Commutative subalgebras of the algebra of smooth operators

Abstract

We consider the Fréchet ${}^*$-algebra ${\mathcal {L}}(s',s)\subseteq \mathcal L(\ell _2)$ of the so-called smooth operators, i.e. continuous linear operators from the dual $s'$ of the space s of rapidly decreasing sequences to s. This algebra is a non-commutative analogue of the algebra s. We characterize closed ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ which are at the same time isomorphic to closed ${}^*$-subalgebras of s and we provide an example of a closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ which cannot be embedded into s.

1 Introduction

The algebra ${\mathcal {L}}(s',s)$ is isomorphic as a Fréchet ${}^*$-algebra to the algebra

$$\begin{aligned} \mathcal K_\infty :=\left\{ (x_{j,k})_{j,k\in \mathbb N}\in \mathbb C^{\mathbb N^2}:\sup _{j,k\in \mathbb N}|x_{j,k}|j^qk^q<\infty \text { for all }q\in \mathbb N_0\right\} \end{aligned}$$

of rapidly decreasing matrices (with matrix multiplication and matrix complex conjugation). Another representation of ${\mathcal {L}}(s',s)$ is the algebra $\mathcal S(\mathbb R^2)$ of Schwartz functions on $\mathbb R^2$ with the Volterra convolution

$$\begin{aligned} (f\cdot g)(x,y):=\int _\mathbb Rf(x,z)g(z,y)\mathrm {d}z \end{aligned}$$

as multiplication and the involution

$$\begin{aligned} f^*(x,y):=\overline{f(y,x)}. \end{aligned}$$

In these forms, the algebra ${\mathcal {L}}(s',s)$ usually appears and plays a significant role in K-theory of Fréchet algebras (see Bhatt and Inoue [1, Ex. 2.12], Cuntz [6, p. 144], [7, p. 64–65], Glöckner and Langkamp [11], Phillips [14, Def. 2.1]) and in $C^*$-dynamical systems (Elliot, Natsume and Nest [9, Ex. 2.6]). Very recently, Piszczek obtained several results concerning closed ideals, automatic continuity (for positive functionals and derivations), amenability and Jordan decomposition in $\mathcal K_\infty$ (see Piszczek [16–19] and his forthcoming paper “The noncommutative Schwartz space is weakly amenable”). Moreover, in the context of algebras of unbounded operators, the algebra ${\mathcal {L}}(s',s)$ appears in the book [20] as

$$\begin{aligned} \mathbb {B}_1(s):=\{x\in \mathcal L(\ell _2):x\ell _2\subseteq s, x^*\ell _2\subseteq s \text { and } \overline{axb} \text { is nuclear for all }a,b\in {\mathcal {L}}^*(s)\}, \end{aligned}$$

where $\mathcal L^*(s)$ is the so-called maximal $O^*$-algebra on s (see also [20, Def. 2.1.6, Prop. 2.1.8, Def. 5.1.3, Cor. 5.1.18, Prop. 5.4.1 and Prop. 6.1.5]).

The algebra of smooth operators can be seen as a noncommutative analogue of the commutative algebra s. The most important features of this algebra are the following:

• it is isomorphic as a Fréchet space to the Schwartz space $\mathcal {S}(\mathbb R)$ of smooth rapidly decreasing functions on the real line;

• it is isomorphic as a Fréchet ${}^*$-algebra to many algebras of operators acting between natural spaces of distributions and functions, e.g. to the algebra of operators from the space $\mathcal S'(\mathbb R)$ of tempered distributions on the real line to the space $\mathcal S(\mathbb R)$ (see also [8, Th. 1.1]);

• it is a dense ${}^*$-subalgebra of the $C^*$-algebra $\mathcal K(\ell _2)$ of compact operators on $\ell _2$;

• it is (properly) contained in the intersection of all Schatten classes $\mathcal S_p(\ell _2)$ over $p>0$; in particular ${\mathcal {L}}(s',s)$ is contained in the class ${\mathcal {HS}}(\ell _2)$ of Hilbert-Schmidt operators, and thus it is a unitary space;

• the operator $C^*$-norm $||\cdot ||_{\ell _2\rightarrow \ell _2}$ is the so-called dominating norm on that algebra (the dominating norm property is a key notion in the structure theory of nuclear Fréchet spaces – see [3, Prop. 3.2] and [13, Prop. 31.5]).

The main result of the present paper is a characterization of closed ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ which are at the same time isomorphic as Fréchet ${}^*$-algebras to closed ${}^*$-subalgebras of the algebra s (Theorem 6.2). It turns out that these are exactly those subalgebras which satisfy the classical condition $(\Omega )$ of Vogt, i.e. which are isomorphic (as Fréchet spaces) to complemented subspaces of s. Then in Theorem 6.9 we give an example of a closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ which does not satisfy this condition.

To prove this result we characterize in Sect. 4 closed ${}^*$-subalgebras of Köthe sequence algebras (Proposition 4.3). In particular, we give such a description for closed ${}^*$-subalgebras of s (Corollary 4.4). In Sect. 5 we describe all closed ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ as suitable Köthe sequence algebras (see Corollary 5.4 and compare with [3, Th.4.8]).

The present paper is a continuation of [3, 8] and it focuses on descriptions of closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ (especially those with the property ($\Omega$)). Most of the results have been already presented in the author’s PhD dissertation [2].

2 Notation and terminology

Throughout the paper, $\mathbb N$ denotes the set of natural numbers $\{1,2,\ldots \}$ and $\mathbb N_0:=\mathbb N\cup \{0\}$.

By a projection on the complex separable Hilbert space $\ell _2$ we always mean a continuous orthogonal (i.e. self-adjoint) projection.

By $e_k$ we denote the vector in $\mathbb C^\mathbb N$ whose k-th coordinate equals 1 and the others equal 0.

By a Fréchet space we mean a complete metrizable locally convex space over $\mathbb C$ (we will not use locally convex spaces over $\mathbb R$). A Fréchet algebra is a Fréchet space which is an algebra with continuous multiplication. A Fréchet ${}^*$it algebra is a Fréchet algebra with continuous involution.

For locally convex spaces E, F, we denote by $\mathcal L(E,F)$ the space of all continuous linear operators from E to F. To shorten notation, we write $\mathcal L(E)$ instead of $\mathcal L(E,E)$.

We use standard notation and terminology. All the notions from functional analysis are explained in [4, 13] and those from topological algebras in [10, 24].

3 Preliminaries

3.1 The space s and its dual

We recall that the space of rapidly decreasing sequences is the Fréchet space

$$\begin{aligned} s:=\left\{ \xi =(\xi _j)_{j\in \mathbb N}\in \mathbb C^\mathbb N:|\xi |_q:=\left( \sum _{j=1}^\infty |\xi _j|^2j^{2q}\right) ^{1/2}<\infty \text { for all }q\in \mathbb N_0\right\} \end{aligned}$$

with the topology corresponding to the system $(|\cdot |_q)_{q\in \mathbb N_0}$ of norms. We may identify the strong dual of s (i.e. the space of all continuous linear functionals on s with the topology of uniform convergence on bounded subsets of s, see e.g. [13, Definition on p. 267]) with the space of slowly increasing sequences

$$\begin{aligned} s':=\left\{ \xi =(\xi _j)_{j\in \mathbb N}\in \mathbb C^\mathbb N:|\xi |_q':=\left( \sum _{j=1}^\infty |\xi _j|^2j^{-2q}\right) ^{1/2}<\infty \text { for some }q\in \mathbb N_0\right\} \end{aligned}$$

equipped with the inductive limit topology given by the system $(|\cdot |_q')_{q\in \mathbb N_0}$ of norms (note that for a fixed q, $|\cdot |'_q$ is defined only on a subspace of $s'$). More precisely, every $\eta \in s'$ corresponds to the continuous linear functional on s:

$$\begin{aligned} \xi \mapsto \langle \xi ,\eta \rangle :=\sum _{j=1}^\infty \xi _j\overline{\eta _j} \end{aligned}$$

(note the conjugation on the second variable). These functionals are continuous, because, by the Cauchy–Schwartz inequality, for all $q\in \mathbb N_0$, $\xi \in s$ and $\eta \in s'$ we have

$$\begin{aligned} |\langle \xi ,\eta \rangle |\le |\xi |_q|\eta |'_q. \end{aligned}$$

Conversely, one can show that for each continuous linear functional y on s there is $\eta \in s'$ such that $y=\langle \cdot ,\eta \rangle$.

Similarly, we identify each $\xi \in s$ with the continuous linear functional on $s'$:

$$\begin{aligned} \eta \mapsto \langle \eta ,\xi \rangle :=\sum _{j=1}^\infty \eta _j\overline{\xi _j}. \end{aligned}$$

In particular, for each continuous linear functional y on $s'$ there is $\xi \in s$ such that $y=\langle \cdot ,\xi \rangle$.

We emphasize that the “scalar product” $\langle \cdot ,\cdot \rangle$ is well-defined on $s\times s'\cup s'\times s$ and, of course, on $\ell _2\times \ell _2$.

3.2 The property (DN) for the space s

Closed subspaces of the space s can be characterized by the so-called property (DN).

Definition 3.1

A Fréchet space $(X,(||\cdot ||_q)_{q\in \mathbb N_0})$ has the property (DN) (see [13, Definition on p. 359]) if there is a continuous norm $||\cdot ||$ on X such that for all $q\in \mathbb N_0$ there is $r\in \mathbb N_0$ and $C>0$ such that

$$\begin{aligned} ||x||_q^2\le C||x||\;||x||_r \end{aligned}$$

for all $x\in X$. The norm $||\cdot ||$ is called a dominating norm.

Vogt (see [23] and [13, Ch. 31]) proved that a Fréchet space is isomorphic to a closed subspace of s if and only if it is nuclear and it has the property (DN).

The property (DN) for the space s reads as follows (see [13, Lemma 29.2(3)] and its proof).

Proposition 3.2

For every $p\in \mathbb N_0$ and $\xi \in s$ we have

$$\begin{aligned} |\xi |_p^2\le ||\xi ||_{\ell _2}|\xi |_{2p}. \end{aligned}$$

In particular, the norm $||\cdot ||_{\ell _2}$ is a dominating norm on s.

3.3 The algebra ${\mathcal {L}}(s',s)$

It is a simple matter to show that ${\mathcal {L}}(s',s)$ with the topology of uniform convergence on bounded sets in $s'$ is a Fréchet space. It is isomorphic to $s\widehat{\otimes }s$, the completed tensor product of s (see [12, §41.7(5)] and note that, s being nuclear, there is only one tensor topology), and thus ${\mathcal {L}}(s',s)\cong s$ as Fréchet spaces (see e.g. [13, Lemma 31.1]). Moreover, it is easily seen that $(||\cdot ||_q)_{q\in \mathbb N_0}$,

$$\begin{aligned} ||x||_q:=\sup _{|\xi |_q^{'}\le 1}{|x\xi |_q}, \end{aligned}$$

is a fundamental sequence of norms on ${\mathcal {L}}(s',s)$.

Let us introduce multiplication and involution on ${\mathcal {L}}(s',s)$. First observe that s is a dense subspace of $\ell _2$. Moreover, $\ell _2$ is a dense subspace of $s'$, and, finally, the inclusion maps $j_1:s\hookrightarrow \ell _2$, $j_2:\ell _2\hookrightarrow s'$ are continuous. Hence,

$$\begin{aligned} \iota :{\mathcal {L}}(s',s)\hookrightarrow \mathcal L(\ell _2),\quad \iota (x):=j_1\circ x \circ j_2, \end{aligned}$$

is a well-defined (continuous) embedding of ${\mathcal {L}}(s',s)$ into the $C^*$-algebra $\mathcal L(\ell _2)$, and thus it is natural to define a multiplication on ${\mathcal {L}}(s',s)$ by

$$\begin{aligned} xy:=\iota ^{-1}(\iota (x)\circ \iota (y)), \end{aligned}$$

i.e.

$$\begin{aligned} xy=x\circ j\circ y, \end{aligned}$$

where $j:=j_2\circ j_1:s\hookrightarrow s'$. Similarly, an involution on ${\mathcal {L}}(s',s)$ is defined by

$$\begin{aligned} x^*:=\iota ^{-1}(\iota (x)^*), \end{aligned}$$

where $\iota (x)^*$ is the hermitian adjoint of $\iota (x)$. One can show that these definitions are correct, i.e. $\iota (x)\circ \iota (y),\iota (x)^*\in \iota ({\mathcal {L}}(s',s))$ for all $x,y\in {\mathcal {L}}(s',s)$ (see also [3, p.148]).

From now on, we will identify $x\in {\mathcal {L}}(s',s)$ and $\iota (x)\in \mathcal L(\ell _2)$ (we omit $\iota$ in the notation).

A Fréchet algebra E is called locally m -convex if E has a fundamental system of submultiplicative seminorms. It is well-known that ${\mathcal {L}}(s',s)$ is locally m-convex (see e.g. [14, Lemma 2.2]), and moreover, the norms $||\cdot ||_q$ are submultiplicative (see [3, Proposition 2.5]). This shows simultaneously that the multiplication introduced above is separately continuous, and thus, by [24, Theorem 1.5], it is jointly continuous. Moreover, by [10, Corollary 16.7], the involution on ${\mathcal {L}}(s',s)$ is continuous.

We may summarize this paragraph by saying that ${\mathcal {L}}(s',s)$ is a noncommutative ${}^*$-subalgebra of the $C^*$-algebra $\mathcal L(\ell _2)$ which is (with its natural topology) a locally m-convex Fréchet ${}^*$-algebra isomorphic as a Fréchet space to s.

4 Köthe algebras

In this section we collect and prove some folklore facts on Köthe algebras which are known for specialists but probably never published.

Definition 4.1

A matrix $A=(a_{j,q})_{j\in \mathbb N,q\in \mathbb N_0}$ of non-negative numbers such that

• (i) for each $j\in \mathbb N$ there is $q\in \mathbb N_0$ such that $a_{j,q}>0$

• (ii) $a_{j,q}\le a_{j,q+1}$ for $j\in \mathbb N$ and $q\in \mathbb N_0$

is called a Köthe matrix.

For $1\le p<\infty$ and a Köthe matrix A we define the Köthe space

$$\begin{aligned} \lambda ^p(A):=\left\{ \xi =(\xi _j)_{j\in \mathbb N}\in \mathbb C^\mathbb N:|\xi |_{p,q}^p:=\sum _{j=1}^\infty |\xi _j|^pa_{j,q}^p<\infty \text { for all }q\in \mathbb N_0\right\} \end{aligned}$$

and for $p=\infty$

$$\begin{aligned} \lambda ^\infty (A):=\left\{ \xi =(\xi _j)_{j\in \mathbb N}\in \mathbb C^\mathbb N:|\xi |_{\infty ,q}:=\sup _{j\in \mathbb N}|\xi _j|a_{j,q}<\infty \text { for all }q\in \mathbb N_0\right\} \end{aligned}$$

with the locally convex topology given by the seminorms $(|\cdot |_{p,q})_{q\in \mathbb N_0}$ (see e.g. [13, Definition p. 326]).

Sometimes, for simplicity, we will write $\lambda ^\infty (a_{j,q})$ (i.e. only the entries of the matrix) instead of $\lambda ^\infty (A)$.

It is well-known (see [13, Lemma 27.1]) that the spaces $\lambda ^p(A)$ are Fréchet spaces and sometimes they are Fréchet ${}^*$-algebras with pointwise multiplication and conjugation (e.g. if $a_{j,q}\ge 1$ for all $j\in \mathbb N$ and $q\in \mathbb N_0$, see also [15, Proposition 3.1]); in that case they are called Köthe algebras.

Clearly, s is the Köthe space $\lambda ^2(A)$ for $A=(j^q)_{j\in \mathbb N,q\in \mathbb N_0}$ and it is a Fréchet ${}^*$-algebra. Moreover, since the matrix A satisfies the so-called Grothendieck–Pietsch condition (see e.g. [13, Proposition 28.16(6)]), s is nuclear, and thus it has also other Köthe space representations (see again [13, Proposition 28.16 and Example 29.4(1)]), i.e. for all $1\le p\le \infty$, $s=\lambda ^p(A)$ as Fréchet spaces.

We use $\ell _2$-norms in the definition of s to clarify our ideas, for example we have $|\xi |_0=||\xi ||_{\ell _2}$ for $\xi \in s$ and $|\eta |_0'=||\eta ||_{\ell _2}$ for $\eta \in \ell _2$. However, in some situations the supremum norms $|\cdot |_{\infty ,q}$ (as they are relatively easy to compute) or the $\ell _1$-norms will be more convenient.

Proposition 4.2

Let $A=(a_{j,q})_{j\in \mathbb N,q\in \mathbb N_0}$, $B=(b_{j,q})_{j\in \mathbb N,q\in \mathbb N_0}$ be Köthe matrices and for a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ let $A_\sigma :=(a_{\sigma (j),q})_{j\in \mathbb N,q\in \mathbb N_0}$. Assume that $\lambda ^\infty (A)$ and $\lambda ^\infty (B)$ are Fréchet ${}^*$-algebras. Then the following assertions are equivalent:

• (i) $\lambda ^\infty (A)\cong \lambda ^\infty (B)$ as Fréchet ${}^*$-algebras;

• (ii) there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that $\lambda ^\infty (A_\sigma )=\lambda ^\infty (B)$ as Fréchet ${}^*$-algebras;

• (iii) there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that $\lambda ^\infty (A_\sigma )=\lambda ^\infty (B)$ as sets;

• (iv) there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that

  • $(\alpha )$ $\forall q\in \mathbb N_0\;\exists r\in \mathbb N_0\;\exists C>0\;\forall j\in \mathbb N\quad a_{\sigma (j),q}\le Cb_{j,r}$,

  • $(\beta )$ $\forall r'\in \mathbb N_0\;\exists q'\in \mathbb N_0\;\exists C'>0\;\forall j\in \mathbb N\quad b_{j,r'}\le C'a_{\sigma (j),q'}$.

Proof

(i) $\Rightarrow$ (ii) Assume that there is an isomorphism $\Phi :\lambda ^\infty (A)\rightarrow \lambda ^\infty (B)$ of Fréchet ${}^*$-algebras. Clearly, if $\xi ^2=\xi$, then $\Phi (\xi )=\Phi (\xi ^2)=(\Phi (\xi ))^2$, and the same is true of $\Phi ^{-1}$, i.e. $\Phi$ maps the idempotents of $\lambda ^\infty (A)$ onto the idempotents of $\lambda ^\infty (B)$. Hence for a fixed $k\in \mathbb N$, there is $I\subset \mathbb N$ such that

$$\begin{aligned} \Phi (e_{k})=e_I, \end{aligned}$$

where $e_I$ is a sequence which has 1 on an index set $I\subset \mathbb N$ and 0 otherwise. Suppose that $|I|\ge 2$ and let $j\in I$. Then $e_I=e_j+e_{I{\setminus }\{j\}}$, where $e_j\in \lambda ^\infty (B)$ and $e_{I{\setminus }\{j\}}=e_I-e_j\in \lambda ^\infty (B)$. Therefore, there are nonempty subsets $I_j,I_{j}'\subset \mathbb N$ such that $\Phi (e_{I_{j}})=e_j$ and $\Phi (e_{I_{j'}})=e_{I{\setminus }\{j\}}$. We have

$$\begin{aligned} e_{I_j}e_{I_j'}=\Phi ^{-1}(e_j)\Phi ^{-1}(e_{I{\setminus }\{j\}})=\Phi ^{-1}(e_je_{I{\setminus }\{j\}})=0, \end{aligned}$$

and thus $I_j\cap I_j'=\emptyset$. Consequently,

$$\begin{aligned}\Phi (e_k)=e_j+e_{I{\setminus }\{j\}}=\Phi (e_{I_j})+\Phi (e_{I_j'})=\Phi (e_{I_j\cup I_j'}),\end{aligned}$$

whence $1=|\{k\}|=|I_j\cup I_j'|\ge 2$, a contradiction. Hence $\Phi (e_k)=e_{n_k}$ for some $n_k\in \mathbb N$, i.e. for the bijection $\sigma :\mathbb N\rightarrow \mathbb N$ defined by $n_{\sigma (k)}:=k$ we have $\Phi (e_{\sigma (k)})=e_k$. Therefore, a Fréchet ${}^*$-isomorphism $\Phi$ is given by $(\xi _{\sigma (k)})_{k\in \mathbb N}\mapsto (\xi _k)_{k\in \mathbb N}$ for $(\xi _{\sigma (k)})_{k\in \mathbb N}\in \lambda ^\infty (A)$, and thus $\lambda ^\infty (A_\sigma )=\lambda ^\infty (B)$ as Fréchet ${}^*$-algebras.

(ii) $\Rightarrow$ (iii) Obvious.

(iii) $\Rightarrow$ (iv) The proof follows from the observation that the identity map $\mathrm{Id}:\lambda ^\infty (A_\sigma )\rightarrow \lambda ^\infty (B)$ is continuous (use the closed graph theorem).

(iv) $\Rightarrow$ (i) It is easy to see that $\Phi :\lambda ^\infty (A)\rightarrow \lambda ^\infty (B)$ defined by $e_{\sigma (k)}\mapsto e_k$ is an isomorphism of Fréchet ${}^*$-algebras. $\square$

In the following proposition we characterize infinite-dimensional closed ${}^*$-subalgebras of nuclear Köthe algebras whose elements tend to zero (note that if a Köthe space is contained in $\ell _\infty$ then it is a Köthe algebra). Consequently, we obtain a characterization of closed ${}^*$-subalgebras of s (Corollary 4.4).

Proposition 4.3

For $\mathcal N\subset \mathbb N$ let $e_\mathcal N$ denote a sequence which has 1 on $\mathcal N$ and 0 otherwise. Let $A=(a_{j,q})_{j\in \mathbb N,q\in \mathbb N_0}$ be a Köthe matrix such that $\lambda ^\infty (A)$ is nuclear and $\lambda ^\infty (A)\subset c_0$. Let E be an infinite-dimensional closed ${}^*$-subalgebra of $\lambda ^\infty (A)$. Then

• (i) there is a family $\{\mathcal N_k\}_{k\in \mathbb N}$ of finite nonempty pairwise disjoint sets of natural numbers such that $(e_{\mathcal N_k})_{k\in \mathbb N}$ is a Schauder basis of E;

• (ii) $E\cong \lambda ^\infty \left( \max _{j\in \mathcal N_k}a_{j,q}\right)$ as Fréchet ${}^*$-algebras and the isomorphism is given by $e_{\mathcal N_k}\mapsto e_k$ for $k\in \mathbb N$. Conversely, if $\{\mathcal N_k\}_{k\in \mathbb N}$ is a family of finite nonempty pairwise disjoint sets of natural numbers and F is the closed ${}^*$-subalgebra of $\lambda ^\infty (A)$ generated by the set $\{e_{\mathcal N_k}\}_{k\in \mathbb N}$, then

• (iii) $(e_{\mathcal N_k})_{k\in \mathbb N}$ is a Schauder basis of F;

• (iv) $F\cong \lambda ^\infty (\max _{j\in \mathcal N_k}a_{j,q})$ as Fréchet ${}^*$-algebras and the isomorphism is given by $e_{\mathcal N_k}\mapsto e_k$ for $k\in \mathbb N$.

Proof

In order to prove (i) and (ii) set

$$\begin{aligned} \mathcal N_0:=\{j\in \mathbb N:\xi _j=0\quad \text {for all}\,\xi \in E \} \end{aligned}$$

and define an equivalence relation $\sim$ on $\mathbb N{\setminus }\mathcal N_0$ by

$$\begin{aligned} i\sim j\Leftrightarrow \xi _i=\xi _j\text { for all}\,\,\,\xi \in E. \end{aligned}$$

Since E is infinite-dimensional, our relation produces infinitely many equivalence classes $\mathcal N_k$, say

$$\begin{aligned} \mathcal N_k:=[\min (\mathbb N{\setminus }\mathcal N_0\cup \cdots \cup \mathcal N_{k-1})]_{/\sim } \end{aligned}$$

for $k\in \mathbb N$.

Fix $\kappa \in \mathbb N$ and take $\xi \in E$ such that $\xi _j\ne 0$ for $j\in \mathcal N_{\kappa }$. Denote $\eta _k:=\xi _j$ if $j\in \mathcal N_k$. Let

$$\begin{aligned} \mathcal M_1:=\left\{ j\in \mathbb N:|\eta _j|=\sup _{i\in \mathbb N}{|\eta _i|}\right\} . \end{aligned}$$

Assume we have already defined $\mathcal M_1,\ldots ,\mathcal M_{l-1}$. If there is $j\in \mathbb N{\setminus }\{\mathcal M_1\cup \cdots \cup \mathcal M_{l-1}\}$ such that $\eta _j\ne 0$ then we define

$$\begin{aligned} \mathcal M_l:=\{j\in \mathbb N:|\eta _j|=\sup \{|\eta _i|:i\in \mathbb N{\setminus }\mathcal M_1\cup \cdots \cup \mathcal M_{l-1}\}\}. \end{aligned}$$

Otherwise, denote $\mathcal I:=\{1,\ldots ,l-1\}$. If this procedure leads to infinitely many sets $\mathcal M_l$ then we set $\mathcal I:=\mathbb N$. It is easily seen that for each $l\in \mathcal {I}$ there is $\mathcal {I}_l\subset \mathbb N$ such that $\mathcal M_l=\bigcup _{k\in \mathcal {I}_l}\mathcal N_k$. By assumption $\xi \in c_0$, hence $(|\eta _k|)_{k\in \mathbb N}\in c_0$ as well, and thus each $\mathcal M_l$ is a finite nonempty set.

We first show that $e_{\mathcal M_l}\in E$ for $l\in \mathcal I$. For $l\in \mathcal I$ fix $m_l\in \mathcal M_l$. If $\mathcal I=\{1\}$, then $\xi _j=0$ for $j\notin \mathcal M_1$, and $e_{\mathcal M_1}=\frac{\xi \overline{\xi }}{|\eta _{m_1}|^2}\in E$. Let us consider the case $|\mathcal I|>1$. Since in nuclear Fréchet spaces every basis is absolute (and thus unconditional), we have

$$\begin{aligned} \sum _{l\in \mathcal I}|\eta _l|^2 e_{\mathcal M_l}=\sum _{j=1}^\infty |\xi _j|^2e_j=\xi \overline{\xi }\in E, \end{aligned}$$

and, consequently,

$$\begin{aligned} x_n:=\sum _{l\in \mathcal I}\left( \frac{|\eta _l|}{|\eta _{m_1}|}\right) ^{2n}e_{\mathcal M_l}=\left( \frac{\xi \overline{\xi }}{|\eta _{m_1}|^2}\right) ^n\in E \end{aligned}$$

for all $n\in \mathbb N$. Then for q and n we get

$$\begin{aligned} |x_n-e_{\mathcal M_1}|_{\infty ,q}&=\left| \sum _{l\in \mathcal I}^\infty \left( \frac{|\eta _l|}{|\eta _{m_1}|}\right) ^{2n}e_{\mathcal M_l}-e_{\mathcal M_1}\right| _{\infty ,q}\\&=\left| \sum _{l\in \mathcal I{\setminus }\{1\}}\left( \frac{|\eta _l|}{|\eta _{m_1}|}\right) ^{2n}e_{\mathcal M_l}\right| _{\infty ,q} \le \sum _{l\in \mathcal I{\setminus }\{1\}}\left( \frac{|\eta _l|}{|\eta _{m_1}|}\right) ^{2n}|e_{\mathcal M_l}|_{\infty ,q}\\&\le \frac{1}{|\eta _{m_1}|}\left( \frac{|\eta _{m_2}|}{|\eta _{m_1}|}\right) ^{2n-1}\sum _{l\in \mathcal I{\setminus }\{1\}}|\eta _l|\,|e_{\mathcal M_l}|_{\infty ,q}. \end{aligned}$$

Since $(e_j)_{j\in \mathbb N}$ is an absolute basis in $\lambda ^\infty (A)$, the above series is convergent. Note also that $|\eta _{m_2}|<|\eta _{m_1}|$. This shows that $x_n\rightarrow e_{\mathcal M_1}$ in $\lambda ^\infty (A)$, and $e_{\mathcal M_1}\in E$. Assume that $e_{\mathcal M_1},\ldots ,e_{\mathcal M_{l-1}}\in E$. If $|\mathcal I|=l-1$ then we are done. Otherwise, $\eta _{m_l}\ne 0$ and

$$\begin{aligned} x_n^{(l)}:=\left( \frac{\xi \overline{\xi }-\xi \overline{\xi }\sum _{j=1}^{l-1}e_{\mathcal M_j}}{|\eta _{m_l}|^2}\right) ^n\in E \end{aligned}$$

for $n\in \mathbb N$. As above we show that $x_n^{(l)}\rightarrow e_{\mathcal M_l}$ in $\lambda ^\infty (A)$, and thus $e_{\mathcal M_l}\in E$. Proceeding by induction, we prove that $e_{\mathcal M_l}\in E$ for $l\in \mathcal I$.

Now, we shall prove that $(e_{\mathcal N_k})_{k\in \mathbb N}$ is a Schauder basis of E. Choose $\iota \in \mathcal I$ such that $\kappa \in \mathcal I_\iota$ and for $k\in \mathcal I_{\iota }$ let $n_k$ be an arbitrary element of $\mathcal N_k$. Then $\sum _{k\in \mathcal I_\iota }\eta _{n_k}e_{\mathcal N_k}=\xi e_{\mathcal M_\iota }\in E$. Consequently, by [3, Lemma 4.1], $e_{\mathcal N_\kappa }\in E$. Since $\kappa$ was arbitrarily choosen, each $e_{\mathcal N_k}$ is in E and it is a simple matter to show that $(e_{\mathcal N_k})_{k\in \mathbb N}$ is a Schauder basis of E.

Moreover, $|e_{\mathcal N_k}|_{\infty ,q}=\max _{j\in \mathcal N_k}{a_{j,q}}$ hence, by [13, Corollary 28.13] and nuclearity, E is isomorphic as a Fréchet space to $\lambda ^\infty (\max _{j\in \mathcal N_k}{a_{j,q}})$. The analysis of the proof of [13, Corollary 28.13] shows that this isomorphism is given by $e_{\mathcal N_k}\mapsto e_k$ for $k\in \mathbb N$, and thus it is also a Fréchet ${}^*$-algebra isomorphism.

Now, we prove (iii) and (iv). First note that every element of F is the limit of elements of the form $\sum _{k=1}^Mc_ke_{\mathcal N_k}$, where $M\in \mathbb N$ and $c_1,\ldots ,c_M\in \mathbb C$. Therefore, if $\xi \in F$, then $\xi _i=\xi _j$ for $k\in \mathbb N$ and $i,j\in \mathcal N_k$. This shows that each $\xi \in F$ has the unique series representation $\xi =\sum _{k=1}^\infty \xi _{n_k}e_{\mathcal N_k}$, where $(n_k)_{k\in \mathbb N}$ is an arbitrarily choosen sequence such that $n_k\in \mathcal N_k$ for $k\in \mathbb N$. Since the series is absolutely convergent, $(e_{\mathcal N_k})_{k\in \mathbb N}$ is a Schauder basis of F. Statement (iv) follows by the same method as in (ii). $\square$

Corollary 4.4

Every infinite-dimensional closed ${}^*$-subalgebra of s is isomorphic as a Fréchet ${}^*$-algebra to $\lambda ^\infty (n_k^q)$ for some strictly increasing sequence $(n_k)_{k\in \mathbb N}$ of natural numbers. Conversely, if $(n_k)_{k\in \mathbb N}$ is a strictly increasing sequence of natural numbers, then $\lambda ^\infty (n_k^q)$ is isomorphic as a Fréchet ${}^*$-algebra to some infinite-dimensional closed ${}^*$-subalgebra of s. Moreover, every closed ${}^*$-subalgebra of s is a complemented subspace of s.

Proof

We apply Proposition 4.3 to the Köthe matrix $(j^q)_{j\in \mathbb N,q\in \mathbb N_0}$. Let $\{\mathcal N_k\}_{k\in \mathbb N}$ be a family of finite nonempty pairwise disjoint sets of natural numbers. We have

$$\begin{aligned} \max _{j\in \mathcal N_k}j^q=(\max \{j:j\in \mathcal N_k\})^q \end{aligned}$$

(1)

for all $q\in \mathbb N_0$ and $k\in \mathbb N$. Let $\sigma :\mathbb N\rightarrow \mathbb N$ be the bijection for which $(\max \{j:j\in \mathcal N_{\sigma (k)}\})_{k\in \mathbb N}$ is (strictly) increasing and let $n_k:=\max \{j:j\in \mathcal N_{\sigma (k)}\}$ for $k\in \mathbb N$. Then, by Proposition 4.2,

$$\begin{aligned} \lambda ^\infty \left( \max _{j\in \mathcal N_k}j^q\right) \cong \lambda ^\infty (n_k^q) \end{aligned}$$

as Fréchet ${}^*$-algebras, and therefore the first two statements follow from Proposition 4.3.

Now, let E be a closed ${}^*$-subalgebra of s. If E is finite dimensional then, clearly, E is complemented in s. Otherwise, by Proposition 4.3(i), E is a closed linear span of the set $\{e_{\mathcal N_k}\}_{k\in \mathbb N}$ for some family $\{\mathcal N_k\}_{k\in \mathbb N}$ of finite nonempty pairwise disjoint sets of natural numbers. Define $\pi :s\rightarrow E$ by

$$\begin{aligned} (\pi x)_j:= \left\{ \begin{array}{ll} x_{n_k} &{}\quad \text {for} j\in \mathcal N_{\sigma (k)}\\ 0 &{}\quad \text {otherwise} \end{array} \right. \end{aligned}$$

where $(n_k)_{k\in \mathbb N}$ and $\sigma$ are as above. From (1) we have for every $q\in \mathbb N_0$

$$\begin{aligned} |\pi x|_{\infty ,q}&=\sup _{j\in \mathbb N}|(\pi x)_j|j^q\le \sup _{k\in \mathbb N}|x_{n_k}|\max _{j\in \mathcal N_{\sigma (k)}}j^q=\sup _{k\in \mathbb N}|x_{n_k}|(\max \{j:j\in \mathcal N_k\})^q\\&=\sup _{k\in \mathbb N}|x_{n_k}|n_k^q\le \sup _{j\in \mathbb N}|x_j|j^q=|x|_{\infty ,q}, \end{aligned}$$

and thus $\pi$ is well-defined and continuous. Since $\pi$ is a projection, our proof is complete. $\square$

5 Representations of closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ by Köthe algebras

The aim of this section is to describe all closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ as Köthe algebras $\lambda ^\infty (A)$ for matrices A determined by orthonormal sequences whose elements belong to the space s (Theorem 5.3 and Corollaries 5.4 and 5.5). For the convenience of the reader, we quote two results from [3] (with minor modifications which do not require extra arguments).

For a subset Z of ${\mathcal {L}}(s',s)$ we will denote by $\mathrm{alg}(Z)$ ($\overline{\mathrm{lin}}(Z)$, resp.) the closed ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ generated by Z (the closed linear span of Z, resp.).

By [3, Lemma 4.4], every closed commutative ${}^*$-subalgebra E of ${\mathcal {L}}(s',s)$ admits a special Schauder basis. This basis consists of all nonzero minimal projections in E ([3, Lemma 4.4] shows that these projections are pairwise orthogonal) and we call it the canonical Schauder basis of E.

Proposition 5.1

[3, Proposition 4.7] Every sequence $\{P_k\}_{k\in \mathcal N}\subset {\mathcal {L}}(s',s)$ of nonzero pairwise orthogonal projections is the canonical Schauder basis of the algebra $\mathrm{alg}(\{P_k\}_{k\in \mathcal N})$. In particular, $\{P_k\}_{k\in \mathcal N}$ is a basic sequence in ${\mathcal {L}}(s',s)$, i.e. it is a Schauder basis of the Fréchet space $\overline{\mathrm{lin}}(\{P_k\}_{k\in \mathcal N})$.

Theorem 5.2

[3, Theorem 4.8] Let E be an infinite-dimensional closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ and let $\{P_k\}_{k\in \mathbb N}$ be the canonical Schauder basis of E. Then

$$\begin{aligned} E=\mathrm{alg}(\{P_k\}_{k\in \mathbb N})\cong \lambda ^\infty (||P_k||_q) \end{aligned}$$

as Fréchet ${}^*$-algebras and the isomorphism is given by $P_k\mapsto e_k$ for $k\in \mathbb N$.

Please note that a projection P is in ${\mathcal {L}}(s',s)$ if and only if it is of the form

$$\begin{aligned} P\xi =\sum _{k\in I}\langle \xi ,f_k\rangle f_k \end{aligned}$$

for some finite set I and an orthonormal sequence $(f_k)_{k\in I}\subset s$.

We will also use the identity

$$\begin{aligned} \lambda ^\infty (||\langle \cdot ,f_k\rangle f_k||_q)=\lambda ^\infty (|f_k|_q) \end{aligned}$$

(2)

which holds for every orthonormal sequence $(f_k)_{k\in \mathbb N}\subset s$. (see [3, Rem. 4.11]).

Now we are ready to state and prove the main result of this section.

Theorem 5.3

Every closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ is isomorphic as a Fréchet ${}^*$-algebra to some closed ${}^*$-subalgebra of the algebra $\lambda ^\infty (|f_k|_q)$ for some orthonormal sequence $(f_k)_{k\in \mathbb N}\subset s$. More precisely, if E is an infinite-dimensional closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ and $(\sum _{j\in \mathcal N_k}\langle \cdot ,f_j\rangle f_j)_{k\in \mathbb N}$ is its canonical Schauder basis for some family of finite pairwise disjoint subsets $(\mathcal N_k)_{k\in \mathbb N}$ of natural numbers and an orthonormal sequence $(f_j)_{j\in \mathbb N}\subset s$, then E is isomorphic as a Fréchet ${}^*$-algebra to the closed ${}^*$-subalgebra of $\lambda ^\infty (|f_k|_q)$ generated by $\{\sum _{j\in \mathcal N_k}e_j\}_{k\in \mathbb N}$ and the isomorphism is given by $\sum _{j\in \mathcal N_k}\langle \cdot ,f_j\rangle f_j\mapsto \sum _{j\in \mathcal N_k}e_j$ for $k\in \mathbb N$.

Conversely, if $(f_k)_{k\in \mathbb N}\subset s$ is an orthonormal sequence, then every closed ${}^*$-subalgebra of $\lambda ^\infty (|f_k|_q)$ is isomorphic as a Fréchet ${}^*$-algebra to some closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$.

Proof

By Theorem 5.2, $E=\mathrm{alg}\left( \left\{ \sum _{j\in \mathcal N_k}\langle \cdot ,f_j\rangle f_j\right\} _{k\in \mathbb N}\right)$ for $(\mathcal N_k)_{k\in \mathbb N}$ and $(f_j)_{j\in \mathbb N}\subset s$ as in the statement. Let F be the closed ${}^*$-subalgebra of $\lambda ^\infty (|f_k|_q)$ generated by $\{\sum _{j\in \mathcal N_k}e_j\}_{k\in \mathbb N}$. Define

$$\begin{aligned}\Phi :\mathrm{alg}(\{\langle \cdot ,f_k\rangle f_k\}_{k\in \mathcal N})\rightarrow \lambda ^\infty (|f_k|_q) \end{aligned}$$

by $\langle \cdot ,f_k\rangle f_k\mapsto e_k$, where $\mathcal N:=\bigcup _{k\in \mathbb N}\mathcal N_k$. By Proposition 5.1, $\{\langle \cdot ,f_k\rangle f_k\}_{k\in \mathcal N}$ is the canonical Schauder basis of $\mathrm{alg}(\{\langle \cdot ,f_k\rangle f_k\}_{k\in \mathcal N})$, and thus Theorem 5.2 and (2) imply that $\Phi$ is a Fréchet ${}^*$-algebra isomorphism. Hence, $(\sum _{j\in \mathcal N_k}e_j)_{k\in \mathbb N}=(\Phi (\sum _{j\in \mathcal N_k}\langle \cdot ,f_j\rangle f_j))_{k\in \mathbb N}$ is a Schauder basis of $\Phi (E)$ and $\Phi (E)$ is a closed ${}^*$-subalgebra of $\lambda ^\infty (|f_k|_q)$. Therefore,

$$\begin{aligned} \Phi (E)=\overline{\mathrm{lin}}\left( \left\{ \sum _{j\in \mathcal N_k}e_j\right\} _{k\in \mathbb N}\right) \subset F\subset \Phi (E), \end{aligned}$$

whence $\Phi (E)=F$. In consequence $\Phi _{\mid E}$ is a Fréchet ${}^*$-algebra isomorphism of E and F, which completes the proof of the first statement.

If now $(f_k)_{k\in \mathbb N}\subset s$ is an arbitrary orthonormal sequence then, according to Proposition 5.1, Theorem 5.2 and identity (2), $\lambda ^\infty (|f_k|_q)$ is isomorphic as a Fréchet ${}^*$-algebra to $\mathrm{alg}(\{\langle \cdot ,f_k\rangle f_k\}_{k\in \mathbb N})$. Consequently, every closed ${}^*$-subalgebra of $\lambda ^\infty (|f_k|_q)$ is isomorphic as a Fréchet ${}^*$-algebra to some closed ${}^*$-subalgebra of $\mathrm{alg}(\{\langle \cdot ,f_k\rangle f_k\}_{k\in \mathbb N})$. $\square$

The following characterization of infinite-dimensional closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ is a straightforward consequence of Proposition 4.3 and Theorem 5.3.

Corollary 5.4

Every infinite-dimensional closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ is isomorphic as a Fréchet ${}^*$-algebra to the algebra $\lambda ^\infty (\max _{j\in \mathcal N_k}|f_j|_q)$ for some orthonormal sequence $(f_k)_{k\in \mathbb N}\subset s$ and some family $\{\mathcal N_k\}_{k\in \mathbb N}$ of finite nonempty pairwise disjoint sets of natural numbers. In fact, if E is an infinite-dimensional closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ and $(\sum _{j\in \mathcal N_k}\langle \cdot ,f_j\rangle f_j)_{k\in \mathbb N}$ is its canonical Schauder basis, then

$$\begin{aligned} E\cong \lambda ^\infty \left( \max _{j\in \mathcal N_k}|f_j|_q\right) \end{aligned}$$

as Fréchet ${}^*$-algebras and the isomorphism is given by $\sum _{j\in \mathcal N_k}\langle \cdot ,f_j\rangle f_j\mapsto e_k$ for $k\in \mathbb N$.

Conversely, if $(f_k)_{k\in \mathbb N}\subset s$ is an orthonormal sequence and $\{\mathcal N_k\}_{k\in \mathbb N}$ is a family of finite nonempty pairwise disjoint sets of natural numbers, then $\lambda ^\infty (\max _{j\in \mathcal N_k}|f_j|_q)$ is isomorphic as a Fréchet ${}^*$-algebra to some infinite-dimensional closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$.

At the end of this section we consider the case of maximal commutative subalgebras of ${\mathcal {L}}(s',s)$. A closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ is said to be maximal commutative if it is not properly contained in any larger closed commutative ${}^*$ -subalgebra of ${\mathcal {L}}(s',s)$.

We say that an orthonormal system $(f_k)_{k\in \mathbb N}$ of $\ell _2$ is s -complete, if every $f_k$ belongs to s and for every $\xi \in s$ the following implication holds: if $\langle \xi ,f_k\rangle =0$ for every $k\in \mathbb N$, then $\xi =0$. A sequence $\{P_k\}_{k\in \mathbb N}$ of nonzero pairwise orthogonal projections belonging to ${\mathcal {L}}(s',s)$ is called ${\mathcal {L}}(s',s)$-complete if there is no nonzero projection P belonging to ${\mathcal {L}}(s',s)$ such that $P_kP=0$ for every $k\in \mathbb N$.

One can easily show that an orthonormal system $(f_k)_{k\in \mathbb N}$ is s-complete if and only if the sequence of projections $(\langle \cdot ,f_k\rangle f_k)_{k\in \mathbb N}$ is ${\mathcal {L}}(s',s)$-complete. Hence, by [3, Theorem 4.10], closed commutative ${}^*$-subalgebra E of ${\mathcal {L}}(s',s)$ is maximal commutative if and only if there is an s-complete sequence $(f_k)_{k\in \mathbb N}$ such that $(\langle \cdot ,f_k\rangle f_k)_{k\in \mathbb N}$ is the canonical Schauder basis of E. Combining this with Corollary 5.4, we obtain the first statement of the following corollary.

Corollary 5.5

Every closed maximal commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ is isomorphic as a Fréchet ${}^*$-algebra to the algebra $\lambda ^\infty (|f_k|_q)$ for some s-complete orthonormal sequence $(f_k)_{k\in \mathbb N}$. More precisely, if E is a closed maximal commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ with the canonical Schauder basis $(\langle \cdot ,f_k\rangle f_k)_{k\in \mathbb N}$, then

$$\begin{aligned} E\cong \lambda ^\infty (|f_k|_q) \end{aligned}$$

as Fréchet ${}^*$-algebras and the isomorphism is given by $\langle \cdot ,f_k\rangle f_k\mapsto e_k$ for $k\in \mathbb N$.

Conversely, if $(f_k)_{k\in \mathbb N}$ is an s-complete orthonormal sequence, then $\lambda ^\infty (|f_k|_q)$ is isomorphic as a Fréchet ${}^*$-algebra to some closed maximal commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$.

Proof

In order to prove the second statement, take an arbitrary s-complete orthonormal sequence $(f_k)_{k\in \mathbb N}$. By Proposition 5.1 and the remark above our Corollary, $\mathrm{alg}(\{\langle \cdot , f_k\rangle f_k\}_{k\in \mathbb N})$ is maximal commutative and from the first statement it follows that it is isomorphic as a Fréchet ${}^*$-algebra to $\lambda ^\infty (|f_k|_q)$. $\square$

It is also worth pointing out the following result.

Proposition 5.6

Every commutative (not necessary closed) ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ is contained in some maximal commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$.

Proof

Let E be a commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$. Clearly,

$$\begin{aligned} \mathcal X:=\{\widetilde{E}: \widetilde{E} \text { commutative} {}^* \text { -subalgebra of} \quad {\mathcal {L}}(s',s)\quad \text {and}\quad E\subset \widetilde{E}\} \end{aligned}$$

with the inclusion relation is a partially ordered set. Consider a nonempty chain $\mathcal C$ in $\mathcal X$ and let $E_{\mathcal C}:=\bigcup _{F\in \mathcal C}F$. It is easy to check that $E_{\mathcal C}\in \mathcal X$, and, of course, $E_{\mathcal C}$ is an upper bound of $\mathcal C$. Hence, by the Kuratowski–Zorn lemma, $\mathcal X$ has a maximal element; let us call it M. By the continuity of the algebra operations, $\overline{M}^{{\mathcal {L}}(s',s)}$ is a closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$, hence from the maximality of M, we have $M=\overline{M}^{{\mathcal {L}}(s',s)}$, i.e. M is a (closed) maximal commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ containing E. $\square$

6 Closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ with the property ($\Omega$)

The main purpose of the last section is to prove that a closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ is isomorphic as a Fréchet ${}^*$-algebra to some closed ${}^*$-subalgebra of s if and only if it is isomorphic as a Fréchet space to some complemented subspace of s (Theorem 6.2), i.e. if it has the so-called property ($\Omega$).

Definition 6.1

A Fréchet space E with a fundamental sequence $(||\cdot ||_q)_{q\in \mathbb N_0}$ of seminorms has the property ($\Omega$) if the following condition holds:

$$\begin{aligned} \forall p\;\exists q\;\forall r\;\exists \theta \in (0,1)\;\exists C>0\;\forall y\in E'\quad ||y||_q'\le C||y||_p'^{1-\theta }||y||_r'^{\theta }, \end{aligned}$$

where $E'$ is the topological dual of E and $||y||_p':=\sup \{|y(x)|:||x||_p\le 1\}$.

The property ($\Omega$) (together with the property (DN)) plays a crucial role in the theory of nuclear Fréchet spaces (for details, see [13, Ch. 29]).

Recall that a subspace F of a Fréchet space E is called complemented (in E) if there is a continuous projection $\pi :E\rightarrow E$ with $\mathrm{im}\pi =F$. Since every subspace of ${\mathcal {L}}(s',s)$ has the property (DN) (and, by [3, Proposition 3.2], the norm $||\cdot ||_{\ell _2\rightarrow \ell _2}$ is already a dominating norm) [13, Proposition 31.7] implies that a closed ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ is isomorphic to a complemented subspace of s if and only if it has the property ($\Omega$). The class of complemented subspaces of s is still not well-understood (e.g. we do not know whether every such subspace has a Schauder basis—the Pełczyński problem) and, on the other hand, the class of closed ${}^*$-subalgebras of s has a simple description (see Corollary 4.4). The following theorem implies that, when restricting to the family of closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$, these two classes of Fréchet spaces coincide.

Theorem 6.2

Let E be an infinite-dimensional closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ and let $(\sum _{j\in \mathcal N_k}\langle \cdot ,f_j\rangle f_j)_{k\in \mathbb N}$ be its canonical Schauder basis. Then the following assertions are equivalent:

• (i) E is isomorphic as a Fréchet ${}^*$-algebra to some closed ${}^*$-subalgebra of s;

• (ii) E is isomorphic as a Fréchet space to some complemented subspace of s;

• (iii) E has the property ($\Omega$);

• (iv) $\exists p\;\forall q\;\exists r\;\exists C>0\;\forall k\quad \max _{j\in \mathcal N_k}|f_j|_q\le C\max _{j\in \mathcal N_k}|f_j|_p^r$.

In order to prove Theorem 6.2, we will need Lemmas 6.3, 6.4 and Propositions 6.5, 6.6.

The following result is a consequence of nuclearity of closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$.

Lemma 6.3

Let $(f_k)_{k\in \mathbb N}\subset s$ be an orthonormal sequence and let $(\mathcal N_k)_{k\in \mathbb N}$ be a family of finite pairwise disjoint subsets of natural numbers. For $r\in \mathbb N_0$ let $\sigma _r:\mathbb N\rightarrow \mathbb N$ be a bijection such that the sequence $(\max _{j\in \mathcal N_{\sigma _r(k)}}|f_j|_r)_{k\in \mathbb N}$ is non-decreasing. Then there is $r_0\in \mathbb N$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{k}{\max _{j\in \mathcal N_{\sigma _r(k)}}|f_j|_r}=0 \end{aligned}$$

for all $r\ge r_0$.

Proof

By Corollary 5.4, $\lambda ^\infty (\max _{j\in \mathcal N_k}|f_j|_q)$ is a nuclear space. Hence, by the Grothendieck–Pietsch theorem (see e.g. [13, Theorem 28.15]), for every $q\in \mathbb N_0$ there is $r\in \mathbb N_0$ such that

$$\begin{aligned} \sum _{k=1}^\infty \frac{\max _{j\in \mathcal N_k}|f_j|_q}{\max _{j\in \mathcal N_k}|f_j|_{r}}<\infty . \end{aligned}$$

In particular (for $q=0$), there is $r_0$ such that for $r\ge r_0$ we have

$$\begin{aligned} \sum _{k=1}^\infty \frac{1}{\max _{j\in \mathcal N_{\sigma _r(k)}}|f_j|_r}=\sum _{k=1}^\infty \frac{1}{\max _{j\in \mathcal N_k}|f_j|_r}<\infty . \end{aligned}$$

Since the sequence $(\max _{j\in \mathcal N_{\sigma _r(k)}}|f_j|_r)_{k\in \mathbb N}$ is non-decreasing, the conclusion follows from the elementary theory of number series. $\square$

Lemma 6.4

Let $(a_k)_{k\in \mathbb N}\subset [1,\infty )$ be a non-decreasing sequence such that $a_k\ge 2k$ for k big enough. Then there exist a strictly increasing sequence $(b_k)_{k\in \mathbb N}$ of natural numbers and $C>0$ such that

$$\begin{aligned} \frac{1}{C}a_k\le b_k\le Ca_k^2 \end{aligned}$$

for every $k\in \mathbb N$.

Proof

Let $k_0\in \mathbb N$ be such that $a_k\ge 2k$ for $k> k_0$ and choose $C\in \mathbb N$ so that

$$\begin{aligned} \frac{1}{C}a_k\le k\le Ca_k^2 \end{aligned}$$

for $k\in \mathcal N_0:=\{1,\ldots ,k_0\}$. Denote also $\mathcal N_1:=\{k\in \mathbb N: a_k=a_{k_0+1}\}$ and, recursively, $\mathcal N_{j+1}:=\{k\in \mathbb N: a_k=a_{\max \mathcal N_j+1}\}$. Clearly, $\mathcal N_j$ are finite, pairwise disjoint, $\bigcup _{j\in \mathbb N_0}\mathcal N_j=\mathbb N$ and $k<l$ for $k\in \mathcal N_j$, $l\in \mathcal N_{j+1}$.

Let $b_k:=k$ for $k\in \mathcal N_0$ and let

$$\begin{aligned} b_{m_j+l-1}:=C\lceil \max \{a_{m_j-1}^2,a_{m_j}\}\rceil +l \end{aligned}$$

for $j\in \mathbb N$ and $1\le l\le |\mathcal N_j|$, where $m_j:=\min \mathcal N_{j}$ and $\lceil x\rceil :=\min \{n\in \mathbb Z:n\ge x\}$ stands for the ceiling of $x\in \mathbb R$. We will show inductively that $(b_k)_{k\in \mathbb N}$ is a strictly increasing sequence of natural numbers such that

$$\begin{aligned} \frac{1}{C}a_k\le b_k\le Ca_k^2 \end{aligned}$$

(3)

for every $k\in \mathbb N$.

Clearly, the condition (3) holds for $k\in \mathcal N_0$. Assume that $(b_k)_{k\in \mathcal N_0\cup \cdots \cup \mathcal N_j}$ is a strictly increasing sequence of natural numbers for which the condition (3) holds. For simplicity, denote $m:=\min \mathcal N_{j+1}$. By the inductive assumption, we obtain $b_{m-1}\le C a_{m-1}^2$, hence

$$\begin{aligned} b_m-b_{m-1}\ge C\lceil \max \{a_{m-1}^2,a_m\}\rceil +1-Ca_{m-1}^2\ge Ca_{m-1}^2+1-Ca_{m-1}^2\ge 1 \end{aligned}$$

so $b_{m-1}<b_m$, and, clearly, $b_m<b_{m+1}<\cdots <b_{\max \mathcal N_{j+1}}$.

Fix $1\le l\le |N_{j+1}|$. We have

$$\begin{aligned} b_{m+l-1}\ge Ca_m=Ca_{m+l-1}\ge \frac{1}{C}a_{m+l-1} \end{aligned}$$

so the first inequality in (3) holds for $k\in \mathcal N_{j+1}$. Next, by assumption, we get

$$\begin{aligned} a_{m+l-1}\ge 2(m+l-1), \end{aligned}$$

(4)

whence

$$\begin{aligned} l\le a_{m-l+1}-m+1. \end{aligned}$$

(5)

Consider two cases. If $a_m\ge a_{m-1}^2$, then, from (5)

$$\begin{aligned} b_{m-l+1}&=C\lceil a_m\rceil +l=C\lceil a_{m+l-1}\rceil +l\le 2Ca_{m+l-1}+a_{m+l-1}-m+1\\&\le (2C+1)a_{m+l-1}\le Ca_{m+l-1}^2, \end{aligned}$$

where the last inequality holds because $C\ge 1$ and, from (4), we have

$$\begin{aligned} a_{m-l+1}\ge 2(m+l-1)\ge 2m\ge 2(k_0+1)\ge 4. \end{aligned}$$

Finally, if $a_{m-1}^2>a_m$, then, from (4), we obtain (note that, by the definition of $\mathcal N_{j}$ and $\mathcal N_{j+1}$, we have $a_{m-1}<a_m$)

$$\begin{aligned} b_{m-l+1}&=C\lceil a_{m-1}^2\rceil +l\\&\le C\lceil (a_{m}-1)^2\rceil +l\\&=C\lceil a_m^2-2a_m+1\rceil +l\\&\le C(a_m^2-2a_m+2)+l\\&\le Ca_m^2-2Ca_m+2C+Cl\\&=Ca_{m+l-1}^2-C(2a_{m+l-1}-2-l)\\&\le Ca_{m+l-1}^2-C(4(m+l-1)-2-l)\\&=Ca_{m+l-1}^2-C(4m+3l-6)\le Ca_{m+l-1}^2. \end{aligned}$$

Hence we have shown that the second inequality in (3) holds for $k\in \mathcal N_{j+1}$, and the proof is complete. $\square$

Proposition 6.5

Let E be an infinite-dimensional closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ and let $(\sum _{j\in \mathcal N_k}\langle \cdot ,f_j\rangle f_j)_{k\in \mathbb N}$ be its canonical Schauder basis. Moreover, let $(n_k)_{k\in \mathbb N}$ be a strictly increasing sequence of natural numbers and let F be the closed ${}^*$-subalgebra of s generated by $\{e_{n_k}\}_{k\in \mathbb N}$. Then the following assertions are equivalent:

• (i) E is isomorphic to F as a Fréchet ${}^*$-algebra;

• (ii) $\lambda ^\infty (\max _{j\in \mathcal N_k}|f_j|_q)\cong \lambda ^\infty (n_k^q)$ as Fréchet ${}^*$-algebras;

• (iii) there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that $\lambda ^\infty (\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_q)=\lambda ^\infty (n_k^q)$ as Fréchet ${}^*$-algebras;

• (iv) there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that $\lambda ^\infty (\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_q)=\lambda ^\infty (n_k^q)$ as sets;

• (v) there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that

  • $(\alpha )$ $\forall q\in \mathbb N_0\;\exists r\in \mathbb N_0\;\exists C>0\;\forall k\in \mathbb N\quad \max _{j\in \mathcal N_{\sigma (k)}}|f_j|_q\le Cn_k^r$,

  • $(\beta )$ $\forall r'\in \mathbb N_0\;\exists q'\in \mathbb N_0\;\exists C'>0\;\forall k\in \mathbb N\quad n_k^{r'}\le C'\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{q'}$.

Proof

This is an immediate consequence of Proposition 4.2 and Corollary 5.4. $\square$

In view of Corollary 4.4, every closed ${}^*$-subalgebra of s is isomorphic as a Fréchet ${}^*$-algebra to $\lambda ^\infty (n_k^q)$ (i.e. the closed ${}^*$-subalgebra of s generated by $\{e_{n_k}\}_{k\in \mathbb N}$) for some strictly increasing sequence $(n_k)_{k\in \mathbb N}\subset \mathbb N$, hence Proposition 6.5 characterizes closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ which are isomorphic as Fréchet ${}^*$-algebras to some ${}^*$-subalgebra of s.

The property (DN) for the space s gives us the following inequality.

Proposition 6.6

For every $p,r\in \mathbb N_0$ there is $q\in \mathbb N_0$ such that for all $\xi \in s$ with $||\xi ||_{\ell _2}=1$ the following inequality holds

$$\begin{aligned} |\xi |_p^r\le |\xi |_q. \end{aligned}$$

Proof

Take $p,r\in \mathbb N_0$ and let $j\in \mathbb N_0$ be such that $r\le 2^j$. Applying iteratively (j-times) the inequality from Proposition 3.2 to $\xi \in s$ with $||\xi ||_{\ell _2}=1$ we get

$$\begin{aligned} |\xi |_p^r\le |\xi |_p^{2^j}\le |\xi |_{2^jp}, \end{aligned}$$

and thus the required inequality holds for $q=2^jp$. $\square$

Now we are ready to prove Theorem 6.2.

Proof of Theorem 6.2

(i)$\Rightarrow$(ii): By Corollary 4.4, each closed ${}^*$-subalgebra of s is a complemented subspace of s.

(ii)$\Leftrightarrow$(iii): See e.g. [13, Proposition31.7].

(iii)$\Rightarrow$(iv): By Corollary 5.4 and nuclearity (see e.g. [13, Proposition28.16]),

$$\begin{aligned} E\cong \lambda ^\infty \left( \max _{j\in \mathcal N_k}|f_j|_q\right) =\lambda ^1\left( \max _{j\in \mathcal N_k}|f_j|_q\right) \end{aligned}$$

as Fréchet ${}^*$-algebras. Hence, by [21, 22, Proposition 5.3], the property $(\Omega )$ yields

$$\begin{aligned} \forall l\;\exists m\;\forall n\;\exists t\;\exists C>0\; \forall k\quad \max _{j\in \mathcal N_k}|f_j|_l^t\max _{j\in \mathcal N_k}|f_j|_n\le C\max _{j\in \mathcal N_k}|f_j|_m^{t+1}. \end{aligned}$$

In particular, taking $l=0$, we get (iv).

(iv)$\Rightarrow$(i): Take p from the condition (iv). By Lemma 6.3(ii), there is $p_1\ge p$ and a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that $(\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{p_1})_{k\in \mathbb N}$ is non-decreasing and $\lim _{k\rightarrow \infty }\frac{k}{\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{p_1}}=0$. Consequently, for k big enough

$$\begin{aligned} \max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{p_1}\ge 2k, \end{aligned}$$

and therefore, by Lemma 6.4, there is a strictly increasing sequence $(n_k)_{k\in \mathbb N}\subset \mathbb N$ and $C_1>0$ such that

$$\begin{aligned} \frac{1}{C_1}\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{p_1}\le n_k\le C_1\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{p_1}^2 \end{aligned}$$

(6)

for every $k\in \mathbb N$. Now, by the conditions (iv) and (6), we get that for all q there is r and $C_2:=CC_1^r$ such that

$$\begin{aligned} \max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{q}\le C\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{p_1}^r\le C_2n_k^r \end{aligned}$$

for all $k\in \mathbb N$, so the condition $(\alpha )$ from Proposition 6.5(v) holds. Finally, by (6) and Proposition 6.6 we obtain that for all $r'$ there is $q'$ and $C_3:=C_1^{r'}$ such that

$$\begin{aligned} n_k^{r'}\le C_3\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{p_1}^{2{r'}}\le C_3\max _{j\in \mathcal N_{\sigma (k)}}|f_j|_{q'} \end{aligned}$$

for every $k\in \mathbb N$. Hence the condition ($\beta$) from Proposition 6.5(v) is satisfied, and therefore, by Proposition 6.5, E is isomorphic as a Fréchet ${}^*$-algebra to the closed ${}^*$-subalgebra of s generated by $\{e_{n_k}\}_{k\in \mathbb N}$. $\square$

Now we shall give an example of some class of closed commutative ${}^*$-subalgebras of ${\mathcal {L}}(s',s)$ which are isomorphic to closed ${}^*$-subalgebras of s.

Example 6.7

Let $\mathbb {H}_1:=[1]$. We define recursively Hadamard matrices

$$\begin{aligned} \mathbb {H}_{2^n}:= \left[ \begin{array}{ll} \mathbb {H}_{2^{n-1}} &{} \phantom {-}\mathbb {H}_{2^{n-1}}\\ \mathbb {H}_{2^{n-1}} &{} -\mathbb {H}_{2^{n-1}} \end{array} \right] \end{aligned}$$

for $n\in \mathbb N$. Then the matrices $\widehat{\mathbb {H}}_{2^n}:=2^{-\frac{n}{2}}\mathbb {H}_{2^n}$ are unitary, and thus their rows form an orthonormal system of $2^n$ vectors. Now fix an arbitrary sequence $(d_n)_{n\in \mathbb N}\subset \mathbb N_0$ and define

$$\begin{aligned} U:= \left[ \begin{array}{l@{\quad }l@{\quad }l@{\quad }l} \widehat{\mathbb {H}}_{2^{d_1}} &{} 0 &{} 0 &{} \ldots \\ 0 &{} \widehat{\mathbb {H}}_{2^{d_2}} &{} 0 &{} \ldots \\ 0 &{} 0 &{} \widehat{\mathbb {H}}_{2^{d_3}} &{} \\ \vdots &{} \vdots &{} &{} \ddots \end{array} \right] . \end{aligned}$$

Let $f_k$ denote the k-th row of the matrix U. Then $(f_k)_{k\in \mathbb N}$ is an orthonormal basis of $\ell _2$ and clearly each $f_k$ belongs to s. We will show that the closed (maximal) commutative ${}^*$-subalgebra $\mathrm{alg}(\{\langle ,\cdot ,f_k\rangle f_k\}_{k\in \mathbb N})$ of ${\mathcal {L}}(s',s)$ is isomorphic to some closed ${}^*$-subalgebra of s. By Theorem 6.2, it is enough to prove that

$$\begin{aligned} \exists p\;\forall q\;\exists r\;\exists C>0\;\forall k\quad |f_k|_{\infty ,q}\le C|f_k|_{\infty ,p}^r. \end{aligned}$$

(7)

Fix $q\in \mathbb N_0$, $k\in \mathbb N$ and find $n\in \mathbb N$ such that $2^{d_1}+\cdots +2^{d_{n-1}}<k\le 2^{d_1}+\cdots +2^{d_n}$. Then

$$\begin{aligned} \frac{|f_k|_{\infty ,q}}{|f_k|_{\infty ,1}^{2q}}=\frac{2^{-\frac{d_n}{2}}(2^{d_1}+\cdots +2^{d_n})^q}{2^{-d_nq}(2^{d_1}+\cdots +2^{d_n})^{2q}} =2^{d_n(q-1/2)}(2^{d_1}+\cdots +2^{d_n})^{-q}\le 1 \end{aligned}$$

and thus the condition (7) holds with $p=C=1$ and $r=2q$.

The next theorem solves in negative [3, Open Problem 4.13]. In contrast to the algebra s, all of whose closed ${}^*$-subalgebras are complemented subspaces of s (Corollary 4.4), Theorems 6.2 and 6.9 imply that there is a closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ which is not complemented in ${\mathcal {L}}(s',s)$ (otherwise it would have the property ($\Omega$), see [13, Proposition 31.7]). In the proof we will use the following identity.

Lemma 6.8

For every increasing sequence $(\alpha _j)_{j\in \mathbb N}\subset (0,\infty )$ and every $p\in \mathbb N$ we have

$$\begin{aligned} \sup _{j\in \mathbb N}\left( \alpha _j^{p-j+1}\cdot \prod _{i=1}^{j-1}\alpha _i\right) =\prod _{i=1}^p \alpha _i. \end{aligned}$$

Proof

For $j\ge p+1$ we get

$$\begin{aligned} \frac{\alpha _j^{p-j+1}\cdot \prod _{i=1}^{j-1}\alpha _i}{\prod _{i=1}^p \alpha _i}=\alpha _j^{p-j+1}\cdot \prod _{i=p+1}^{j-1}\alpha _i =\frac{\prod _{i=p+1}^{j-1}\alpha _i}{\alpha _j^{j-p-1}}\le 1 \end{aligned}$$

and, similarly, for $j\le p-1$ we obtain

$$\begin{aligned} \frac{\alpha _j^{p-j+1}\cdot \prod _{i=1}^{j-1}\alpha _i}{\prod _{i=1}^p \alpha _i}=\frac{\alpha _j^{p-j+1}}{\prod _{i=j}^p\alpha _i}\le 1. \end{aligned}$$

Since $\alpha _p^{p-p+1}\cdot \prod _{i=1}^{p-1}\alpha _i=\prod _{i=1}^p \alpha _i$, the supremum is attained for $j=p$, and we are done. $\square$

Theorem 6.9

There is a closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ which is not isomorphic to any closed ${}^*$-subalgebra of s.

Proof

Let $m_k$ be the k-th prime number, $N_{k,1}:=m_k$, $N_{k,j+1}:=m_k^{N_{k,j}}$ for $j,k\in \mathbb N$. Define $a_{k,1}:=c_k$ and

$$\begin{aligned} a_{k,j}:=c_k\frac{\prod _{i=1}^{j-1}N_{k,i}}{N_{k,j}^{j-1}} \end{aligned}$$

for $j\ge 2$, where the sequence $(c_k)_{k\in \mathbb N}$ is choosen so that $||(a_{k,j})_{j\in \mathbb N}||_{\ell _2}=1$, i.e.

$$\begin{aligned} c_k:=\left( \sum _{j=1}^\infty \left( \frac{\prod _{i=1}^{j-1}N_{k,i}}{N_{k,j}^{j-1}}\right) ^2\right) ^{-1/2}. \end{aligned}$$

The numbers $c_k$ make sense, because, by Lemma 6.8,

$$\begin{aligned} \sum _{j=1}^\infty \left( \frac{\prod _{i=1}^{j-1}N_{k,i}}{N_{k,j}^{j-1}}\right) ^2&=\sum _{j=1}^\infty \left( N_{k,j}^{-j+1}\cdot \prod _{i=1}^{j-1}N_{k,i}\right) ^2\\&=\sum _{j=1}^\infty \frac{1}{N_{k,j}^2}\left( N_{k,j}^{1-j+1}\cdot \prod _{i=1}^{j-1}N_{k,i}\right) ^2\\&\le \sup _{j\in \mathbb N}\left( N_{k,j}^{1-j+1}\cdot \prod _{i=1}^{j-1}N_{k,i}\right) ^2\sum _{j=1}^\infty \frac{1}{N_{k,j}^2}\\&=N_{k,1}^2\sum _{j=1}^\infty \frac{1}{N_{k,j}^2}<N_{k,1}^2\sum _{j=1}^\infty \frac{1}{j^2}<\infty . \end{aligned}$$

Finally, define an orthonormal sequence $(f_k)_{k\in \mathbb N}$ by

$$\begin{aligned} f_k:=\sum _{j=1}^\infty a_{k,j}e_{N_{k,j}}. \end{aligned}$$

We will show that $\mathrm{alg}(\{\langle \cdot ,f_k\rangle f_k\}_{k\in \mathbb N})$ is a closed ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ which is not isomorphic as an algebra to any closed ${}^*$-subalgebra of s. By Theorem 6.2 and nuclearity, it is enough to show that each $f_k$ belongs to s and for every $p,r\in \mathbb N$ the following condition holds

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{|f_k|_{\infty ,p+1}}{|f_k|_{\infty ,p}^r}=\infty , \end{aligned}$$

where $|\xi |_{\infty ,q}:=\sup _{j\in \mathbb N}|\xi _j|j^q$.

Note first that $|f_k|_{\infty ,p}=a_{k,p}N_{k,p}^p$. In fact, by Lemma 6.8, we get

$$\begin{aligned} |f_k|_{\infty ,p}&=\sup _{j\in \mathbb N}a_{k,j}N_{k,j}^p=c_k\sup _{j\in \mathbb N}\left( N_{k,j}^p\cdot \frac{\prod _{i=1}^{j-1}N_{k,i}}{N_{k,j}^{j-1}}\right) \\&=c_k\sup _{j\in \mathbb N}\left( N_{k,j}^{p-j+1}\cdot \prod _{i=1}^{j-1}N_{k,i}\right) \\&=c_k\prod _{i=1}^p N_{k,i}=c_kN_{k,p}^p\cdot \frac{\prod _{i=1}^{p-1}N_{k,i}}{N_{k,p}^{p-1}}=a_{k,p}N_{k,p}^p. \end{aligned}$$

In particular, $f_k\in s$ for $k\in \mathbb N$. Next, for $j,k\in \mathbb N$, we have

$$\begin{aligned} \frac{a_{k,j+1}N_{k,j+1}^j}{a_{k,j}} =\frac{c_k N_{k,j+1}^j\cdot \frac{\prod _{i=1}^{j}N_{k,i}}{N_{k,j+1}^{j}}}{c_k\frac{\prod _{i=1}^{j-1}N_{k,i}}{N_{k,j}^{j-1}}} =\frac{\prod _{i=1}^{j}N_{k,i}}{\frac{\prod _{i=1}^{j-1}N_{k,i}}{N_{k,j}^{j-1}}}=N_{k,j}^j. \end{aligned}$$

Moreover, for every $j,r\in \mathbb N$ we get

$$\begin{aligned} \frac{N_{k,j+1}}{N_{k,j}^r}=\frac{m_k^{N_{k,j}}}{N_{k,j}^r}\ge \frac{2^{N_{k,j}}}{N_{k,j}^r}\xrightarrow [k\rightarrow \infty ]{}\infty , \end{aligned}$$

and clearly $a_{k,j}\le 1$ for $j,k\in \mathbb N$. Hence, for $p,r\in \mathbb N$ we obtain

$$\begin{aligned} \frac{|f_k|_{\infty ,p+1}}{|f_k|_{\infty ,p}^r}&=\frac{a_{k,p+1}N_{k,p+1}^{p+1}}{a_{k,p}^rN_{k,p}^{pr}} =\frac{a_{k,p+1}N_{k,p+1}^{p}}{a_{k,p}}\cdot \frac{1}{a_{k,p}^{r-1}}\cdot \frac{N_{k,p+1}}{N_{k,p}^{pr}}\\&=N_{k,p}^p\cdot \frac{1}{a_{k,p}^{r-1}}\cdot \frac{N_{k,p+1}}{N_{k,p}^{pr}}\ge \frac{N_{k,p+1}}{N_{k,p}^{pr}}\xrightarrow [k\rightarrow \infty ]{}\infty , \end{aligned}$$

which is the desired conclusion. $\square$

We end this section with two consequences of Theorem 6.2.

For a monotonically increasing sequence $\alpha =(\alpha _k)_{k\in \mathbb N}$ in $[0,\infty )$ such that $\lim _{j\rightarrow \infty }\alpha _j=\infty$ we define the power series space of infinite type

$$\begin{aligned} \Lambda _\infty (\alpha ):=\{(\xi _j)_{j\in \mathbb N}\subset \mathbb C^\mathbb N:\sum _{k=1}^\infty |\xi _k|^2e^{2q\alpha _k}<\infty \quad \text {for all}\, q\in \mathbb N_0\}. \end{aligned}$$

Corollary 6.10

Let E be a closed commutative ${}^*$-subalgebra of ${\mathcal {L}}(s',s)$ isomorphic as Fréchet space to $\Lambda _\infty (\alpha )$. Then E is isomorphic to $\Lambda _\infty (\alpha )$ as a Fréchet ${}^*$-algebra.

Proof

Let $(P_k)_{k\in \mathbb N}$ be the canonical Schauder basis of E. In view of Proposition 4.2, we should show that there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that

• $(\alpha )$ $\forall q\in \mathbb N_0\;\exists r\in \mathbb N_0\;\exists C>0\;\forall k\in \mathbb N\quad ||P_{\sigma (k)}||_q\le Ce^{r\alpha _k}$,

• ($\beta$) $\forall r'\in \mathbb N_0\;\exists q'\in \mathbb N_0\;\exists C'>0\;\forall k\in \mathbb N\quad e^{r'\alpha _k}\le C'||P_{\sigma (k)}||_{q'}$.

By Theorem 6.2, E is isomorphic as a Fréchet ${}^*$-algebra to some infinite-dimensional closed ${}^*$-subalgebra of s, and thus by Corollary 4.4, E is isomorphic as a Fréchet ${}^*$-algebra to $\lambda ^\infty (n_k^q)$ for some strictly increasing sequence $(n_k)_{k\in \mathbb N}$ in $\mathbb N$. Hence, by Proposition 4.2, there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that

$$\begin{aligned}&\forall q\in \mathbb N_0\;\exists r\in \mathbb N_0\;\exists C>0\;\forall k\in \mathbb N\quad ||P_{\sigma (k)}||_q\le C n_k^r,\end{aligned}$$

(8)

$$\begin{aligned}&\forall r'\in \mathbb N_0\;\exists q'\in \mathbb N_0\;\exists C'>0\;\forall k\in \mathbb N\quad n_k^{r'}\le C'||P_{\sigma (k)}||_{q'}. \end{aligned}$$

(9)

Since $\lambda ^\infty (n_k^q)=\Lambda _\infty (\log n_k)$, it follows from [13, Theorem 29.1] that there is $q\in \mathbb N$ and $k_0$ such that for $k\ge k_0$

$$\begin{aligned} \frac{1}{q}\alpha _k\le \log n_k\le q\alpha _k. \end{aligned}$$

Consequently, there is $q\in \mathbb N$ and $D>0$ such that

$$\begin{aligned} e^{\alpha _k}\le Dn_k^q\quad \text {and}\quad n_k\le De^{q\alpha _k} \end{aligned}$$

for all $k\in \mathbb N$. Now (8) and (9) yield the desired conclusion. $\square$

By the theorem of Crone and Robinson [5] it follows that all bases of the space s are quasi-equivalent, i.e. given any two bases $(f_k)_{k\in \mathbb N}$ and $(g_k)_{k\in \mathbb N}$ of s, there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ and a sequence $(c_k)_{k\in \mathbb N}$ of non-zero scalars such that the operator $T:s\rightarrow s$ defined by $Te_k=c_kf_{\sigma (k)}$ is a Fréchet space isomorphism. Our last result shows that in the case of bases of s which form an orthonormal sequence of $\ell _2$, the sequence $(c_k)_{k\in \mathbb N}$ can always be taken constant and equal to 1.

Corollary 6.11

For every Schauder basis $(f_k)_{k\in \mathbb N}$ of the space s which is at the same time an orthonormal sequence of $\ell _2$ there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that $T:s\rightarrow s$ defined by $Te_k:=f_{\sigma (k)}$, $k\in \mathbb N$, is a Fréchet space isomorphism.

Proof

Clearly, the closed ${}^*$-subalgebra E of ${\mathcal {L}}(s',s)$ generated by the sequence of one-dimensional projections $(\langle \cdot ,f_k\rangle f_k)_{k\in \mathbb N}$ is isomorphic as a Fréchet space to s. Hence, by Corollaries 5.5 and 6.10, $\lambda ^\infty (|f_k|_q)\cong E\cong s$ as Fréchet ${}^*$-algebras. Now, by Proposition 4.2, there is a bijection $\sigma :\mathbb N\rightarrow \mathbb N$ such that

$$\begin{aligned}&\forall q\in \mathbb N_0\;\exists r\in \mathbb N_0\;\exists C>0\;\forall k\in \mathbb N\quad |f_{\sigma (k)}|_r\le Ck^r,\\&\forall r'\in \mathbb N_0\;\exists q'\in \mathbb N_0\;\exists C'>0\;\forall k\in \mathbb N\quad k^{r'}\le C'|f_{\sigma (k)}|_{q'}. \end{aligned}$$

This shows that the map $T:s\rightarrow s$ which sends $e_k$ to $f_{\sigma (k)}$, $k\in \mathbb N$, defines an automorphism of the Fréchet space s. $\square$