Gleason parts for algebras of holomorphic functions in infinite dimensions

Abstract

For a complex Banach space X with open unit ball $B_X,$ consider the Banach algebras $\mathcal {H}^\infty (B_X)$ of bounded scalar-valued holomorphic functions and the subalgebra $\mathcal {A}_u(B_X)$ of uniformly continuous functions on $B_X.$ Denoting either algebra by $\mathcal {A},$ we study the Gleason parts of the set of scalar-valued homomorphisms $\mathcal {M}(\mathcal {A})$ on $\mathcal {A}.$ Following remarks on the general situation, we focus on the case $X = c_0,$ giving a complete characterization of the Gleason parts of $\mathcal {M}(\mathcal {A}_u(B_{c_0}))$ and, among other things, showing that every fiber in $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$ over a point in $B_{\ell _\infty }$ contains $2^c$ discs lying in different Gleason parts.

1 Introduction

Let X be a complex Banach space with open unit ball $B_X$ and unit sphere $S_X.$ Using standard notation, $\mathcal {A}_u(B_X)$ denotes the Banach algebra of holomorphic (complex-analytic) functions $f:B_X \rightarrow \mathbb {C}$ that are uniformly continuous on $B_X.$ This algebra is clearly a subalgebra of $\mathcal {H}^\infty (B_X),$ the Banach algebra of all bounded holomorphic mappings on $B_X$ both endowed with the supremum norm $\Vert f\Vert =\sup \{|f(x)|\ |\ \Vert x\Vert <1\}$. Also each function in $\mathcal {A}_u(B_X)$ extends continuously to $\overline{B}_X$. Then, the maximal ideal space (the spectrum for short) of $\mathcal {A}_u(B_X),$ that is the set of all nonzero $\mathbb {C}-$valued homomorphisms $\mathcal {M}(\mathcal {A}_u(B_X))$ on $\mathcal {A}_u(B_X),$ contains the point evaluations $\delta _x$ for all $x \in X, \ \Vert x\Vert \le 1.$ Our primary interest here will be in the structure of the set of such homomorphisms, and our specific focus will be on the Gleason parts of $\mathcal {M}(\mathcal {A}_u(B_X))$ and $\mathcal {M}(\mathcal {H}^\infty (B_X))$ when $X = c_0.$ Classically, in the case of Banach algebras of holomorphic functions on a finite dimensional space, the study of Gleason parts was motivated by the search for analytic structure in the spectrum. That remains true in our case, in which the holomorphic functions have as their domain the (infinite dimensional) ball of X. However, in infinite dimensions the situation is more complicated and more interesting. For instance, in this case, we will exhibit non-trivial examples of Gleason parts intersecting more than one fiber; this phenomenon holds in the finite dimensional case in only simple, uninteresting cases. Unlike the situation when dim $X < \infty ,$ it is well-known (see, e.g., [3]) that $\mathcal {M}(\mathcal {A}_u(B_X))$ usually contains much more than mere evaluations at points of $\overline{B}_X$. As we will see, the study of Gleason parts of $\mathcal {M}(\mathcal {A}_u(B_X))$ in the case of an infinite dimensional X is considerably more difficult than in the easy, finite dimensional situation. Now, when the algebra considered is $\mathcal {H}^\infty (\mathbb {D})$ the seminal paper of Hoffman [17] evidences the complicated nature of the Gleason parts for its spectrum (see also [16, 19, 22]). So, it is not surprising that our results when $\mathbb {D}$ is replaced by $B_X$ are incomplete. However, as we will see, much information about Gleason parts for both the $\mathcal {A}_u$ and $\mathcal {H}^\infty$ cases can be obtained when $X = c_0.$

As just mentioned, we will concentrate on the case $X=c_0$, which is the natural extension of the polydisc $\mathbb {D}^n$. After a review in Sect. 1 of necessary background and some general results, the description of Gleason parts for $\mathcal {M}(\mathcal {A}_u(B_{c_0}))$ will constitute Sect. 2. Finally, in Sect. 3 we will discuss what we have learned about Gleason parts for $\mathcal {M}(\mathcal {H}^\infty (B_{c_0})).$

For general theory of holomorphic functions we refer the reader to the monograph of Dineen [12] and for further information on uniform algebras and Gleason parts we suggest the books of Bear [5], Gamelin [14], Garnett [15] and Stout [21].

2 Background and general results

In this section, we will discuss some simple results concerning Gleason parts for $\mathcal {M}(\mathcal {A})$ where $\mathcal {A}$ is an algebra of holomorphic functions defined on the open unit ball of a general Banach space X. Namely, $\mathcal {A}$ will denote either $\mathcal {A}_u(B_X)$ or $\mathcal {H}^\infty (B_X)$. For a Banach space X, as usual $X^*$ and $X^{**}$ denote the dual and the bidual spaces, respectively. We begin with very short reviews of:

• (i) Gleason parts ([5, 14]) and

• (ii) the particular Banach algebras of holomorphic functions that we are interested in.

  • (i) Let $\mathcal {A}$ be a uniform algebra and let $\mathcal {M}(\mathcal {A})$ denote the compact set of non-trivial homomorphisms $\varphi :\mathcal {A} \rightarrow \mathbb {C}$ endowed with the $w(\mathcal {A}^*,\mathcal {A})$ topology. For $\varphi , \psi \in \mathcal {M}(\mathcal {A}),$ we set the pseudo-hyperbolic distance

    $$\begin{aligned} \rho (\varphi , \psi ) := \sup \{|\varphi (f)| \ | \ f \in \mathcal {A}, \Vert f\Vert \le 1, \psi (f) = 0 \}. \end{aligned}$$

    Recall that when $\mathcal {A}= \mathcal {A}(\mathbb {D})$ or $\mathcal {A}=\mathcal {H}^\infty (\mathbb {D})$, the pseudo-hyperbolic metric for $\lambda$ and $\mu$ in the unit disc $\mathbb {D}$ is given by

    $$\begin{aligned} \rho (\delta _\lambda , \delta _\mu ) = \Big | \frac{\lambda - \mu }{1 - \overline{\lambda }\mu }\Big |. \end{aligned}$$

    Also, the formula given above remains true if $\mathcal {A}=\mathcal {A}(\mathbb {D})$ for $\lambda , \mu \in \overline{\mathbb {D}}$, if $|\lambda |=1$ and $\lambda \ne \mu$. Clearly, in this case, $\rho (\delta _\lambda , \delta _\mu ) =1$.

    The following very useful relation is well known (see, for instance, [5, Theorem 2.8]):

    $$\begin{aligned} \Vert \varphi - \psi \Vert = \frac{2 - 2\sqrt{1 - \rho (\varphi , \psi )^2}}{\rho (\varphi ,\psi )}. \end{aligned}$$

    (1.1)

    Noting that it is always the case that $\Vert \varphi - \psi \Vert \ (\equiv \sup _{\Vert f\Vert \le 1} |\varphi (f) - \psi (f)|) \le 2,$ the main point here being that $\Vert \varphi - \psi \Vert < 2$ if and only if $\rho (\varphi , \psi ) <1.$ From this (with some work), it follows that by defining $\varphi \sim \psi$ to mean that $\rho (\varphi , \psi ) < 1$ leads to a partition of $\mathcal {M}(\mathcal {A})$ into equivalence classes, called Gleason parts. Specifically, for each $\varphi \in \mathcal {M}(\mathcal {A}),$ the Gleason part containing $\varphi$ is the set

    $$\begin{aligned} \mathcal {GP}(\varphi ) := \{ \psi \ | \ \rho (\varphi ,\psi ) < 1\}. \end{aligned}$$

    We remark that it was perhaps König [18] who coined the phrase Gleason metric for the metric $\Vert \varphi - \psi \Vert .$

  • (ii) We first recall [10] that any $f \in \mathcal {H}^\infty (B_X)$ can be extended in a canonical way to $\tilde{f} \in \mathcal {H}^\infty (B_{X^{**}}).$ Moreover, the extension $f \leadsto \tilde{f}$ is a homomorphism of Banach algebras. A standard argument shows that the canonical extension takes functions in $\mathcal {A}_u(B_X)$ to functions in $\mathcal {A}_u(B_{X^{**}}).$ Consequently, each point $z_0 \in B_{X^{**}}$ (resp. $\overline{B}_{X^{**}}$) gives rise to an element $\tilde{\delta }_{z_0}\in \mathcal {M}(\mathcal {H}^\infty (B_X))$ (resp. $\mathcal {M}(\mathcal {A}_u(B_{X}))).$ Here, for a given function $f, \tilde{\delta }_{z_0}(f) = \tilde{f}(z_0).$ Note that for $f\in \mathcal {A}_u(B_{X})$ and $z_0\in X^{**}$, with $\Vert z_0\Vert =1$, we are allowed to compute $\tilde{f}(z_0)$ and we will use this fact without further mention. Also, in order to avoid unwieldy notation we will omit the tilde over the $\delta$, simply writing $\delta _{z_0}(f).$ We recall that either for $\mathcal {A}=\mathcal {A}_u(B_X)$ or $\mathcal {A}=\mathcal {H}^\infty (B_X)$ there is a mapping $\pi :\mathcal {M}(\mathcal {A}) \rightarrow \overline{B}_{X^{**}}$ given by $\pi (\varphi ) := \varphi |_{X^*}.$ Note that this makes sense since $X^*\subset \mathcal {A}.$ It is not difficult to see that $\pi$ is surjective [3]. As usual, for any $z \in \overline{B}_{X^{**}},$ the fiber over z,  will be denoted by

    $$\begin{aligned} \mathcal {M}_z := \{ \varphi \in \mathcal {M}(\mathcal {A}) \ | \ \pi (\varphi ) = z\}. \end{aligned}$$

    As we will see, knowledge of the fiber structure is useful in the study of Gleason parts, in the context of the Banach algebras $\mathcal {A}_u(B_X)$ and $\mathcal {H}^\infty (B_X).$ The first instance of this occurs in part (b) of Proposition 1.1 below.

Proposition 1.1

Let X be a Banach space and $\mathcal {M} = \mathcal {M}(\mathcal {A})$ be as above.

• (a) The set $\{\delta _z:z\in B_{X^{**}}\}$ is contained in $\mathcal {GP}(\delta _0)$. In fact, $\rho (\delta _0,\delta _z) = \Vert z\Vert$ for each $z \in B_{X^{**}}.$

• (b) Let $z \in S_{X^{**}}$ and $w\in B_{X^{**}}$. Then, for any $\varphi \in \mathcal {M}_z$ and $\psi \in \mathcal {M}_w$, $\rho (\varphi , \psi ) = 1.$ That is, $\varphi$ and $\psi$ lie in different Gleason parts.

Proof

• (a) Fix $z \in B_{X^{**}}, z \ne 0,$ and $f \in \mathcal {A},$ such that $\Vert f\Vert \le 1$ and $f(0) = \delta _0(f) = 0.$ By an application of the Schwarz lemma to $\tilde{f} \in \mathcal {A}(X^{**}),$ we see that $|\delta _z(f)| = |\tilde{f}(z)| \le \Vert z\Vert .$ Therefore $\rho (\delta _0, \delta _z) \le \Vert z\Vert < 1,$ or in other words $\delta _z$ is in the same Gleason part as $\delta _0.$ In addition, if we apply the definition of $\rho$ to a sequence $(x^*_n) \subset \overline{B}_{X^{*}} \subset \mathcal {A}$ such that $|z(x^*_n)| \rightarrow \Vert z\Vert ,$ we get that $\rho (\delta _0, \delta _z) \ge \Vert z\Vert .$

• (b) As in part (a) and using that $\varphi \in \mathcal {M}_z$, we may choose a sequence $(x_n^*)$ of norm one functionals on X such that $\varphi (x^*_n) = z(x^*_n) \rightarrow \Vert z\Vert = 1.$ Observe that $| \psi (x^*_n) | = |w(x^*_n)| \le \Vert w\Vert < 1.$ For each $n, m \in \mathbb {N},$ the function $g_{n,m}:B_X\rightarrow \mathbb {C}$ defined as

  $$\begin{aligned} g_{n,m}(\cdot ) = \frac{(x^*_n(\cdot ))^m - w(x^*_n)^m}{\Vert (x^*_n)^m - w(x^*_n)^m \Vert } \end{aligned}$$

  is in $\mathcal {A} = \mathcal {A}_u(B_X)$ or $\mathcal {H}^\infty (B_X).$ Evidently, $\Vert g_{n,m} \Vert = 1$ and $\psi (g_{n,m}) = 0.$ In addition,

  $$\begin{aligned} |\varphi (g_{n,m})| \ge \frac{ |z(x^*_n)|^m - \Vert w\Vert ^m}{1 + \Vert w\Vert ^m}, \end{aligned}$$

  which approaches 1 with n and m. Then, $\rho (\psi , \varphi ) = 1$ and $\psi$ and $\varphi$ are in different parts.$\square$

In the classical situation of $\mathcal {M}(\mathcal {H}^\infty (\mathbb {D})),$ the Gleason part containing the evaluation at the origin, $\delta _0,$ consists of the set $\{ \delta _z \ | \ z \in \mathbb {D}\}.$ This known fact is made evident in view of Proposition 1.1 and the fact that fibers over points in $\mathbb {D}$ are singletons. In the case of an infinite dimensional space X, it can happen that fibers (over interior points) are bigger than single evaluations and also the Gleason part of $\delta _0$ could properly contain $B_{X^{**}}$. The following, which uses part (a) of Proposition 1.1, gives a glimpse at this situation.

Proposition 1.2

Let X be a Banach space. Fix r, $0< r < 1$ and consider $B_{X^{**}}(0,r) \approx \{ \delta _z \ | \ z \in X^{**}, \Vert z\Vert < r\} \subset \mathcal {M}(\mathcal {A}).$ Then the closure of $B_{X^{**}}(0,r)$ in $\mathcal {M}(\mathcal {A})$ is contained in $\mathcal {GP}(\delta _0).$

Proof

Fix $\varphi \in \mathcal {M}(\mathcal {A})$, $\varphi$ in the closure of $B_{X^{**}}(0,r)$, and choose any $f \in \mathcal {A}, f(0)=0, \Vert f\Vert = 1.$ By definition, for fixed $\varepsilon >0$ such that $r + \varepsilon <1$ there is $z \in B_{X^{**}}(0,r)$ such that $| \varphi (f) - \delta _z(f)| < \varepsilon .$ Then,

$$\begin{aligned} |\varphi (f) - \delta _0(f)| \le \varepsilon + |\delta _0(f) - \delta _z(f)|\le \varepsilon + \rho (\delta _0, \delta _z) < \varepsilon + r. \end{aligned}$$

Thus, $\rho (\varphi ,\delta _0) < 1,$ which concludes the proof. $\square$

In many common situations, there are norm-continuous polynomials P acting on the Banach space X whose restriction to $B_X$ is not weakly continuous. To give one very easy example, the $2-$homogeneous polynomial $P:\ell _2 \rightarrow \mathbb {C}, \ P(x) = \sum _n x_n^2$ is such that $1 = P(\frac{\sqrt{2}}{2}[e_1 + e_n]) \ne 1/2 = P(\frac{\sqrt{2}}{2}e_1).$ In these cases, the following corollary shows that the exact composition of $\mathcal {GP}(\delta _0)$ is somewhat more complicated.

Corollary 1.3

Let X be a Banach space which admits a (norm) continuous polynomial that is not weakly continuous when restricted to the unit ball. Then $B_{X^{**}} \subsetneqq \mathcal {GP}(\delta _0).$

Proof

Combining [6, Corollary 2] and [6, Proposition 3] if X admits a polynomial which is not weakly continuous when restricted to the unit ball, then there is a homogeneous polynomial P on X whose canonical extension $\tilde{P}$ to $X^{**}$ is not weak-star continuous at 0 when restricted to any ball $B_{X^{**}}(0,r),\ 0<r<1.$ Fix any r and choose a net $(z_\alpha ) \subset B_{X^{**}}(0,r)$ that is weak-star convergent to 0 and $\tilde{P}(z_\alpha ) \nrightarrow 0.$ Choosing a subnet if necessary, we may assume that $\tilde{P}(z_\alpha ) \rightarrow b \ne 0.$ Applying Proposition 1.2, if $\varphi \in \mathcal {M}(\mathcal {A})$ is a limit point of $\{\delta _{z_\alpha }\},$ then $\varphi \in \mathcal {GP}(\delta _0).$ Note that $\delta _0(P) = 0 \ne b = \varphi (P),$ so that $\delta _0 \ne \varphi .$ Finally, $\varphi \in \mathcal {M}_0,$ since $\pi (\varphi ) = \varphi |_{X^*},$ which shows that $\varphi \in \mathcal {GP}(\delta _0) \backslash B_{X^{**}}.$$\square$

Remark 1.4

Note that, under the hypothesis of the above result, by Proposition 1.1, each homomorphism $\varphi \in \mathcal {GP}(\delta _0)\backslash B_{X^{**}}$ should be in some fiber over points in $B_{X^{**}}$.

In the rest of this section, we will focus on the calculation of the pseudo-hyperbolic distance in some special, albeit important, situations. Here, we will have to distinguish between the cases $\mathcal {A} = \mathcal {A}_u(B_X)$ and $\mathcal {A} = \mathcal {H}^\infty (B_X).$

Proposition 1.5

Let X be a Banach space and $\mathcal {A}=\mathcal {A}_u(B_X)$ or $\mathcal {A}=\mathcal {H}^\infty (B_X)$. Suppose that there exists an automorphism $\Phi :B_X \rightarrow B_X$ and in addition for the case of $\mathcal {A}_u(B_X)$, assume $\Phi$ is uniformly continuous. Then, given $x\in B_X$ such that $\Phi (x)=0$, for any $y\in B_X$ we have

$$\begin{aligned} \rho (\delta _x,\delta _y) = \Vert \Phi (y)\Vert . \end{aligned}$$

Proof

We only prove the case $\mathcal {A}=\mathcal {A}_u(B_X)$. Let $f \in \mathcal {A}_u(B_X), \Vert f\Vert \le 1,$ such that $\delta _x(f) = f(x) = 0.$ As $f \circ \Phi ^{-1}$ is in $\mathcal {H}^\infty (B_X)$, we can apply the Schwarz lemma to obtain

$$\begin{aligned} |\delta _y(f)| = |f(y)| = |f \circ \Phi ^{-1}(\Phi (y))| \le \Vert \Phi (y)\Vert . \end{aligned}$$

Thus, from the definition of $\rho ,$ we see that $\rho (\delta _x,\delta _y) \le \Vert \Phi (y)\Vert .$

For the reverse inequality, choose a norm one functional $x^*\in X^*$ such that $x^*(\Phi (y)) = \Vert \Phi (y)\Vert ,$ and set $f = x^*\circ \Phi .$ Since $f \in \mathcal {A}_u(B_X)$ has norm at most 1 and satisfies $f(x) = 0,$ we get that

$$\begin{aligned} \rho (\delta _x, \delta _y) \ge |\delta _y(f)| = \Vert \Phi (y)\Vert . \end{aligned}$$

$\square$

Note that the proof of Proposition 1.5 shows that $\rho (\delta _x,\delta _y)$ is independent of the particular choice of the automorphism $\Phi .$

For subsequent embedding results, for a Banach space X and $\mathcal {A}=\mathcal {A}_u(B_X)$ or $\mathcal {A}=\mathcal {H}^\infty (B_X)$ we will use the Gleason metric on $\mathcal {M}(\mathcal {A})$. As we have already noted in (i) at the beginning of this section, this metric is the restriction of the usual distance given by the norm on $\mathcal {A}^*$. When we refer to the Gleason metric for elements of $B_{X^{**}}$, the open unit ball $B_{X^{**}}$ will be regarded as a subset of $\mathcal {M}(\mathcal {A})$. As we will see in the next proposition, under certain conditions, the automorphism $\Phi$ of Proposition 1.5 induces an isometry (for the Gleason metric) in the spectrum that sends some fibers onto different fibers. This type of isometry allows us to transfer information relative to Gleason parts intersecting one fiber to other fibers. Recall that a finite type polynomial on X is a function in the algebra generated by $X^*$. Also, a Banach space X is said to be symmetrically regular if every continuous linear mapping $T:X\rightarrow X^*$ which is symmetric (i. e. $T(x_1)(x_2)=T(x_2)(x_1)$ for all $x_1, x_2\in X$) turns out to be weakly compact.

Proposition 1.6

Let X be a Banach space and $\mathcal {A}=\mathcal {A}_u(B_X)$ or $\mathcal {A}=\mathcal {H}^\infty (B_X)$. Suppose that there exists an automorphism $\Phi :B_X \rightarrow B_X$ and in addition for the case of $\mathcal {A}_u(B_X)$, assume $\Phi$ and $\Phi ^{-1}$ are uniformly continuous.

• (i) The mapping $\Phi$ induces a composition operator $C_\Phi :\mathcal {A}\rightarrow \mathcal {A}$, $C_\Phi (f)=f\circ \Phi$ such that $\Lambda _\Phi := C_\Phi ^t|_{\mathcal {M}(\mathcal {A})}:\mathcal {M}(\mathcal {A})\rightarrow \mathcal {M}(\mathcal {A})$, the restriction of its transpose to $\mathcal {M}(\mathcal {A})$, is an onto isometry for the Gleason metric with inverse $\Lambda _\Phi ^{-1}=\Lambda _{\Phi ^{-1}}$.

• (ii) If for every $x^*\in X^*$, $x^*\circ \Phi$ and $x^*\circ \Phi ^{-1}$ are uniform limits of finite type polynomials then for any $x\in \overline{B}_{X}$, $\Lambda _\Phi (\mathcal {M}_x)= \mathcal {M}_{\Phi (x)}$. If in addition X is symmetrically regular, then, for any $z\in \overline{B}_{X^{**}}$, $\Lambda _\Phi (\mathcal {M}_z)= \mathcal {M}_{\widetilde{\Phi }(z)}$.

Proof

To prove (i), just notice that for $f\in \mathcal {A}$ and $\varphi \in \mathcal {M}(\mathcal {A})$,

$$\begin{aligned} \Lambda _{\Phi ^{-1}}(\Lambda _\Phi (\varphi ))(f)=\Lambda _\Phi (\varphi ) \big (f\circ \Phi ^{-1}\big )=\varphi (f). \end{aligned}$$

Through this equality it is easily seen that $\Vert \Lambda _\Phi (\varphi )-\Lambda _\Phi (\psi )\Vert =\Vert \varphi -\psi \Vert$, for all $\varphi , \psi \in \mathcal {M}(\mathcal {A})$.

It is enough to prove (ii) in the case X is symmetrically regular. Fix $z\in \overline{B}_{X^{**}}$ and take $\varphi \in \mathcal {M}_z$. Given $x_1^*, \ldots , x_n^*$ in $X^*$ as $\varphi$ is multiplicative, we have that

$$\begin{aligned} \varphi (x_1^*\ldots x_n^*)=\varphi (x_1^*)\ldots \varphi (x_n^*)=z(x_1^*)\ldots z(x_n^*). \end{aligned}$$

Thus, since any polynomial Q of finite type is a linear combination of elements as above, we have

$$\begin{aligned} \varphi (Q)=\widetilde{Q}(z). \end{aligned}$$

By hypothesis, for any $x^*\in X^*$ there exists a sequence $(Q_k)$ of polynomials of finite type that converges uniformly to $x^*\circ \Phi$ on $B_X$. Hence, the sequence $(\widetilde{Q}_k)$ converges to $\widetilde{x}^*\circ \widetilde{\Phi }$ uniformly on $B_{X^{**}}$ and $\widetilde{\Phi }$ admits a unique extension to $\overline{B}_{X^{**}}$ through weak-star continuity. Thus,

$$\begin{aligned} \Lambda _\Phi (\varphi )(x^*)= \varphi (x^*\circ \Phi )= \lim _k \varphi (Q_k)=\lim _k \widetilde{Q}_k(z)=(\widetilde{\Phi } (z))(x^*). \end{aligned}$$

Consequently, $\Lambda _{\Phi }(\mathcal {M}_z)\subset \mathcal {M}_{\widetilde{\Phi }(z)}$. Now, the reverse inclusion follows from (i) because, since X is symmetrically regular and arguing as in the proof of [8, Corollary 2.2], we know that $\widetilde{\Phi ^{-1}}\circ \widetilde{\Phi }=Id$. Therefore, $\Lambda _{\Phi }(\mathcal {M}_z)= \mathcal {M}_{\widetilde{\Phi }(z)}$. $\square$

To conclude this section, we give three examples of these results.

Example 1.7

Let $X = c_0$ and fix a point $x=(x_n) \in B_{c_0}$. Define the mapping $\Phi _x:B_{c_0} \rightarrow B_{c_0}$ as follows:

$$\begin{aligned} \Phi _x(y) = (\eta _{x_1}(y_1), \eta _{x_2}(y_2),\dots ), \end{aligned}$$

where $\eta _\alpha (\lambda ) = \frac{\alpha - \lambda }{1 - \overline{\alpha }\lambda }, \ \alpha , \lambda \in \mathbb {D}.$ In this case $\Phi _x$ is a uniformly continuous automorphism ($\Phi _x^{-1} = \Phi _x$) with $\Phi _x(x)=0$ and so, for any $y\in B_{c_0}$,

$$\begin{aligned} \rho (\delta _x,\delta _y) = \Vert \Phi _x(y)\Vert = \sup _{n \ge 1 }\Big | \frac{x_n - y_n}{1 - \overline{x_n}y_n}\Big | = \sup _{n \ge 1} \rho (\delta _{x_n},\delta _{y_n}). \end{aligned}$$

Also, $\Lambda _{\Phi _x}$ is an onto isometry for the Gleason metric in $\mathcal {M}(\mathcal {A})$ both for $\mathcal {A}= \mathcal {A}_u(B_{c_0})$ or $\mathcal {A}=\mathcal {H}^\infty (B_{c_0})$. Moreover, $\Lambda _{\Phi _x}(\mathcal {M}_z)=\mathcal {M}_{\widetilde{\Phi }_x(z)}$ for any $z\in \overline{B}_{\ell _\infty }$.

In the next section, we will discuss the more complicated, more interesting extension of the previous example to $z\in \overline{B}_{\ell _\infty }$; see Theorem 2.4.

Example 1.8

([2, Lemma 4.4]) Let $X = \ell _2$ and fix a point $x \in B_{\ell _2}.$ Define the mapping $\beta _x:B_{\ell _2}\rightarrow B_{\ell _2}$ as follows:

$$\begin{aligned} \beta _x(y) = \frac{1}{1 + \sqrt{1 - \Vert x\Vert ^2}} \left\langle \frac{x - y}{1 - \langle y,x\rangle }, x\right\rangle x + \sqrt{1 - \Vert x\Vert ^2} \frac{x-y}{1 - \langle y,x\rangle } \end{aligned}$$

$(y \in B_{\ell _2}).$ From [20, Proposition 1, p.132], we know that $\beta _x$ is an automorphism from $B_{\ell _2}$ onto itself, with inverse map $\beta _x^{-1} = \beta _x$ and $\beta _x(x)=0$.

Also, by expanding $1/[1-\langle y,x\rangle ]$ as a geometric series $\sum \langle y,x\rangle ^n$ and noting that the series converges uniformly on $\overline{B}_{\ell _2}$, we see that $\beta _x(y) = g(y)x+h(y)y$, where the functions g and h are in $\mathcal {A}_u(B_{\ell _2})$. Thus, $\beta _x$ is uniformly continuous. Applying Proposition 1.5, we see that for all $x, y \in B_{\ell _2}, \ \rho (\delta _x,\delta _y) = \Vert \beta _x(y)\Vert .$ Also, by Proposition 1.6, $\Lambda _{\beta _x}$ is an onto isometry for the Gleason metric in $\mathcal {M}(\mathcal {A})$, both for $\mathcal {A}= \mathcal {A}_u(B_{\ell _2})$ or $\mathcal {A}=\mathcal {H}^\infty (B_{\ell _2})$. Moreover, as Proposition 1.6 (ii) holds (see [2, Lemma 4.3]) $\Lambda _{\beta _x}(\mathcal {M}_y)=\mathcal {M}_{\beta _x(y)}$ for all $y\in \overline{B}_{\ell _2}$.

Example 1.9

Let H be an infinite dimensional Hilbert space and let $X = \mathcal {L}(H)$ be the Banach space of all bounded linear operators from H into itself. Fix $R\in B_{\mathcal {L}(H)}$ and denote by $R^*$ its adjoint operator. Define the mapping $\Phi _R$ on $B_{\mathcal {L}(H)}$ as follows:

$$\begin{aligned} \Phi _R(T)=(I-RR^*)^{\frac{1}{2}}(T - R)(I - R^*T)^{-1}(I-R^*R)^{\frac{1}{2}}, \end{aligned}$$

($T\in B_{\mathcal {L}(H)}$). Note that $\Phi _R:B_{\mathcal {L}(H)}\rightarrow B_{\mathcal {L}(H)}$ is an automorphism with inverse map $\Phi _{-R}$ and $\Phi _R(R)=0$. As in the example above, it can be seen that $\Phi _R$ is uniformly continuous. Then, by Proposition 1.5, for $R, S\in B_{\mathcal {L}(H)}$ we obtain $\rho (\delta _R,\delta _S) = \Vert \Phi _R(S)\Vert .$ Again, by Proposition 1.6, $\Lambda _{\Phi _R}$ is an onto isometry for the Gleason metric in $\mathcal {M}(\mathcal {A})$, both for $\mathcal {A}= \mathcal {A}_u(B_{\mathcal {L}(H)})$ or $\mathcal {A}=\mathcal {H}^\infty (B_{\mathcal {L}(H)})$.

3 Gleason parts for $\mathcal {M}(\mathcal {A}_u(B_{c_0}))$

Compared to other infinite dimensional Banach spaces, what is unusual about $X = c_0$ is that, in relative terms, there are very few continuous polynomials $P:c_0 \rightarrow \mathbb {C}.$ All such polynomials are norm limits of finite linear combinations of elements of $c_0^*= \ell _1.$ As a consequence, there are very few holomorphic functions on $c_0$ [12]. In particular, every $f \in \mathcal {A}_u(B_{c_0})$ is a uniform limit of such polynomials. Thus, since any homomorphism is automatically continuous, its action on $\mathcal {A}_u(B_{c_0})$ is completely determined by its action on $c_0^*.$ In other words, $\mathcal {M}(\mathcal {A}_u(B_{c_0}))$ is precisely $\{ \delta _{z} \ | \ \ z \in \overline{B}_{\ell _\infty } \}.$ Note that if $c_0$ were replaced by $\ell _p,$ this approximation result would be false, and in fact $\mathcal {M}(\mathcal {A}_u(B_{\ell _p}))$ is considerably larger and more complicated than $\overline{B}_{\ell _p} \approx \{\delta _z \ | \ z \in \overline{B}_{\ell _p}\}$ (see, e.g., [13]).

Our aim here will be to get a reasonably complete description of the Gleason parts of $\mathcal {M}(\mathcal {A}_u(B_{c_0})).$ As just mentioned, our work is greatly helped by the fact that we know exactly what $\mathcal {M}(\mathcal {A}_u(B_{c_0}))$ is, namely that it can be associated with $\overline{B}_{\ell _\infty }.$ A special role is played by homomorphisms $\delta _z$ where z belongs to the distinguished boundary $\mathbb {T}^\mathbb {N}$, the set of all elements $z=(z_n)$ such that $|z_n|=1$ for all n. Also, notice that compared with the finite dimensional situation, there is a new and interesting “wrinkle” here in that there are unit vectors $z = (z_n)\in \overline{B}_{\ell _\infty }$ all of whose coordinates have absolute value smaller than 1. We begin with a straightforward lemma.

Lemma 2.1

For any $\varnothing \ne \mathbb {N}_0 \subset \mathbb {N},$ let $\Gamma :\ell _\infty \rightarrow \ell _\infty (\mathbb {N}_0)$ be the projection mapping taking $z = (z_j)_{j \in \mathbb {N}} \mapsto \Gamma (z) = (z_j)_{j \in \mathbb {N}_0}.$ Then for all $z, w \in \overline{B}_{\ell _\infty },$ the following inequality holds:

$$\begin{aligned} \Vert \delta _{\Gamma (z)} - \delta _{\Gamma (w)}\Vert \le \Vert \delta _z - \delta _w\Vert . \end{aligned}$$

Proof

Clearly, $\Gamma$ is a linear operator having norm 1, and $\Gamma (c_0) = c_0(\mathbb {N}_0).$ Thus each $f \in \mathcal {A}_u(B_{c_0(\mathbb {N}_0)})$ generates a function $g \in \mathcal {A}_u(B_{c_0})$ given by $g = f \circ \Gamma |_{c_0}$ having the same norm as f. An easy verification shows that the extension of g to $\mathcal {A}_u(B_{\ell _\infty })$ is given by $\tilde{g} = \tilde{f} \circ \Gamma .$ Therefore for all $z, w \in \ell _\infty , \Vert z\Vert , \Vert w\Vert \le 1,$

$$\begin{aligned} \Vert \delta _{\Gamma (z)} - \delta _{\Gamma (w)}\Vert= & {} \sup \{| \tilde{f}(\Gamma (z)) - \tilde{f}(\Gamma (w))| \ | \ f \in \mathcal {A}_u(B_{c_0(\mathbb {N}_0)}), \ \Vert f\Vert \le 1\} \\\le & {} \sup \{ | \tilde{g}(z) - \tilde{g}(w)| \ | \ g \in \mathcal {A}_u(B_{c_0}),\ \Vert g\Vert \le 1\} = \Vert \delta _z - \delta _w\Vert . \end{aligned}$$

$\square$

Another way to restate Lemma 2.1 is as follows: if $\delta _z \in \mathcal {GP}(\delta _w),$ then $\delta _{\Gamma (z)} \in \mathcal {GP}(\delta _{\Gamma (w)}).$ Since $\mathbb {N}_0$ is allowed to be finite, say of cardinal k, if $\delta _z$ and $\delta _w$ are in the same Gleason part, then their projections onto finite coordinates (viewed as being in $\mathbb {D}^k$) are also in the same Gleason part. Our next result examines the situation: Suppose that $z, w \in \overline{B}_{\ell _{\infty }}$ are such that $\delta _z$ and $\delta _w$ are in the same Gleason part. What can we say about the coordinates where these points differ and where these points are identical?

Lemma 2.2

For $z, w \in \overline{B}_{\ell _\infty }$, let $\mathbb {N}_0 = \{n \in \mathbb {N}\ | \ z_n \ne w_n\}$ and $\Gamma :\ell _\infty \rightarrow \ell _\infty (\mathbb {N}_0)$ be the projection as in Lemma 2.1. Then

$$\begin{aligned} \Vert \delta _z - \delta _w\Vert =\Vert \delta _{\Gamma (z)} - \delta _{\Gamma (w)}\Vert . \end{aligned}$$

Proof

Fix $z\in \overline{B}_{\ell _\infty }$ and define $\Theta _z:\ell _\infty (\mathbb {N}_0) \rightarrow \ell _\infty$ by:

$$\begin{aligned} (\Theta _z(u))_n = {\left\{ \begin{array}{ll} u_n &{} \quad \mathrm{\ if \ } n \in \mathbb {N}_0,\\ z_n &{} \quad \mathrm{\ if \ } n \notin \mathbb {N}_0. \end{array}\right. } \end{aligned}$$

Given $g \in \mathcal {A}_u(B_{c_0}), \ \Vert g\Vert \le 1,$ let $f = \tilde{g} \circ \Theta _z|_{c_0(\mathbb {N}_0)}.$ Note that f is well-defined since whenever $u \in \overline{B}_{\ell _\infty (\mathbb {N}_0)}$ then $\Theta _z(u) \in \overline{B}_{\ell _\infty }.$ It is easy to check that $f \in \mathcal {A}_u(B_{c_0(\mathbb {N}_0)}),$$\Vert f\Vert \le 1,$ and that $\tilde{f} = \tilde{g} \circ \Theta _z \in \mathcal {A}_u(B_{\ell _\infty (\mathbb {N}_0)}).$ From the definition of $\mathbb {N}_0,$ we see that

$$\begin{aligned} \Vert \delta _z - \delta _w\Vert= & {} \sup \{ |\tilde{g}(z) - \tilde{g}(w)| \ | \ g \in \mathcal {A}_u(B_{c_0}), \Vert g\Vert \le 1\}\\= & {} \sup \{ |\tilde{g}(\Theta _z \circ \Gamma (z)) - \tilde{g}(\Theta _z \circ \Gamma (w))| \ | \ g \in \mathcal {A}_u(B_{c_0}), \Vert g\Vert \le 1\}\\\le & {} \sup \{ | \tilde{f}(\Gamma (z)) - \tilde{f}(\Gamma (w))| \ | \ f \in \mathcal {A}_u(B_{c_0(\mathbb {N}_0)}), \Vert f\Vert \le 1\}\\= & {} \Vert \delta _{\Gamma (z)} - \delta _{\Gamma (w)}\Vert , \end{aligned}$$

and this, with the previous lemma, completes the proof. $\square$

One consequence of this result is that if $z\in \overline{B}_{\ell _\infty }$ with $|z_n| < 1$, for some n, then any $w\in \overline{B}_{\ell _\infty }$ such that $w_j = z_j,$ for all $j \ne n$, and $|w_n| < 1$, satisfies that $\delta _z$ and $\delta _w$ are in the same Gleason part. In particular, the only Gleason parts that are singleton points are the evaluations at points in the distinguished boundary $\mathbb {T}^\mathbb {N}$ of $\overline{B}_{\ell _\infty },$ i.e. the points in the Shilov boundary of $\mathcal {M}(\mathcal {A}_u(B_{c_0}))$.

Lemma 2.3

For each $n \in \mathbb {N},$ let $\Gamma _n:\ell _\infty \rightarrow \ell _{\infty }(\{1,2,\dots ,n\})$ be the natural projection. If z and w are both in $\overline{B}_{\ell _\infty },$ then

$$\begin{aligned} \Vert \delta _z - \delta _w\Vert = \lim _{n \rightarrow \infty } \Vert \delta _{\Gamma _n(z)} - \delta _{\Gamma _n(w)}\Vert = \sup _{n \in \mathbb {N}} \Vert \delta _{\Gamma _n(z)} - \delta _{\Gamma _n(w)}\Vert . \end{aligned}$$

Proof

First, Lemma 2.1 implies that the sequence $(\Vert \delta _{\Gamma _n(z)} - \delta _{\Gamma _n(w)}\Vert )$ is increasing and bounded by $\Vert \delta _z - \delta _w\Vert .$ Note also that for each $u \in \overline{B}_{\ell _\infty }, \ \Gamma _n(u) \overset{w(\ell _\infty ,\ell _1)}{\longrightarrow } u$, and if f is in $\mathcal {A}_u(B_{c_0}),$ it follows that $\tilde{f} \in \mathcal {A}_u(B_{\ell _\infty })$ is weak-star continuous. Consequently, $\tilde{f}(\Gamma _n(u)) \rightarrow \tilde{f}(u)$ as $n \rightarrow \infty .$ Therefore, for any $\varepsilon > 0$ take $f \in \mathcal {A}_u(B_{c_0}), \Vert f\Vert \le 1,$ such that $|\tilde{f}(z) - \tilde{f}(w)|>\Vert \delta _z -\delta _w\Vert -\frac{\varepsilon }{2}$. Then, we can find $n_0 \in \mathbb {N}$ such that both of the following hold:

$$\begin{aligned} |\tilde{f}(\Gamma _{n_0}(z)) - \tilde{f}(z)|< \frac{\varepsilon }{4} \quad \mathrm{and }\quad |\tilde{f}(\Gamma _{n_0}(w)) - \tilde{f}(w)| < \frac{\varepsilon }{4}. \end{aligned}$$

Hence, we see that

$$\begin{aligned} |\tilde{f}(z) - \tilde{f}(w)| \le \frac{\varepsilon }{4} + |\tilde{f}(\Gamma _{n_0}(z)) - \tilde{f}(\Gamma _{n_0}(w))| + \frac{\varepsilon }{4} \le \Vert \delta _{\Gamma _{n_0}(z)} - \delta _{\Gamma _{n_0}(w)}\Vert + \frac{\varepsilon }{2}. \end{aligned}$$

From this, we obtain that $\Vert \delta _z - \delta _w\Vert \le \Vert \delta _{\Gamma _{n_0}(z)} - \delta _{\Gamma _{n_0}(w)}\Vert + \varepsilon ,$ and the lemma follows. $\square$

For the subsequent description of the Gleason parts for $\mathcal {M}(\mathcal {A}_u(B_{c_0}))$ we introduce the following notation. For each $\lambda \in \mathbb {D}$ and $0<r<1$, we denote the pseudo-hyperbolic r-disc centered at $\lambda$ by

$$\begin{aligned} \mathcal {D}_r(\lambda )=\Big \{\mu \in \mathbb {D}\ |\ \rho (\delta _\lambda , \delta _\mu ) = \Big | \frac{\lambda - \mu }{1 - \overline{\lambda }\mu }\Big | < r\Big \}. \end{aligned}$$

Theorem 2.4

Let $z = (z_n)$ and $w = (w_n)$ be vectors in $\overline{B}_{\ell _\infty }.$ Then

$$\begin{aligned} \Vert \delta _z - \delta _w\Vert = \sup _{n \in \mathbb {N}} \Vert \delta _{z_n} - \delta _{w_n}\Vert . \end{aligned}$$

(2.1)

Moreover, if $\mathbb {N}_0=\{n \in \mathbb {N}\ | \ z_n \ne w_n\}$ then

$$\begin{aligned} \rho (\delta _z, \delta _w) = \sup _{n \in \mathbb {N}} \rho (\delta _{z_n}, \delta _{w_n})=\sup _{n \in \mathbb {N}_0} \Big |\frac{z_n - w_n}{1 - \overline{z_n}w_n}\Big |. \end{aligned}$$

(2.2)

Hence, given $z = (z_n)\in \overline{B}_{\ell _\infty }$ we have

$$\begin{aligned} \mathcal {GP}(\delta _z)=\bigcup _{0<r<1}\{\delta _w\ |\ w_n=z_n \ \text {if}\ |z_n|=1 \ \text {and}\ w_n\in \mathcal {D}_r(z_n)\ \text {if}\ |z_n|<1\,\}. \end{aligned}$$

Proof

By Lemma 2.3, it is enough to see that $\Vert \delta _{\Gamma _n(z)} - \delta _{\Gamma _n(w)}\Vert =\sup _{1\le k \le n}\Vert \delta _{z_k}- \delta _{w_k}\Vert$ for all n, where $\Gamma _n:\ell _\infty \rightarrow \ell _{\infty }(\{1,2,\dots ,n\})$ is the natural projection. By Lemma 2.2, we may also assume that $z_k\ne w_k$ for $k=1,\ldots , n$.

First, suppose that there exists k, $1\le k\le n$, such that $|z_k|=1$ or $|w_k|=1$. Then, $\Vert \delta _{z_k}- \delta _{w_k}\Vert =2$ and Lemma 2.1 gives the equality. Now, assume that $|z_k|, |w_k| <1$ for all $1\le k\le n$. Note that (1.1) describes $\Vert \delta _{\Gamma _n(z)} - \delta _{\Gamma _n(w)}\Vert$ in terms of $\rho (\delta _{\Gamma _n(z)}, \delta _{\Gamma _n(w)})$ by an increasing function. Using Example 1.7 we see that $\rho (\delta _{\Gamma _n(z)}, \delta _{\Gamma _n(w)})=\sup _{1\le k \le n} \rho (\delta _{z_k}, \delta _{w_k})$ and both equalities (2.1) and (2.2) follow from this.

Now, from $\rho (\delta _z, \delta _w) = \sup _{n \in \mathbb {N}} \rho (\delta _{z_n}, \delta _{w_n})$, we have

$$\begin{aligned} \mathcal {GP}(\delta _z)=\bigcup _{0<r<1} \{\delta _w\ |\ \rho (\delta _{z_n}, \delta _{w_n})<r, \text { for all } n\}. \end{aligned}$$

The conclusion trivially holds. $\square$

Notice that if the algebra is $\mathcal {H}^\infty (B_{c_0})$ and the vectors z, w belong to the open unit ball $B_{\ell _\infty }$, equation (2.1) coincides with equation (6.1) of [9, Theorem 6.6]. The next example illustrates how Theorem 2.4 can be used.

Example 2.5

Consider the following points in the sphere of $\ell _\infty :$$z = (1 - \frac{1}{n})_n, w = (1 - \frac{1}{n^2})_n,$ and $u = (1 - \frac{1}{2n})_n.$ Then $\delta _z$ and $\delta _w$ are in different Gleason parts, while $\delta _z$ and $\delta _u$ are in the same part.

To see this, observe that

$$\begin{aligned} \rho (\delta _z, \delta _w) = \sup _{n \in \mathbb {N}} \rho (\delta _{z_n},\delta _{w_n}) = \sup _{n \in \mathbb {N}} \left| \frac{z_n - w_n}{1 - \overline{z_n}w_n}\right| = \sup _{n \in \mathbb {N}} \left| \frac{n - n^2}{n^2 + n - 1}\right| = 1, \end{aligned}$$

which shows the first assertion. Similarly,

$$\begin{aligned} \rho (\delta _z, \delta _u) =\sup _{n \in \mathbb {N}} \rho (\delta _{z_n},\delta _{u_n}) = \sup _{n \in \mathbb {N}} \left| \frac{z_n - u_n}{1 - \overline{z_n}u_n}\right| =\sup _{n \in \mathbb {N}} \frac{n}{3n - 1} = \frac{1}{2}. \end{aligned}$$

Thus, $\delta _z$ and $\delta _u$ belong to the same Gleason part.

In order to give a more descriptive insight of the size of the Gleason parts, let us introduce some notation. Given $z= (z_n)\in \overline{B}_{\ell _\infty }$, let $\mathbb {N}_1$ be the (possibly empty) set $\mathbb {N}_1=\{n\in \mathbb {N}\ |\ |z_n|=1\}$. Now, $\mathbb {N}\setminus \mathbb {N}_1$ can be split into two disjoint sets $\mathbb {N}_2\cup \mathbb {N}_3$ such that

$$\begin{aligned} \sup _{n\in \mathbb {N}_2} |z_n|<1 \qquad \text { and }\qquad \sup _{n\in \mathbb {N}_3} |z_n|=1. \end{aligned}$$

Note that $\mathbb {N}_2$ and $\mathbb {N}_3$ could be empty and that they are not uniquely determined. For instance, if $\mathbb {N}_3$ is infinite and $\mathbb {N}_2$ is finite, we may redefine $\mathbb {N}_3$ as the union of $\mathbb {N}_3$ and $\mathbb {N}_2$ and redefine $\mathbb {N}_2$ to be empty. Also, $\mathbb {N}_3$ cannot be finite.

In this way we write $\mathbb {N}$ as a disjoint union satisfying the above conditions: $\mathbb {N}=\mathbb {N}_1\cup \mathbb {N}_2\cup \mathbb {N}_3$ and, therefore, the Gleason part containing $\delta _z$ satisfies:

$$\begin{aligned} \mathcal {GP}(\delta _z)=\left\{ \delta _w\ |\ w_n=z_n\ \text {if}\ n\in \mathbb {N}_1,\ \sup _{n\in \mathbb {N}_2}|w_n|<1 \ \text {and}\ \sup _{n\in \mathbb {N}_3}\left| \frac{z_n-w_n}{1-\overline{z_n}w_n}\right| <1 \,\right\} . \end{aligned}$$

Now, taking into account all the possibilities for the sets $\mathbb {N}_1$, $\mathbb {N}_2$ and $\mathbb {N}_3$ we obtain a more specific description of the different Gleason parts.

Corollary 2.6

Given $z\in \overline{B}_{\ell _\infty }$ and $\mathbb {N}_1$, $\mathbb {N}_2$, $\mathbb {N}_3$ defined as above, the Gleason part $\mathcal {GP}(\delta _z)$ satisfies one of the following:

• (i) If $\mathbb {N}=\mathbb {N}_2$ then $z\in B_{\ell _\infty }$ and $\mathcal {GP}(\delta _z)=\mathcal {GP}(\delta _0)=\{\delta _w\ |\ w \in B_{\ell _\infty } \}$. This produces the identification $\mathcal {GP}(\delta _z) \thickapprox B_{\ell _\infty }.$

• (ii) If $\mathbb {N}=\mathbb {N}_1$ then $z = (z_n)\in \mathbb {T}^\mathbb {N}$. So, $\mathcal {GP}(\delta _z)=\{\delta _z\}.$

• (iii) If $\mathbb {N}_3=\varnothing$ and $\mathbb {N}_1,\,\mathbb {N}_2\not =\varnothing$ then $\mathcal {GP}(\delta _z)=\{\delta _w\ |\ w_n=z_n \ \text {if}\ n\in \mathbb {N}_1 \ \text {and}\ \sup _{n\in \mathbb {N}_2} |w_n| < 1\,\}$. So,

  • if $\#(\mathbb {N}_2)=k$ then $\mathcal {GP}(\delta _z) \thickapprox \mathbb {D}^k$,

  • if $\mathbb {N}_2$ is infinite then, $\mathcal {GP}(\delta _z) \thickapprox B_{\ell _\infty }$.

  Both identifications are isometries with respect to the Gleason metric.

• (iv) If $\mathbb {N}_3$ is infinite and $\mathbb {N}_2=\varnothing$, then $\mathcal {GP}(\delta _z)$ contains $\mathbb {D}^k$ for every $k\in \mathbb {N}$ and this inclusion is an isometry for the Gleason metric. There is also a continuous injection of $B_{\ell _\infty }$ into $\mathcal {GP}(\delta _z)$.

• (v) If both $\mathbb {N}_2$ and $\mathbb {N}_3$ are infinite, then $\mathcal {GP}(\delta _z)$ contains an isometric copy of $B_{\ell _\infty }$, for the Gleason metric.

Proof

The results concerning isometries follow from Lemma 2.3 and Theorem 2.4. We only have to show the continuous injection of $B_{\ell _\infty }$ in item (iv). If we write $\mathbb {N}_3=\{n_k\}_k$, for each k there exists $r_k>0$ such that whenever $|z_{n_k}-w_{n_k}|<r_k$ we have $w_{n_k}\in \mathbb {D}$ and

$$\begin{aligned} \left| \frac{z_{n_k} - w_{n_k}}{1 - \overline{z_{n_k}}w_{n_k}}\right| <\frac{1}{2}. \end{aligned}$$

Then, denoting $C_{n_k}=r_k\mathbb {D}$ and $C_n=\{0\}$ for $n\not \in \mathbb {N}_3$ we obtain that if $w\in z+ \prod _{n=1}^\infty C_n$ then $\delta _w\in \mathcal {GP}(\delta _z)$. Since it is clear how to inject $B_{\ell _\infty }$ onto the set $z+ \prod _{n=1}^\infty C_n$, we derive the injection of $B_{\ell _\infty }$ into $\mathcal {GP}(\delta _z)$. $\square$

4 Gleason parts for $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$

Some of our knowledge about the Gleason parts of $\mathcal {M}(\mathcal {A}_u(B_{X}))$ passes to $\mathcal {M}(\mathcal {H}^\infty (B_{X}))$ if we consider the restriction mapping $\Upsilon _u:\mathcal {M}(\mathcal H^\infty (B_X))\rightarrow \mathcal {M}(\mathcal {A}_u(B_X))$. With obvious notation, it is clear that for any $\varphi , \psi \in \mathcal {M}(\mathcal H^\infty (B_X)),$

$$\begin{aligned} \rho (\varphi , \psi )\ge \rho _u(\Upsilon _u(\varphi ), \Upsilon _u(\psi )). \end{aligned}$$

Therefore, if $\mathcal {GP}_{\mathcal {A}_u}(\Upsilon _u(\varphi ))\not =\mathcal {GP}_{\mathcal {A}_u}(\Upsilon _u(\psi ))$ we also have $\mathcal {GP}_{\mathcal {H}^\infty }(\varphi )\not = \mathcal {GP}_{\mathcal {H}^\infty }(\psi )$.

Remark 3.1

Let $X=c_0$ and consider $z,w\in S_{\ell _\infty }$ such that $\mathcal {GP}_{\mathcal {A}_u}(\delta _z)\not = \mathcal {GP}_{\mathcal {A}_u}(\delta _w)$. Then, for any $\varphi \in \mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$ and $\psi \in \mathcal {M}_w(\mathcal {H}^\infty (B_{c_0}))$, as $\Upsilon _u(\varphi )=\delta _z$ and $\Upsilon _u(\psi )=\delta _w$, we also have $\mathcal {GP}_{\mathcal {H}^\infty }(\varphi )\not = \mathcal {GP}_{\mathcal {H}^\infty }(\psi )$. In particular, if $z\in \overline{B}_{\ell _\infty }$ belongs to the distinguished boundary $\mathbb {T}^\mathbb {N}$, every $\varphi \in \mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$ satisfies $\mathcal {GP}_{\mathcal {H}^\infty }(\varphi )\subset \mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$. That is, the Gleason part of $\varphi$ is contained in the fiber over z.

The following is somehow a counterpart to the above remark.

Proposition 3.2

Let $z,w\in S_{\ell _\infty }$ be such that $\mathcal {GP}_{\mathcal {A}_u}(\delta _z)= \mathcal {GP}_{\mathcal {A}_u}(\delta _w)$. Then there exist $\varphi \in \mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$ and $\psi \in \mathcal {M}_w(\mathcal {H}^\infty (B_{c_0}))$ satisfying $\mathcal {GP}_{\mathcal {H}^\infty }(\varphi )= \mathcal {GP}_{\mathcal {H}^\infty }(\psi )$.

Proof

Fix real numbers $(r_k)$, with $|r_k|<1$ and $r_k\nearrow 1$. Consider the sequences in $B_{\ell _\infty }$:

$$\begin{aligned} x^k=r_k z\rightarrow z\quad \text { and }\quad y^k=r_k w\rightarrow w. \end{aligned}$$

Now, as $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$ is weak-star compact, both $(\delta _{x^k})$ and $(\delta _{y^k})$ admit weak-star convergent subnets $(\delta _{x^{k(\alpha )}})_\alpha$, $(\delta _{y^{k(\alpha )}})_\alpha$ in $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$. Say

$$\begin{aligned} \delta _{x^{k(\alpha )}}{\longrightarrow }\varphi ; \qquad \qquad \delta _{y^{k(\alpha )}} {\longrightarrow }\psi . \end{aligned}$$

It is clear that $\varphi \in \mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$ and $\psi \in \mathcal {M}_w(\mathcal {H}^\infty (B_{c_0}))$. Now, as $\mathcal {GP}_{\mathcal {A}_u}(\delta _z)= \mathcal {GP}_{\mathcal {A}_u}(\delta _w)$, by Theorem 2.4 we have

$$\begin{aligned} C = \sup _n \Vert \delta _{z_n}-\delta _{w_n}\Vert _{\mathcal {M}(\mathcal {A}_u(\mathbb {D}))} =\Vert \delta _z-\delta _w\Vert _{\mathcal {M}(\mathcal {A}_u(B_{c_0}))} < 2. \end{aligned}$$

Then, given $f\in \mathcal {H}^\infty (B_{c_0})$, $\Vert f\Vert \le 1$, we can find $\alpha _0$ so that for any $\alpha \ge \alpha _0$,

$$\begin{aligned} \left| \delta _{x^{k(\alpha )}}(f)-\varphi (f)\right|<\frac{2-C}{4}\quad \text { and }\quad \left| \delta _{y^{k(\alpha )}}(f)-\psi (f)\right| <\frac{2-C}{4}. \end{aligned}$$

Therefore,

$$\begin{aligned} |\varphi (f)-\psi (f)|\le & {} \frac{2-C}{2} + \left| \delta _{x^{k(\alpha )}}(f)-\delta _{y^{k(\alpha )}}(f)\right| \\\le & {} \frac{2-C}{2} + \left\| \delta _{x^{k(\alpha )}}-\delta _{y^{k(\alpha )}}\right\| _{\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))}\\= & {} \frac{2-C}{2} + \sup _n \left\| \delta _{x_n^{k(\alpha )}}-\delta _{y_n^{k(\alpha )}}\right\| , \end{aligned}$$

where the last equality, which is a version of the statement of Theorem 2.4 for the spectrum $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$, appears in the proof of [9, Theorem 6.5]. Now, using the pseudo-hyperbolic distance for the unit disc $\mathbb {D}$ and the Schwarz–Pick theorem applied to the function $f(z)=r_{k(\alpha )}z$, for each fixed n such that $z_n\not = w_n$ we have

$$\begin{aligned} \rho (\delta _{x_n^{k(\alpha )}},\delta _{y_n^{k(\alpha )}})= & {} \left| \frac{x_n^{k(\alpha )} -y_n^{k(\alpha )}}{1-\overline{x_n^{k(\alpha )}}y_n^{k(\alpha )}}\right| = \left| \frac{r_{k(\alpha )}(z_n-w_n)}{1-r_{k(\alpha )}^2\overline{z_n}w_n}\right| \\\le & {} \left| \frac{z_n-w_n}{1-\overline{z_n}w_n}\right| \le \rho _u (\delta _z,\delta _w). \end{aligned}$$

Then, by (1.1) $\left\| \delta _{x^{k(\alpha )}}-\delta _{y^{k(\alpha )}}\right\| _{\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))} \le \Vert \delta _z-\delta _w\Vert _{\mathcal {M}(\mathcal {A}_u(B_{c_0}))} =C$.

Finally, $|\varphi (f)-\psi (f)|\le \frac{2-C}{2} + C= \frac{2+C}{2}$, for any $f\in \mathcal {H}^\infty (B_{c_0})$ with $\Vert f\Vert \le 1$. Therefore, $\Vert \varphi - \psi \Vert _{\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))}\le \frac{2+C}{2}<2$ and the proof is complete. $\square$

We next prove a kind of extension of the previous proposition. In [4, Lemma 2.9] it is shown that for $w\in \overline{B}_{\ell _\infty }$ and $b\in \mathbb {D}$ the fibers over w and (b, w) are homeomorphic. To recall the homeomorphism let us consider $\Lambda _b:B_{c_0}\rightarrow B_{c_0}$ given by $\Lambda _b(z)=(b,z)$ and let us denote by $S:B_{c_0}\rightarrow B_{c_0}$, the shift mapping $S(z)=(z_2, z_3,\dots )$. Now, the homomorphism between the fibers is given by

$$\begin{aligned} R_b:\mathcal {M}_w\rightarrow & {} \mathcal {M}_{(b,w)}\\ \varphi\mapsto & {} (f\in \mathcal {H}^\infty (B_{c_0})\mapsto \varphi (f\circ \Lambda _b)). \end{aligned}$$

Since both $\Lambda _b$ and S map the unit ball into the unit ball and $S\circ \Lambda _b=Id$ it is easy to see that $R_b$ is an isometry for the Gleason metric. Therefore, the fiber over w and the fiber over (b, w) (for any $w\in \overline{B}_{\ell _\infty }$) intersect the same “number” of Gleason parts.

From Remark 3.1 we know that if $z\in \mathbb {T}^{\mathbb {N}}$, then every $\varphi \in \mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$ satisfies that the Gleason part of $\varphi$ is contained in the fiber over z. The next proposition will show us not only that this does not hold for the fibers over points outside $\mathbb {T}^{\mathbb {N}}$, but also that any Gleason part outside $\mathbb {T}^{\mathbb {N}}$ must have elements from different fibers (in fact, at least from a disc of fibers).

Proposition 3.3

Given $b\in \mathbb {D}$, there exists $r_b>0$ such that if $|c-b|<r_b$ then, for all $\varphi \in \mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$, $R_b(\varphi )$ and $R_c(\varphi )$ are in the same Gleason part.

Proof

By the Cauchy integral formula, $\overline{B}_{\mathcal {H}^\infty (\mathbb {D})}$ is an equicontinuous set of functions. Therefore, there exists $r_b>0$ such that, if $|c-b|<r_b$ then $c\in \mathbb {D}$ and $|g(b)-g(c)|<1$, for all $g\in B_{\mathcal {H}^\infty (\mathbb {D})}$.

Hence, for $f\in \mathcal {H}^\infty (B_{c_0})$ with $\Vert f\Vert \le 1$ we have

$$\begin{aligned} |f(b,z)-f(c,z)|<1, \qquad \text { if } |c-b|<r_b,\ z\in B_{c_0}. \end{aligned}$$

Therefore, for every $\varphi \in \mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$,

$$\begin{aligned} \Vert R_b(\varphi ) - R_c(\varphi )\Vert= & {} \sup _{ \Vert f\Vert \le 1} |R_b(\varphi )(f) - R_c(\varphi )(f)|\\= & {} \sup _{ \Vert f\Vert \le 1} |\varphi (f\circ \Lambda _b - f\circ \Lambda _c)| \\\le & {} \sup _{ \Vert f\Vert \le 1} \Vert f\circ \Lambda _b - f\circ \Lambda _c \Vert \\= & {} \sup _{ \Vert f\Vert \le 1} \sup _{z\in B_{c_0}} |f(b,z) - f(c,z)| \le 1. \end{aligned}$$

$\square$

It is clear that the previous result is also valid between the fibers over w and over $(w_1, b, w_2,\dots )$ or $(w_1, w_2, b, w_3,\dots )$ and so on. That means that the Gleason part of any morphism in the fiber over a point outside $\mathbb {T}^{\mathbb {N}}$, must have elements from other fibers. In particular, there cannot be singleton Gleason parts outside the fibers over the points in $\mathbb {T}^{\mathbb {N}}$.

Thus far, the above results show that in $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$ there are Gleason parts intersecting different fibers (Propositions 3.2 and 3.3) and there are Gleason parts completely contained in a fiber (Remark 3.1). These results do not provide information on the size of the Gleason parts. In order to understand this feature a usual tool is the following result whose statement covers several versions appearing for instance in [15, Lemma 1.1, p. 393], [17, Lemma 2.1] and [21, p. 162].

Proposition 3.4

Let X, Y be Banach spaces and $\Omega _X\subset X, \Omega _Y\subset Y$ be open convex subsets. Let $\mathcal {A}$ be a uniform algebra of analytic functions defined on $\Omega _X$. Suppose that $\Phi :\Omega _Y\rightarrow \mathcal {M}(\mathcal {A})$ is an analytic inclusion. Then $\Phi (\Omega _Y)$ is contained in only one Gleason part.

Remark 3.5

Using [4] and [9] it was recently proved (independently) in [7] and in [11] that for each $z\in \overline{B}_{\ell _\infty }$ the fiber over z contains an analytic copy of $B_{\ell _\infty }$. Moreover, this injection is a Gleason isometry. Even by the previous proposition or simply using the Gleason isometry it follows that each of these copies of $B_{\ell _\infty }$ should be in a single Gleason part. Hence, for every $z\in \overline{B}_{\ell _\infty }$, there is a thick intersection of the fiber over z with a Gleason part.

Recall that given a compact set K and a uniform algebra $\mathcal {A}$ contained in C(K) a point $x\in K$ is called a strong boundary point for $\mathcal {A}$ if for every neighborhood V of x there exists $f\in \mathcal {A}$ such that $\Vert f\Vert =f(x)=1$ and $|f(y)|<1$ if $y\in K\setminus V$. We see in the next result that in the fiber over each $z\in \mathbb {T}^{\mathbb {N}}$ there is a strong boundary point. Since the Gleason part of a strong boundary point is just a singleton set, by the above remark, we derive that the fiber over any $z\in \mathbb {T}^{\mathbb {N}}$ intersects a thick Gleason part and also a singleton Gleason part.

Proposition 3.6

If $\mathcal {S}$ is the set of strong boundary points of $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$ then $\pi (\mathcal {S})=\mathbb {T}^{\mathbb {N}}$.

Proof

Denoting by $\mathcal {SB}$ the Shilov boundary of $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$, we have that $\mathcal {S}\subset \mathcal {SB}$ (see, e.g., [21, Corollary 7.24]) and thus $\pi (\mathcal {S})\subset \pi (\mathcal {SB})$. Therefore, in order to prove $\pi (\mathcal {S})=\mathbb {T}^{\mathbb {N}}$ it is enough to see $\pi (\mathcal {SB})\subset \mathbb {T}^{\mathbb {N}}$ and $\mathbb {T}^{\mathbb {N}}\subset \pi (\mathcal {S})$.

To prove the first inclusion, for each $n\in \mathbb {N}$, let us consider the map $j_n:\overline{B}_{\ell _\infty }\rightarrow \overline{\mathbb {D}}$ given by $j_n(z)=z_n$. Then, $P_n=j_n\circ \pi$ is a weak-star continuous mapping from $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$ into $\overline{\mathbb {D}}$.

Given $a\in \overline{B}_{\ell _\infty }\setminus \mathbb {T}^{\mathbb {N}}$, we want to show that $a\not \in \pi (\mathcal {SB})$. Since $a\not \in \mathbb {T}^{\mathbb {N}}$, there is n such that $|a_n|<1$. The set $C_n=\overline{\mathbb {D}}\setminus \mathbb {D}(a_n, \frac{1-|a_n|}{2})$ is a closed subset of $\mathbb {C}$, so $P_n^{-1}(C_n)$ is weak-star closed in $\mathcal {M}(\mathcal {H}^\infty (B_{c_0}))$. Also, since $C_n$ contains spheres of radius r, with r approaching to 1, for each $f\in \mathcal {H}^\infty (B_{c_0})$ we should have

$$\begin{aligned} \sup _{z\in B_{c_0}}|f(z)| = \sup _{\varphi \in P_n^{-1}(C_n)}|\varphi (f)|. \end{aligned}$$

Hence, $P_n^{-1}(C_n)$ is a boundary, which implies that $\mathcal {SB}\subset P_n^{-1}(C_n)$. Thus, $\pi (\mathcal {SB})\subset \pi (P_n^{-1}(C_n))$. Since $a\not \in \pi (P_n^{-1}(C_n))$, we obtain that $a\not \in \pi (\mathcal {SB})$.

For the second inclusion, let $a=(a_n)\in \mathbb {T}^{\mathbb {N}}$ be given by $a_n=e^{i\theta _n}$, for all n. As $(\frac{e^{-i\theta _n}}{2^n})\in \ell _1$ its associated function

$$\begin{aligned} x^*(x)=\sum _{n=1}^\infty \frac{e^{-i\theta _n}}{2^n}x_n \end{aligned}$$

belongs to $c_0^*$. Hence $f(x)=1+x^*(x)$ is holomorphic on $c_0$, bounded and uniformly continuous when restricted to $\overline{B}_{\ell _\infty }$. Observe that

$$\begin{aligned} | \tilde{f}(a)|=2;\qquad \text { while }\qquad | \tilde{f}(z)|<2,\ \text { for all }z\in \overline{B}_{\ell _\infty }, \ z\not = a. \end{aligned}$$

Associating f with its Gelfand transform $\widehat{f}$ and noting that $\widehat{f}$ attains its norm at a strong boundary point [21, Theorem 7.21], there is $\varphi \in \mathcal {S}$ such that $|\widehat{f}(\varphi )|=|\varphi (f)|=2$. Finally

$$\begin{aligned} \varphi (f)= \varphi (1)+\varphi (x^*)=1+x^*(\pi (\varphi ))=\tilde{f}(\pi (\varphi )). \end{aligned}$$

Therefore, $\pi (\varphi )=a$, and so $a\in \pi (\mathcal {S})$. $\square$

Up to now our study about the relationships between fibers and Gleason parts gives information about in which fibers there are singleton Gleason parts, which fibers intersect thick Gleason parts and which Gleason parts contain elements of different fibers. To complete this picture we now wonder about how many Gleason parts intersect a particular fiber. Should it always be more than one?

With respect to this question note that we have already seen that in the fiber over any $z\in \mathbb {T}^\mathbb {N}$ there is a singleton Gleason part and also a copy of $B_{\ell _\infty }$. So, at least two Gleason parts are inside each of these fibers. By translations through mappings $R_b$ (as in Proposition 3.3 and the subsequent comment) we also obtain that there are at least two Gleason parts intersecting the fiber over z for each $z\in S_{\ell _\infty }$ with all but finitely many coordinates of modulus 1.

The following results show that the fiber over any $z\in B_{\ell _\infty }$ intersects $2^c$ Gleason parts. First, relying on the proof of [9, Theorem 5.1] (see also [9, Corollary 5.2]) we obtain the desired result for the fiber over 0. For our purposes, we use the construction and notation given in [9].

Theorem 3.7

Let X be an infinite dimensional Banach space. Then there is an embedding $\Psi :(\beta (\mathbb {N})\setminus \mathbb {N})\times \mathbb {D}\rightarrow \mathcal {M}_0$ that is analytic on each slice $\{\theta \}\times \mathbb {D}$ and satisfies:

• (a) $\Psi (\theta ,\lambda )\not \in \mathcal {GP}(\delta _0)$ for each $(\theta ,\lambda )$.

• (b) $\mathcal {GP}(\Psi (\theta , \lambda ))\cap \mathcal {GP}(\Psi (\tilde{\theta }, \tilde{\lambda }))=\varnothing$ for each $\theta , \tilde{\theta }\in \beta (\mathbb {N})\setminus \mathbb {N}$ with $\theta \ne \tilde{\theta }$ and any $\lambda , \tilde{\lambda }\in \mathbb {D}$.

Proof

The existence of the analytic embedding $\Psi :(\beta (\mathbb {N})\setminus \mathbb {N})\times \mathbb {D}\rightarrow \mathcal {M}_0$ is given in [9, Theorem 5.1]. Below, we summarize the main ingredients used in its construction.

• There exists a sequence $(z_k)\subset B_{X^{**}}$ such that $\Vert z_k\Vert < \Vert z_{k+1}\Vert$ and $\Vert z_k\Vert$ is convergent to 1.

• The sequence of norms $(\Vert z_k\Vert )$ increases so rapidly that there exists an increasing sequence $(r_k)$, such that $0<r_k <\Vert z_k\Vert$ and $\sum (1-r_k)$ is finite.

• For a fixed sequence $(a_k)$ so that $0<a_k<1$ and $(a_k)\in \ell _1$, there exists $(L_k)\subset X^*$ such that $\Vert L_k\Vert <1$ and

  • $\cdot$$L_k(z_k)=r_k$,   for all k,

  • $\cdot$$L_j(z_k)=0$,    $1<k<j$,

  • $\cdot$$|L_j(z_k)| < a_j$,   for all $k>j$.

• There exists $0<r<1$ such that for all k, if $w_k:\mathbb {D}\rightarrow X$ is defined as $w_k(\lambda )=\big (\frac{r_k-\lambda }{1-r_k\,\lambda }\big )\frac{z_k}{r_k}$, then $\Vert w_k(\lambda )\Vert <1$ for all $|\lambda | < r$.

• The Blaschke product $G:B_{X^{**}}\rightarrow \mathbb {C}$, given by $G(z)=\prod _{j=1}^\infty \frac{r_j-L_j(z)}{1-r_j\,L_j(z)}$ belongs to $\mathcal {H}^\infty ( B_{X^{**}})$ and $|G(z)| <1$ if $\Vert z\Vert <1$.

• For $|\lambda |< r/2$ and each k there exists a unique $\xi _k(\lambda )$ such that $|\xi _k(\lambda )|<r$ and $G(w_k(\xi _k(\lambda )))=\lambda$ for all $|\lambda |< r/2$.

• For every k the function $z_k(\lambda ):=w_k(\xi _k(\lambda ))$ for $|\lambda |<r/2$ is a multiple of $z_k$, depends analytically on $\lambda$ and satisfies $\Vert z_k(\lambda )\Vert <1$ if $|\lambda |<r/2$ with $z_k(0)=z_k$.

Note that replacing $\mathbb {D}$ by $D=\{\lambda \in \mathbb {C}\ |\ \ |\lambda |<r/2 \}$, it is enough to show the result for $\beta (\mathbb {N})\setminus \mathbb {N}\times D$. The function $\Psi :\mathbb {N}\times D\rightarrow \mathcal {M}$ defined by $\Psi (k, \lambda )=\delta _{z_k(\lambda )}$ extends to a map $\Psi :\beta (\mathbb {N})\times D\rightarrow \mathcal {M}$ which is continuous on $\beta (\mathbb {N})$ for each fixed $\lambda$. Moreover, by [9, Theorem 5.1], we know that $\Psi (\beta (\mathbb {N})\setminus \mathbb {N}\times D)$ lies in the fiber over 0, $\mathcal {M}_0$.

Now, let us prove that (a) holds. As $\Psi$ is analytic on each slice, to show that $\Psi (\theta ,\lambda )\not \in \mathcal {GP}(\delta _0)$ for each $(\theta ,\lambda )$ it is enough to see that $\Psi (\theta , 0)\not \in \mathcal {GP}(\delta _0)$, for any $\theta$. Given $N\in \mathbb {N}$, consider $f_N\in \mathcal {H}^\infty ( B_{X^{**}})$ defined by

$$\begin{aligned} f_N(z){:=} \prod _{j>N}^\infty \frac{r_j-L_j(z)}{1-r_j\,L_j(z)}. \end{aligned}$$

Note that the restriction of $f_N$ to $B_X$ (which we still denote by $f_N$) belongs to $\mathcal {H}^\infty (B_X)$ and the canonical extension to $B_{X^{**}}$ of this restriction coincides with the original function.

Then, $\delta _0(f_N)=\prod _{j>N} r_j \rightarrow 1$ as $N\rightarrow \infty$. On the other hand, as $\Psi (k,0)=\delta _{z_k}$, for $k>N$,

$$\begin{aligned} \Psi (k,0)(f_N)=\prod _{j>N}^\infty \frac{r_j-L_j(z_k)}{1-r_j\,L_j(z_k)}=0. \end{aligned}$$

Now, take $\theta \in \beta (\mathbb {N})\setminus \mathbb {N}$. Then, there is a net $(j(\alpha ))\subset \mathbb {N}$, such that $\theta =\lim _\alpha j(\alpha )$. Thus,

$$\begin{aligned} \Psi (\theta , 0)(f_N)=\lim _\alpha \Psi (j(\alpha ),0)(f_N)=0. \end{aligned}$$

Therefore,

$$\begin{aligned} \rho (\delta _0, \Psi (\theta , 0)) \ge \sup _N \{|\delta _0(f_N)|\} =\sup _N \{\, \textstyle {\prod }_{j>N} r_j\, \}=1, \end{aligned}$$

which shows that $\Psi (\theta , 0)\not \in \mathcal {GP}(\delta _0)$.

To prove (b) let us see that if $\theta \ne \tilde{\theta }$ then $\mathcal {GP}(\Psi (\theta , D))\cap \mathcal {GP}(\Psi (\tilde{\theta }, D))=\varnothing$. Indeed, for $\theta \ne \tilde{\theta }$ there exists an infinite set $J\subset \mathbb {N}$ such that $\mathbb {N}\setminus J$ is also infinite and $\theta \in \overline{\{j:j\in J\}}$, $\tilde{\theta } \in \overline{\{j:j\in \mathbb {N}\setminus J\}}$.

Here, for $N\in \mathbb {N}$ consider $f_{(J,N)}\in \mathcal {H}^\infty ( B_{X^{**}})$ given by

$$\begin{aligned} f_{(J,N)}(z){:=} \prod _{\underset{j> N}{j \in J}}\, \frac{r_j-L_j(z)}{1-r_j\,L_j(z)}. \end{aligned}$$

Then, $\Vert f_{(J,N)}\Vert \le 1$ and $f_{(J,N)}(z_k)=0$ for all $k\in J, k>N$. Hence, as before, we obtain that $\Psi (\theta ,0)(f_{(J,N)})=0$.

On the other hand, $\tilde{\theta } =\lim _{\tilde{\alpha }} k(\tilde{\alpha })$. For these indexes $k(\tilde{\alpha })\not \in J$ with $k(\tilde{\alpha })>N$, the corresponding factor does not appear in $f_{(J,N)}$ and

$$\begin{aligned} \Psi (k(\tilde{\alpha }),0)(f_{(J,N)})= \prod _{\underset{N<j<k(\tilde{\alpha })}{j \in J}}\frac{r_j-L_j(z_{k(\tilde{\alpha })})}{1-r_j\,L_j(z_{k(\tilde{\alpha })})} \cdot \prod _{\underset{j> k(\tilde{\alpha })}{j \in J}} r_j. \end{aligned}$$

Notice that $\Big | \frac{r_j-L_j(z_{ k(\tilde{\alpha })})}{1-r_jL_j(z_{k(\tilde{\alpha })})}\Big | >\frac{r_j-a_j}{1+r_ja_j}$, for $k(\tilde{\alpha })>j$. By the inequality $1 -\frac{r_j-a_j}{1+ r_j\,a_j } < (1-r_j) + 2a_j$, the series $\sum _{j\ge 1} (1 -\frac{r_j-a_j}{1+ r_j\,a_j})$ converges, implying that the infinite product $\prod _{j\ge 1} \frac{r_j-a_j}{1+ r_j\,a_j}$ is convergent as well as the infinite product over $\{j\in J\}$.

Now, given $0<\varepsilon <1$ we can find $k_0\in \mathbb {N}$ such that for all $k\ge k_0$,

$$\begin{aligned} \prod _{\underset{j>k}{j \in J}} r_j>1-\varepsilon \qquad \text { and }\qquad \prod _{\underset{j>k}{j \in J}} \frac{r_j-a_j}{1+ r_j\,a_j} >1-\varepsilon . \end{aligned}$$

Then, for $N>k_0$ and $\tilde{\alpha }$ such that $k(\tilde{\alpha })>k_0$, we have

$$\begin{aligned} \prod _ {\underset{N< j< k(\tilde{\alpha })}{j \in J}} \Big |\frac{r_j-L_j(z_{k(\tilde{\alpha })})}{1-r_jL_j(z_{k(\tilde{\alpha })})}\Big |> \prod _ {\underset{N< j < k(\tilde{\alpha })}{j \in J}} \frac{r_j-a_j}{1+r_ja_j}> \prod _ {\underset{j>N}{j \in J}} \frac{r_j-a_j}{1+r_ja_j} > 1 -\varepsilon . \end{aligned}$$

Hence,

$$\begin{aligned} |\Psi (k(\tilde{\alpha }),0)(f_{(J,N)})| > (1 -\varepsilon )^2, \end{aligned}$$

and $|\Psi (\tilde{\theta },0)(f_{(J,N)})| \ge (1 -\varepsilon )^2$. Finally, for any $0<\varepsilon <1$

$$\begin{aligned} \rho (\Psi (\theta ,0)), \Psi (\tilde{\theta },0)) \ge \sup _N \{| \Psi (\tilde{\theta },0)(f_{(J,N)})|\} \ge (1 -\varepsilon )^2, \end{aligned}$$

and the result follows. $\square$

Next, we will see that there is a bijective biholomorphic mapping from $B_{\ell _\infty }$ into $B_{\ell _\infty }$ which is an isometry for the Gleason metric and transfers each fiber over an interior point to a different fiber. We use this fact to extend the conclusions in Theorem 3.7 to the fiber $\mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$ for any $z\in B_{\ell _\infty }$.

Lemma 3.8

Let $\alpha \in \mathbb {D}$ and let $\eta _\alpha :\mathbb {D}\rightarrow \mathbb {D}$ be the Moebius transformation,

$$\begin{aligned} \eta _\alpha (\lambda )=\frac{\alpha -\lambda }{1-{\overline{\alpha }}\lambda }. \end{aligned}$$

Given $|\alpha |\le s<1$, for any $\lambda \in \mathbb {D}$ with $|\lambda |\le s$ the following inequality holds:

$$\begin{aligned} |\eta _\alpha (\lambda )|\le \frac{2s}{1+s^2}. \end{aligned}$$

Proof

Notice that

$$\begin{aligned} 1-\Big |\frac{\alpha -\lambda }{1-\overline{\alpha } \lambda }\Big |^2 =\frac{|1-\overline{\alpha }\lambda |^2-|\alpha -\lambda |^2}{|1-\overline{\alpha }\lambda |^2} = \frac{(1-|\lambda |^2)(1-|\alpha |^2)}{|1-\overline{\alpha }\lambda |^2}. \end{aligned}$$

Hence, the result follows for any $|\lambda |\le s$ since

$$\begin{aligned} 1- \Big |\frac{\alpha -\lambda }{1-\overline{\alpha }\lambda }\Big |^2\ge \Big (\frac{1-s^2}{1+s^2}\Big )^2\ \text { and }\ \sqrt{1- \Big (\frac{1-s^2}{1+s^2}\Big )^2}=\frac{2s}{1+s^2}. \end{aligned}$$

$\square$

Proposition 3.9

Fix $a=(a_n)\in B_{\ell _\infty }$. The mapping $\Phi _a :B_{\ell _\infty } \rightarrow B_{\ell _\infty }$, defined by

$$\begin{aligned} \Phi _a(z)=(\eta _{a_n}(z_n)) \end{aligned}$$

is bijective and biholomorphic. Moreover, for any $x^*\in \ell _1$, the function $x^*\circ \Phi _a$ is uniformly continuous.

Proof

First, let us check that $\Phi _a(B_{\ell _\infty })\subset B_{\ell _\infty }$. Fix $z=(z_n)\in B_{\ell _\infty }$ and take $s=\max \{\Vert a\Vert ,\Vert z\Vert \} <1$. Using Lemma 3.8 we obtain

$$\begin{aligned} \Vert \Phi _a(z)\Vert =\sup _{n} |\eta _{a_n}(z_n)|\le \frac{2s}{1+s^2}<1. \end{aligned}$$

To check that $\Phi _a$ is holomorphic, by Dunford’s theorem it is enough to check that $\Phi _a$ is weak-star holomorphic, i.e. that $x^*\circ \Phi _a\in \mathcal {H}(B_{\ell _\infty })$ for every $x^*=(b_n)\in \ell _1$. Notice that $x^*\circ \Phi _a(z)=\sum _{n=1}^{\infty }b_n\eta _{a_n}(z_n)$, and

$$\begin{aligned} |b_n \eta _{a_n}(z_n)|\le |b_n|, \end{aligned}$$

for every $z\in B_{\ell _\infty }$ and every n. By the Weierstrass M-test, the series $\sum _{n=1}^{\infty }b_n\eta _{a_n}(z_n)$ converges absolutely and uniformly on $\overline{B}_{\ell _\infty }$ and as each $z\mapsto \eta _{a_n}(z_n)$ belongs to $\mathcal {A}_u(B_{\ell _\infty })$ we have actually proved that $x^*\circ \Phi _a\in \mathcal {A}_u(B_{\ell _\infty })$, for every $x^*\in \ell _1$. Thus $\Phi _a\in \mathcal {H}(B_{\ell _\infty },B_{\ell _\infty })$.

Finally as $\Phi _a\circ \Phi _{a}(z)=z$ for every $z\in B_{\ell _\infty }$, we obtain that $\Phi _a$ has inverse $\Phi _a^{-1}=\Phi _{a}$ and $\Phi _a$ is biholomorphic. $\square$

Remark 3.10

Observe that if we consider $a\in B_{c_0}$ and we restrict $\Phi _a$ to $z\in B_{c_0}$, then we obtain the biholomorphic mapping of Example 1.7.

Given $a\in B_{\ell _\infty }$ the restriction of $\Phi _a$ to $B_{c_0}$ will be denoted by ${\Phi _a}\big |_{c_0}$.

Theorem 3.11

Given $a\in B_{\ell _\infty }$, the mapping $C_{\Phi _a}:\mathcal {H}^\infty ( B_{c_0})\rightarrow \mathcal {H}^\infty ( B_{c_0})$ defined by

$$\begin{aligned} C_{\Phi _a}(f)=\tilde{f}\circ {\Phi _a}\big |_{c_0}, \end{aligned}$$

where $\tilde{f}:B_{\ell _\infty }\rightarrow \mathbb {C}$ is the canonical extension of each $f\in \mathcal {H}^\infty (B_{c_0})$, is an isometric isomorphism of Banach algebras.

Moreover, $\Lambda _{\Phi _a}:= C_{\Phi _a}^t|_{\mathcal {M}(\mathcal {H}^\infty ( B_{c_0}))}:\mathcal {M}(\mathcal {H}^\infty ( B_{c_0}))\rightarrow \mathcal {M}(\mathcal {H}^\infty ( B_{c_0}))$, the restriction of its transpose to $\mathcal {M}(\mathcal {H}^\infty ( B_{c_0}))$, is a surjective isometry for the Gleason metric with inverse $\Lambda _{\Phi _a}^{-1}=\Lambda _{\Phi _a}$ that satisfies

$$\begin{aligned} \Lambda _{\Phi _a}(\mathcal {M}_z)= \mathcal {M}_{\Phi _a(z)}, \end{aligned}$$

for every $z\in B_{\ell _\infty }$.

Proof

Clearly $C_{\Phi _a}$ is well-defined, $\Vert C_{\Phi _a}\Vert \le 1$ and it is an algebra homomorphism. Next we claim that given $f\in \mathcal {H}^\infty ( B_{c_0})$,

$$\begin{aligned} \widetilde{\tilde{f}\circ {\Phi _a}\big |_{c_0}}=\tilde{f}\circ \Phi _a. \end{aligned}$$

(3.1)

Let us observe that $\ell _\infty =C(\beta \mathbb {N})$ is a symmetrically regular space. Moreover, by Lemma 3.8, if $0<s<1$, then $m=\sup _{\Vert z\Vert \le s}\Vert \Phi _a(z)\Vert <1$. With this in mind, by the method of proof of [8, Corollary 2.2], we have

$$\begin{aligned} \widetilde{\tilde{f}\circ {\Phi _a}\big |_{c_0}}=\tilde{\tilde{f}}\circ \widetilde{ {\Phi _a}\big |_{c_0}}= \tilde{f}\circ \widetilde{ {\Phi _a}\big |_{c_0}} . \end{aligned}$$

By Proposition 3.9, ${\Phi _a}\big |_{c_0}$ is $w(c_0,\ell _1)$-uniformly continuous on $B_{c_0}$. Hence it has a unique extension to $B_{\ell _\infty }$ that is $w(\ell _\infty ,\ell _1)$-uniformly continuous on $B_{\ell _\infty }$ and it coincides with its canonical extension $\widetilde{ {\Phi _a}\big |_{c_0}}$. On the other hand, also by Proposition 3.9, $\Phi _a$ is $w(\ell _\infty ,\ell _1)$-uniformly continuous on $B_{\ell _\infty }$ and it is obviously an extension of ${\Phi _a}\big |_{c_0}$ to $B_{\ell _\infty }$. Thus, $\widetilde{ {\Phi _a}\big |_{c_0}}(z)=\Phi _a(z)$, for all $z\in B_{\ell _\infty }$.

From this equality we derive that $C_{\Phi _a}\circ C_{\Phi _a}(f)=f$ for every $f \in \mathcal {H}^\infty ( B_{c_0})$. Indeed,

$$\begin{aligned} C_{\Phi _a}\big ( C_{\Phi _a}(f)\big )(z)=\left( \widetilde{\tilde{f}\circ {\Phi _a}\big |_{c_0}}\circ \Phi _{a}{\big |_{c_0}}\right) (z)={\tilde{f}}\circ \widetilde{ {\Phi _a}\big |_{c_0}} \circ \Phi _a(z)={\tilde{f}}(z)=f(z), \end{aligned}$$

for every $z\in B_{c_0}$. As a consequence $C_{\Phi _a}$ is an isomorphism of algebras. Also we have $\Vert f\Vert \le \Vert C_{\Phi _a}\Vert \Vert C_{\Phi _a}(f)\Vert \le \Vert C_{\Phi _a}(f)\Vert$ for every f, and therefore $C_{\Phi _a}$ is an isometry.

Hence its transpose $C_{\Phi _a}^t$ when restricted to $\mathcal {M}(\mathcal {H}^\infty ( B_{c_0}))$ is well-defined and its range is again in $\mathcal {M}(\mathcal {H}^\infty ( B_{c_0}))$. Moreover, $\Lambda _{\Phi _a}\circ \Lambda _{\Phi _a}(\varphi )=\varphi$ for every $\varphi \in \mathcal {M}(\mathcal {H}^\infty ( B_{c_0}))$. Finally, for each $x^*\in \ell _1$, the function $\widetilde{x^*}\circ {\Phi _a}\big |_{c_0}$ belongs to $\mathcal {A}_u(B_{c_0})$ (as we have already observed) and so it is a uniform limit of finite type polynomials. Hence, as in the proof of Proposition 1.6, we obtain that $\Lambda _{\Phi _a}(\mathcal {M}_z)= \mathcal {M}_{\Phi _a(z)}$, for every $z\in B_{\ell _\infty }$. $\square$

Combining this last theorem with Theorem 3.7 we obtain that for each $z\in B_{\ell _\infty }$, the fiber $\mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$ contains $2^c$discs lying in different Gleason parts.

Corollary 3.12

Let $z\in B_{\ell _\infty }$. Then, there is an embedding of $\Psi :(\beta (\mathbb {N})\setminus \mathbb {N})\times \mathbb {D}\rightarrow \mathcal {M}_z(\mathcal {H}^\infty (B_{c_0}))$ that is analytic on each slice $\{\theta \}\times \mathbb {D}$ and satisfies:

• (a) $\Psi (\theta ,\lambda )\not \in \mathcal {GP}(\delta _z)$ for each $(\theta ,\lambda )$.

• (b) $\mathcal {GP}(\Psi (\theta , \lambda ))\cap \mathcal {GP}(\Psi (\tilde{\theta }, \tilde{\lambda }))=\varnothing$ for each $\theta , \tilde{\theta }\in \beta (\mathbb {N})\setminus \mathbb {N}$ with $\theta \ne \tilde{\theta }$ and any $\lambda , \tilde{\lambda }\in \mathbb {D}$.