega)$ is defined by the equation $T_{\phi}(f)=P(\phi f)$. The commuting Toeplitz operator problem is to characterize those functions $\phi,\psi\in L^{\infty}(\Omega)$ such that $T_{\phi}$ and $T_{\phi}$ commute. With the Hardy space in place of the Bergman space, the commuting Toeplitz operator problem was solved by Arlen Brown and Paul Halmos in [BH64]. On the Bergman space, the problem is still open even on the disk. There are, however, various partial solutions including the following result due to Sheldon Axler, Željko Čučkovič, and Nagisetti Rao [AČR00].

Axler, Cučkovič, and Rao obtained this theorem as a consequence of Theorem 2 of Bishop. Sahutoğlu and Tikaradze used their partial extension of Bishop’s theorem to several variables (Theorem 1 above), to give a generalization of the Axler-Cučkovič-Rao theorem to several variables.

As an intermediate step in the proof of Theorem 6, Sahutoğlu and Tikaradze used Theorem 1 to prove an $L^{p}$-approximation theorem.

Repeating the proofs of Theorems 6 and 7 given in [ST19] with our Theorem 3 in place of Theorem 1 shows that Theorems 6 and 7 continue to hold with the hypothesis that $\Omega$ is an $L^{\infty}$-domain replaced by the hypothesis that $\Omega$ is open in $\mathfrak{M}_{{H^{\infty}(\Omega)}}$. (See, however, Remark 10 at the end of our paper).

We will need the following elementary lemma whose proof we include for completeness.

When $\Omega$ is bounded, the proof of this theorem is rather easy. The case of unbounded $\Omega$ is more difficult.

We thank Akaki Tikaradze for useful conversations.