\documentclass[12pt]{amsart} \usepackage{amssymb} \usepackage{euscript} %\def\marginpar#1{} \makeatletter \@mparswitchfalse \makeatother \textheight=574pt \textwidth=432pt \oddsidemargin=18.88pt \evensidemargin=18.88pt \topmargin=14.21pt \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{question}{Question} \begin{document} % bring back \eqalign from plain TeX \def\eqalign#1{\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,} \let\eps\varepsilon \def\R{\mathbb R} \let\xi\zeta \let\sharp\# \let\cal\EuScript \let\\\cr \let\phi\varphi \let\union\bigcup \let\inter\bigcap \let\kappa=k \def\supp{\operatorname{supp}} \def\loc{\operatorname{loc}} \def\fin{\operatorname{fin}} \let\blip\rho \let\pee=P \let\emptyset\varnothing \let\curly=r \let\jay=j \let\capitalu=U \let\lceil\Gamma \title[Algebras associated with Blaschke products of type $G$]{Algebras associated with Blaschke products of type $\boldsymbol G$} \author{Carroll Guillory} \address{\hskip-\parindent Carroll Guillory\\ Department of Mathematics\\ University of Southwestern Loui\-si\-ana\\ Lafayette, LA 70504} \email{cjg2476@@usl.edu} \author{Kin Y. Li} \address{\hskip-\parindent Kin Y. Li\\ Department of Mathematics\\ Hong Kong University of Science and Technology\\ Clear Water Bay, Kowloon, Hong Kong} \email{makyli@@uxmail.ust.hk} \begin{abstract} Let $\Omega$ and $\Omega_{\fin}$ be the sets of all interpolating Blaschke products of type $G$ and of finite type $G$, respectively. Let $E$ and $E_{\fin}$ be the Douglas algebras generated by $H^\infty$ together with the complex conjugates of elements of $\Omega$ and $\Omega_{\fin}$, respectively. We show that the set of all invertible inner functions in $E$ is the set of all finite products of elements of $\Omega$ , which is also the closure of $\Omega$ among the Blaschke products. Consequently, finite convex combinations of finite products of elements of $\Omega$ are dense in the closed unit ball of the subalgebra of $H^\infty$ generated by $\Omega$. The same results hold when we replace $\Omega$ by $\Omega_{\fin}$ and $E$ by $E_{\fin}$. \end{abstract} \maketitle %\setcounter{section}{-1} \section{Introduction} Let $D$ be the open unit disk and $T$ be the unit circle on the complex plane. Let $H^\infty$ be the Banach algebra of bounded analytic functions on the open unit disk $D$. Via radial limits we can consider $H^\infty$ as a closed subalgebra of $L^\infty$, where $L^\infty$ is the family of all essential bounded measurable functions on $T$. Any function $h$ in $H^\infty$ with $|h|=1$ {\it a.e.} on $T$ is called an {\em inner} function. Let $\{z_n\}$ be a sequence in $D$ with $\sum_n(1-|z_n|) < \infty$. Then the function $$ b(z)=\prod_n \frac{\bar{z}_n}{|z_n|} \frac{z_n-z}{1-\bar{z}_n z}\quad\hbox{for $z\in D$,} $$ is called a {\em Blaschke product} with roots $\{z_n\}$. Let $$ \delta (b) = \inf_{\kappa} \prod_{n\neq \kappa}\left | \frac{z_\kappa - z_n}{1- \bar{z}_n z_\kappa }\right |. $$ If $\delta(b) > 0$, then $b$ and $\{z_n\}$ are called {\em interpolating}. By \cite{Carl1}, if $\delta(b) > 0$, then for every bounded sequence $\{a_n\}$, there exists $f$ in $H^\infty$ such that $f(z_n)=a_n$ for every $n$. If $$ \lim_{k\to \infty}\prod_{n\neq \kappa} \left | \frac{z_\kappa - z_n} {1-\bar{z}_n z_\kappa} \right | = 1, $$ then $b$ and $\{z_n\}$ are called {\em thin} or {\em sparse}. We denote by $M(H^\infty)$ the maximal ideal space of $H^\infty$. A closed subalgebra $B$ between $H^\infty$ and $L^\infty$ is called a {\em Douglas algebra.} We denote by $M(B)$ the maximal ideal space of the Douglas algebra $B$. For an interpolating Blaschke product $b$, we denote by $H^\infty[\bar{b}]$ the Douglas algebra generated by $H^\infty$ and the complex conjugate of $b$. For a function $f$ in $H^\infty$, let $$ Z(f)=\{x\in M(H^\infty): f(x) = 0\} $$ and for $0 \frac{2|\xi|}{1 + |\xi|^2} \text{ for } \jay =1,2,\ldots, \kappa.$$ By Lemma~5.3 on page 310 of \cite{Garnett}, each $Z(b_\jay)\cap D = \{z_{\jay,n}\}$ is interpolating. Since $$ \overline{\{z_n\}} = \bigcup^\kappa_{\jay =1}\overline{Z(b_\jay)} $$ and the $Z(b_\jay)$'s have disjoint closures \cite[p.~422]{Garnett}, it follows that $ m_0 \in \overline{Z(b_\jay)} = \overline{\{z_{\jay,n}\}} $ for some $\jay$ and the net $\{z_{\alpha}\}$ is eventually in $Z(b_\jay)$ because $$ M(H^\infty) \setminus \bigcup_{i\neq\jay}\overline{Z(b_i)} $$ % %[Silvio: check this formula] % is an open neighborhood of $\overline{Z(b_\jay)}$ and $m_0\in \overline{Z(b_\jay)}$. By Proposition~1, each $b_j(z)$ is of type $G$. By Proposition~2 (and the above argument), the Blaschke product with roots $\{\xi_{\jay, n}\}$ is of type $G$. Finally, $$ \lim_\alpha \xi_{\jay,\alpha} = \lim_\alpha L_{z_{\jay,\alpha}}(z) = L_{m_0}(\xi) = m $$ and our assertion follows. \end{proof} \begin{proposition} An interpolating Blaschke product $b$ of type $G$ has modulus $1$ on those Gleason parts of $M(H^\infty)$ that do not contain a point in $Z(b)$. Since $Z(b)$ equals the closure of $Z(b)\cap D$ and $\cal{F}$ is the union of a family of Gleason parts, $b$ has in particular modulus $1$ on $M(H^\infty)\setminus \cal{F}$. \end{proposition} \begin{proof} By Lemma~1.1 of \cite{Gu-Iz-II} (or Theorem 1 of \cite{Gu-Iz-I}) we have $$\{|b|< 1\} = \bigcup_{m\in Z(b)}P(m).$$ Thus $$|b|=1 \text{ on } m(H^\infty)\setminus \bigcup_{m\in Z(b)}P(m).$$ The second statement follows from the proof of Proposition 3. \end{proof} \begin{corollary} $M(E) = M(H^\infty)\setminus \cal{F}$. \end{corollary} \begin{proof} By Theorem 1.3 on page 375 of \cite{Garnett}, $$ M(E)= \{m\in M(H^\infty): |b(m)|=1 \text{ for all } b\in \Omega \}. $$ Now the results follows immediately from Proposition~4. \end{proof} \begin{corollary} $A$ is a proper subalgebra of $H^\infty$. \end{corollary} \begin{proof} This follows because $M(H^\infty)\setminus (\cal{F}\cup M(L^\infty))$ is not empty. \end{proof} \begin{theorem} \label{thm:main} Every invertible inner function in $E$ is a finite product of interpolating Blaschke products of type $G$. \end{theorem} \begin{proof} Let $u$ be an arbitrary invertible inner function in $E$, then $\bar{u}=u^{-1}$ in $E\subset L^\infty$. So $H^\infty[\bar{u}]\subset E$. By Corollary~5 above and Theorem 1.3 on page 375 of \cite{Garnett} $$M(H^\infty)\setminus \cal{F}=M(E)\subset M(H^\infty[\bar{u}])=\{|u|=1\}.$$ Hence $\{|u|<1\}\subset \cal{F}\subset G$. This implies $Z(u)$ cannot contain any trivial parts. By Corollary~24 of \cite{McD-Su}, $u=b_1 b_2\cdots b_n$, where each $b_j$ is an interpolating Blaschke product. Finally, for each $j$, $$\{|b_j|<1\}\subset \bigcup_{k=1}^n \{|b_k|<1\}=\{|u|<1\}\subset G.$$ \end{proof} \begin{corollary} $B=C_E$, the $C^*$-subalgebra of $L^\infty$ generated by the inner functions invertible in $E$ and their complex conjugates. \end{corollary} \begin{corollary} Let $b$ be a finite product of interpolating Blaschke products of type $G$. If $f\in H^\infty$ is such that $\|f\|_\infty < 1$ and $\bar{f}b$ equals on $H^\infty$ a function $g$ almost everywhere on $T$, then the function $$ b_f(z) = \frac{b(z) - f(z)}{1- g(z)},\quad\hbox{for } z\in D, $$ is a finite product of interpolating Blaschke products of type $G$. \end{corollary} \begin{proof} Just observe that $b_f$ is an invertible inner function in $E$. \end{proof} In \cite{Ch-Ma}, Chang and Marshall showed that for an arbitrary Douglas algebra $J$, the closed unit ball of $H^\infty \cap C_J$ is the norm-closed convex hull of the Blaschke products in $H^\infty\cap C_J$, where $C_J$ is the $C^*$-subalgebra of $L^\infty$ generated by the invertible inner functions in $J$ and their complex conjugates. They also showed that $J=H^\infty + C_J$ and that $D$ is dense in the maximal ideal space of $H^\infty\cap C_J$. In our case $J=E$, $C_J=B$ and we have the following corollary. (Note that an inner function in $B \cap H^\infty$ is invertible in $E$.) \begin{corollary} \begin{enumerate} \item[(a)] $A=B\cap H^\infty$, and finite convex combinations of finite products of interpolating Blaschke products of type $G$ are dense in the closed unit ball of $A$. \item[(b)] $E = H^\infty + B$. \item[(c)] $D$ is dense in the maximal ideal space $M(A)$ of $A$. \end{enumerate} \end{corollary} \section{Results For Finite Type $G$} Next we will turn to the main results for interpolating Blaschke product of finite type $G$ analogous to those established in section~2. Let $\Omega_{\fin}$ be the family of all interpolating Blaschke products of finite type $G$ and $A_{\fin}$ be the closed subalgebra of $H^\infty$ generated by $\Omega_{\fin}$. Let $E_{\fin} = [H^\infty, \bar{A}_{\fin}]$, then $E_{\fin}$ is the Douglas algebra generated by $H^\infty$ and the complex conjugate of elements of $\Omega_{\fin}$. We will show that Theorem~\ref{thm:main} holds if $E$ is replaced by $E_{\fin}$. \begin{theorem} \label{thm:was6} Every invertible inner function in $E_{\fin}$ is a finite product of interpolating Blaschke products of finite type $G$. \end{theorem} \begin{proof} We will first show that the analog of Proposition~2 is true for interpolating Blaschke products of finite type $G$. Let $b$ be an interpolating Blaschke product of finite type $G$ with zeros $\{z_n\}$ in $D$, and let $q$ be an interpolating Blaschke product with zeros $\{w_n\}$ in $D$ such that $\blip(z_n, w_n) \leq \curly $ for all $n$ and some $\curly < 1$. The proof of Proposition~2 shows that $\{|q|<1\}\subset \{|b|<1\}$. Since $b$ is of finite type $G$, by Theorem~2.1 of \cite{Gu-Iz-II}, there is a subproduct $b_0$ of $b$ such that $\{|b_0| < 1\} = \{|q| < 1\}$. We will show that $q$ is of finite type $G$. For $x\in Z(q)$, $|b_0(x)|<1$. By Lemma 1.1 of \cite{Gu-Iz-II}, there is an $x_0\in Z(b_0)$ such that $x\in P(x_0)$. Suppose the set $Z(q)\cap P(x)=Z(q)\cap P(x_0)$ is infinite. Then, by Theorem~3.1(i) of \cite{Gu-Iz-II}, there exist $y$ and $y_0$ in $Z(q)$ such that $\supp \mu_y \subsetneq \supp \mu_{y_0}$. Hence there are $m$ and $m_0$ in $Z(b_0)$ such that $y\in P(m)$ and $y_0 \in P(m_0)$, but then $\supp \mu_m \subsetneq \supp \mu_{m_0}$ (because by page 143 of \cite{Gamelin}, $\supp \mu_y = \supp \mu_m$ and $\supp \mu_{y_0} = \supp \mu_{m_0}$). Since $b_0$ is of finite type $G$, this contradicts Theorem~3.2(ii) of \cite{Gu-Iz-II}. Thus $Z(q)\cap P(x_0)=Z(q)\cap P(x)$ must be finite. Next we remark that the analogs of Propositions 3,4 and Corollaries 5,6 for finite type $G$ also hold by the same reasoning because of the analog of Proposition 2 for finite type $G$. Now let $u$ be an invertible inner function in $E_{\fin}\subset E$. By Theorem~7, $u=u_1 u_2\cdots u_m$, where each $u_i$ is of type $G$. Observe that if $\cal{F}_{\fin}$ is the analog of $\cal{F}$ for finite type $G$, then $$Z(u_i)\subset\{|u|<1\}\subset \cal{F}_{\fin}=\bigcup_{b\in \Omega_{\fin}} \{|b|<\frac{1}{2}\}.$$ Let $\delta_i=\inf\{\blip(w,z): w,z\in Z(u_i)\cap D, w\neq z\}>0$. Since $Z(u_i)$ is compact, $$Z(u_i)\subset \bigcup_{j=0}^{n_i} \{|b_j|<\frac{1}{2}\},$$ for some $b_1, b_2, \ldots, b_{n_i}$ of finite type $G$. Let $$S_{ij}=Z(u_i)\cap \{|b_j|<\frac{1}{2}\}\cap D,$$ then $$Z(u_i)\cap D=\bigcup_{j=1}^{n_i} S_{ij}.$$ By removing overlapping elements, we may assume the $S_{ij}$'s are disjoint. Since $b_j$ is of type $G$, by Lemma 2.1 of \cite{Gu-Iz-II}, there is $\delta<1$ such that $$S_{ij}\subset\{|b_j|<\frac{1}{2}\}\subset\{ z\in D:\blip(z,\{z_{j,n}\})\leq \delta\},$$ where $\{z_{j,n}\}$ is the root sequence of $b_j$ in $D$. For each disk $B(z_{j,n},\delta)=\{z\in D :\blip(z, z_{j,n})\leq \delta\}$, there is at most $k_i$ elements of $S_{ij}$ in $B(z_{j,n},\delta)$, where $k_i$ depends only on $\delta_i$. So $S_{ij}$ is the union of at most $k_i$ sequences, each of which has at most one element in each $B(z_{j,n},\delta)$. By the analog of Proposition~2 for finite type $G$, proved above, the Blaschke product with root sequence $S_{ij}$ is a product of at most $k_i$ interpolating Blaschke product of finite type $G$. So $u_i$ is a finite product of at most $n_i k_i$ interpolating Blaschke product of finite type $G$. Therefore, $u$ is a finite product of interpolating Blaschke product of finite type $G$. \end{proof} In general, it is a difficult problem to determine the closure of an infinite set of interpolating Blaschke products among the family of all Blaschke products (see for example \cite{Li}). However, for Blaschke products of type $G$ and finite type $G$, their closures can be identified because of Proposition~1 and Theorems~7, 11. \begin{theorem} Let $\cal{B}$ be the family of all Blaschke products with essential sup-norm. The closure of all interpolating Blaschke products of type $G$ in $\cal{B}$ is the set of all finite products of interpolating Blaschke products of type $G$. Also, the closure of all interpolating Blaschke products of finite type $G$ in $\cal{B}$ is the set of all finite products of interpolating Blaschke products of finite type $G$. \end{theorem} \begin{proof} Suppose $B$ is in the closure of type $G$ products. Take $B$ of type $G$ such that $\|B-b\|_\infty=\|1-B\bar{b}\|_\infty<1$. It follows that $B\bar{b}$ is invertible in $E$ and so is $B=(B\bar{b})b$. By Theorem~7, $B$ is a finite product of interpolating Blaschke products of type $G$. For the converse, it suffices to show for $B_1, B_2$ of type $G$ and $\varepsilon >0$, there is $B$ of type $G$ such that $\|B_1 B_2-B\|_\infty<\varepsilon$. Let the root sequences of $B_1$ and $B_2$ be $\{z_n\}$ and $\{w_n\}$, respectively. Since each of these sequences are separated, $$\delta_I=\inf \{\blip(z_m,z_n), \blip(w_m,w_n) : m\neq n\}>0.$$ By Hoffman's lemma (see Lemma 1.4 on pages 404-5 of \cite{Garnett}), there are $\delta_0, \varepsilon_0 <\delta_I/3$ such that $$V_n\subset\{z\in D : \blip(z,z_n)<\varepsilon_0\},$$ $$W_n\subset\{z\in D : \blip(z,w_n)<\varepsilon_0\}$$ and $$\left\| z-\frac{z-(\delta_0/2)}{1-(\delta_0/2)z}\right\|_\infty<\varepsilon,$$ where $V_n$ and $W_n$ are the components of $\{z\in D: |B_1(z)|<\delta_0\}$ and $\{z\in D : |B_2(z)|<\delta_0\}$ containing $z_n$ and $w_n$, respectively. Since $3\varepsilon_0<\delta_I$, we have $$\blip(V_n,V_m)>\delta_I/3 \text{ and } \blip(W_n,W_m)>\delta_I/3 \text{ for} n\neq m.$$ Factor $B_2=B_3 B_4$ so that $w_n\in Z(B_3)$ if $\blip(w_n,Z(B_1))\geq \delta_0/4$ and $w_n\in Z(B_4)$ if $\blip(w_n,Z(B_1))<\delta_0/4.$ Let $$B_5=\frac{B_4-(\delta_0/2)}{1-(\delta_0/2)B_4}$$ and $B=B_1 B_3 B_5$, then $\|B_1 B_2-B\|_\infty=\|B_4-B_5\|_\infty<\varepsilon$. Next we will show $B$ is an interpolating Blaschke product. Since $\blip(Z(B_1),Z(B_3))\geq \delta_0/4$, $B_1 B_3$ is an interpolating Blaschke product. By Lemma 1.4 on pages 404-5 of \cite{Garnett}, $B_5$ is an interpolating Blaschke product. To see $\blip(Z(B_1 B_3),Z(B_5))>0$, let $w\in Z(B_5)$ and $z\in Z(B_1 B_3)$. Then $w\in W_n$ for some $n$ and $w_n\in Z(B_4)$. If $m\neq n$, then $$\frac{\delta_0}{2}=\blip(B_5(w),B_5(w_n))\leq \blip(w,w_n)<\varepsilon_0<\blip(w,w_m).$$ Since $w_n\in Z(B_4)$, we have $\blip(w_n,z_k)<\delta_0/4$ for some $z_k\in Z(B_1)$. In the case $z\in Z(B_1)$ and $\blip(z,Z(B_4))<\varepsilon_0/4$, there is $w_m\in Z(B_4)$ such that $\blip(z,w_m)<\delta_0/4$ and so $$\blip(w,z)\geq \blip(w,w_m)-\blip(w_m,z)\geq \frac{\delta_0}{2}-\frac{\delta_0}{4}=\frac{\delta_0}{4}.$$ In the case $z\in Z(B_1)$ and $\blip(z,Z(B_4))\geq \delta_0/4$, we have $z\neq z_k$, hence $\blip(z,z_k)\geq \delta_I$. So $$\blip(w,z)\geq \blip(z,z_k)-\blip(z_k,w_n)-\blip(w_n,w)\geq\delta_I-\frac{\delta_0}{4}-\varepsilon_0\geq\frac{\delta_I}{2}.$$ In the case $z\in Z(B_3)$, we have $$\blip(w,z)\geq \blip(z,w_n)-\blip(w_n,w)\geq \delta_I-\varepsilon_0\geq\frac{2\delta_I}{3}.$$ So $\blip(Z(B_1 B_3),Z(B_5))>0$. Therefore, $B=B_1 B_3 B_5$ is an interpolating Blaschke product. Finally since $B_1, B_2$ are of type $G$, by Proposition~1, $B$ is of type $G$. This completes the proof of the statement for the closure of type $G$. For the closure of finite type $G$, use Theorem~11 instead of Theorem~7 and repeat the above proof. \end{proof} \section{Questions} Let $A_{\loc}$ be the closed subalgebra of $H^\infty$ generated by the locally thin Blaschke product and let $E_{\loc} = [H^\infty; \bar{A}_{\loc}]$. \begin{enumerate} \item[1.] Does Theorem~\ref{thm:main} hold if $E$ is replaced with $E_{\loc}$? \item[2.] Does $E=E_{\loc}$ or does $E_{\loc} = E_{\fin}$, or neither? \end{enumerate} If we let $A^*$ be the closed subalgebra of $H^\infty$ generated by the thin Blaschke products, and set $E^* = [H^\infty; \bar{A}^*]$,the main result of Hedenmalm \cite[Theorem~2.6]{He} asserts that Theorem~\ref{thm:main} holds for $E^*$. By Proposition~1.1 and Example~3.1 of \cite{Gu-Iz-II}, there exists a Blaschke product $b$ of finite type $G$, which is not a finite product of thin Blaschke products. 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