Commutative subalgebras of the algebra of smooth operators
aa r X i v : . [ m a t h . F A ] M a r COMMUTATIVE SUBALGEBRAS OF THE ALGEBRA OF SMOOTHOPERATORS
TOMASZ CIAŚAbstract.
We consider the Fr´echet ∗ -algebra L ( s ′ , s ) ⊆ L ( ℓ ) of the so-called smooth oper-ators, i.e. continuous linear operators from the dual s ′ of the space s of rapidly decreasingsequences into s . This algebra is a non-commutative analogue of the algebra s . We characterizeall closed commutative ∗ -subalgebras of 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 L ( s ′ , s )which cannot be embedded into s . Introduction
The algebra L ( s ′ , s ) can be represented as the algebra K ∞ := { ( x j,k ) j,k ∈ N ∈ C N : sup j,k ∈ N | x j,k | j q k q < ∞ for all q ∈ N } of rapidly decreasing matrices (with matrix multiplication and matrix complex conjugation).Another representation of L ( s ′ , s ) is the algebra S ( R ) of Schwartz functions on R with theVolterra convolution ( f · g )( x, y ) := Z R f ( x, z ) g ( z, y )d z as multiplication and the involution f ∗ ( x, y ) := f ( y, x ) . In these forms, the algebra L ( s ′ , s ) usually appears and plays a significant role in K -theory ofFr´echet algebras (see Bhatt and Inoue [1, Ex. 2.12], Cuntz [5, p. 144], [6, p. 64–65], Gl¨ockner andLangkamp [10], Phillips [13, Def. 2.1]) and in C ∗ -dynamical systems (Elliot, Natsume and Nest[8, Ex. 2.6]). Very recently, Piszczek obtained several rusults concerning closed ideals, automaticcontinuity (for positive functionals and derivations), amenability and Jordan decomposition in K ∞ (see Piszczek [16, 15] and his forthcoming papers ‘Automatic continuity and amenabilityin the non-commutative Schwartz space’ and ‘The noncommutative Schwartz space is weaklyamenable’). Moreover, in the context of algebras of unbounded operators, the algebra L ( s ′ , s )appears in the book [17] as B ( s ) := { x ∈ L ( ℓ ) : xℓ ⊆ s, x ∗ ℓ ⊆ s and axb is nuclear for all a, b ∈ L ∗ ( s ) } , where L ∗ ( s ) is the so-called maximal O ∗ -algebra on s (see also [17, 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 commu-tative algebra s . The most important features of this algebra are the following: • it is isomorphic as a Fr´echet space to the Schwartz space S ( R ) of smooth rapidly de-creasing functions on the real line; Primary: 46H35, 46J40. Secondary: 46A11, 46A63.
Key words and phrases:
Topological algebras of operators, structure and classification of commutative topo-logical algebras with involution, nuclear Fr´echet spaces, smooth operators.The research of the author was supported by the National Center of Science, grant no. 2013/09/N/ST1/04410. TOMASZ CIAŚ • it has several representations as algebras of operators acting between natural spaces ofdistributions and functions (see [7, Th. 1.1]); • it is a dense ∗ -subalgebra of the C ∗ -algebra K ( ℓ ) of compact operators on ℓ ; • it is (properly) contained in the intersection of all Schatten classes S p ( ℓ ) over p >
0; inparticular L ( s ′ , s ) is contained the class HS ( ℓ ) of Hilbert-Schmidt operators, and thusit is a unitary space; • the operator C ∗ -norm || · || ℓ → ℓ is the so-called dominating norm on that algebra (thedominating norm property is a key notion in the structure theory of nulcear Fr´echetspaces – see [3, Prop. 3.2] and [12, Prop. 31.5]).The main result of the present paper is a characterization of closed ∗ -subalgebras of L ( s ′ , s )which are at the same time isomorphic as Fr´echet ∗ -algebras to closed ∗ -subalgebras of s (The-orem 6.2). It turns out that these are exactly those subalgebras which satisfy the classicalcondition (Ω) of Vogt. Then in Theorem 6.10 we give an example of a closed commutative ∗ -subalgebra of L ( s ′ , s ) which does not satisfy this condition.In order to prove this result we characterize in Section 4 closed ∗ -subalgebras of K¨othe sequencealgebras (Proposition 4.3). In particular, we give such a description for closed ∗ -subalgebras of s (Corollary 4.4). In Section 5 we describe all closed ∗ -subalgebras of L ( s ′ , s ) as suitable K¨othesequence algebras (see Corollary 5.4 and compare with [3, Th. 4.8])The present paper is a continuation of [3] and [7] and it focuses on description of closedcommutative ∗ -subalgebras of L ( s ′ , s ) (especially those with the property (Ω)). Most of theresults have been already presented in the author PhD dissertation [2].2. Notation and terminology
Throughout the paper, N will denote the set of natural numbers { , , . . . } and N := N ∪ { } .By a projection on the complex separable Hilbert space ℓ we always mean a continuousorthogonal (i.e. self-adjoint) projection.By e k we denote the vector in C N whose k -th coordinate equals 1 and the others equal 0.By a Fr´echet space we mean a complete metrizable locally convex space over C (we will notuse locally convex spaces over R ). A Fr´echet algebra is a Fr´echet space which is an algebra withcontinuous multiplication. A
Fr´echet ∗ -algebra is a Fr´echet algebra with continuous involution.For locally convex spaces E, F , we denote by L ( E, F ) the space of all continuous linearoperators from E to F . To shorten notation, we write L ( E ) instead of L ( E, E ).We use the standard notation and terminology. All the notions from functional analysis areexplained in [4] or [12] and those from topological algebras in [9] or [20].3.
PreliminariesThe space s and its dual . We recall that the space of rapidly decreasing sequences is theFr´echet space s := (cid:26) ξ = ( ξ j ) j ∈ N ∈ C N : | ξ | q := (cid:18) ∞ X j =1 | ξ j | j q (cid:19) / < ∞ for all q ∈ N (cid:27) with the topology corresponding to the system ( | · | q ) q ∈ N of norms. We may identify the strongdual of s (i.e. the space of all continuous linear functionals on s with the topology of uniformconvergence on bounded subsets of s , see e.g. [12, Def. on p. 267]) with the space of slowlyincreasing sequences s ′ := (cid:26) ξ = ( ξ j ) j ∈ N ∈ C N : | ξ | ′ q := (cid:18) ∞ X j =1 | ξ j | j − q (cid:19) / < ∞ for some q ∈ N (cid:27) OMMUTATIVE SUBALGEBRAS OF THE ALGEBRA OF SMOOTH OPERATORS equipped with the inductive limit topology given by the system ( | · | ′ q ) q ∈ N of norms (note thatfor a fixed q , | · | ′ q is defined only on a subspace of s ′ ). More precisely, every η ∈ s ′ correspondsto the continuous linear functional on s : ξ
7→ h ξ, η i := ∞ X j =1 ξ j η j (note the conjugation on the second variable). These functionals are continuous, because, bythe Cauchy-Schwartz inequality, for all q ∈ N , ξ ∈ s and η ∈ s ′ we have |h ξ, η i| ≤ | ξ | q | η | ′ q . Conversely, one can show that for each continuous linear functional y on s there is η ∈ s ′ suchthat y = h· , η i .Similarly, we identify each ξ ∈ s with the continuous linear functional on s ′ : η
7→ h η, ξ i := ∞ X j =1 η j ξ j . In particular, for each continuous linear functional y on s ′ there is ξ ∈ s such that y = h· , ξ i .We emphasize that the ”scalar product” h· , ·i is well-defined on s × s ′ ∪ s ′ × s and, of course,on ℓ × ℓ . Property (DN) for the space s . Closed subspaces of the space s can be characterized bythe so-called property (DN). Definition 3.1.
A Fr´echet space ( X, ( || · || q ) q ∈ N ) has the property (DN) (see [12, Def. on p.359]) if there is a continuous norm || · || on X such that for all q ∈ N there is r ∈ N and C > || x || q ≤ C || x || || x || r for all x ∈ X . The norm || · || is called a dominating norm .Vogt (see [19] and [12, Ch. 31]) proved that a Fr´echet space is isomorphic to a closed subspaceof s if and only if it is nuclear and it has the property (DN).The (DN) condition for the space s reads as follows (see [12, Lemma 29.2(3)] and its proof). Proposition 3.2.
For every p ∈ N and ξ ∈ s we have | ξ | p ≤ || ξ || ℓ | ξ | p . In particular, the norm || · || ℓ is a dominating norm on s . The algebra L ( s ′ , s ). It is a simple matter to show that L ( s ′ , s ) with the topology of uniformconvergence on bounded sets in s ′ is a Fr´echet space. It is isomorphic to s b ⊗ s , the completedtensor product of s (see [11, § s being nuclear, there is only one tensortopology), and thus L ( s ′ , s ) ∼ = s as Fr´echet spaces (see e.g. [12, Lemma 31.1]). Moreover, it iseasily seen that ( || · || q ) q ∈ N , || x || q := sup | ξ | ′ q ≤ | xξ | q , is a fundamental sequence of norms on L ( s ′ , s ).Let us introduce multiplication and involution on L ( s ′ , s ). First observe that s is a densesubspace of ℓ , ℓ is a dense subspace of s ′ , and, moreover, the embedding maps j : s ֒ → ℓ , j : ℓ ֒ → s ′ are continuous. Hence, ι : L ( s ′ , s ) ֒ → L ( ℓ ) , ι ( x ) := j ◦ x ◦ j , is a well-defined (continuous) embedding of L ( s ′ , s ) into the C ∗ -algebra L ( ℓ ), and thus it isnatural to define a multiplication on L ( s ′ , s ) by xy := ι − ( ι ( x ) ◦ ι ( y )) , TOMASZ CIAŚ i.e. xy = x ◦ j ◦ y, where j := j ◦ j : s ֒ → s ′ . Similarly, an involution on L ( s ′ , s ) is defined by x ∗ := ι − ( ι ( x ) ∗ ) , where ι ( x ) ∗ is the hermitian adjoint of ι ( x ). One can show that these definitions are correct, i.e. ι ( x ) ◦ ι ( y ) , ι ( x ) ∗ ∈ ι ( L ( s ′ , s )) for all x, y ∈ L ( s ′ , s ) (see also [3, p. 148]).From now on, we will identify x ∈ L ( s ′ , s ) and ι ( x ) ∈ L ( ℓ ) (we omit ι in the notation).A Fr´echet algebra E is called locally m -convex if E has a fundamental system of submulti-plicative seminorms. It is well-known that L ( s ′ , s ) is locally m -convex (see e.g. [13, Lemma2.2]), and moreover, the norms || · || q are submultiplicative (see [3, Prop. 2.5]). This showssimultaneously that the multiplication introduced above is separately continuous, and thus, by[20, Th. 1.5], it is jointly continuous. Moreover, by [9, Cor. 16.7], the involution on L ( s ′ , s ) iscontinuous.We may summarize this paragraph by saying that L ( s ′ , s ) is a noncommutative ∗ -subalgebraof the C ∗ -algebra L ( ℓ ) which is (with its natural topology) a locally m -convex Fr´echet ∗ -algebraisomorphic as a Fr´echet space to s . 4. K¨othe algebras
In this section we collect and prove some results on K¨othe algebras which are known forspecialists but probably never published.
Definition 4.1.
A matrix A = ( a j,q ) j ∈ N ,q ∈ N of non-negative numbers such that(i) for each j ∈ N there is q ∈ N such that a j,q > a j,q ≤ a j,q +1 for j ∈ N and q ∈ N is called a K¨othe matrix .For 1 ≤ p < ∞ and a K¨othe matrix A we define the K¨othe space λ p ( A ) := (cid:26) ξ = ( ξ j ) j ∈ N ∈ C N : | ξ | p,q := (cid:18) ∞ X j =1 | ξ j | p a j,q | p (cid:19) /p < ∞ for all q ∈ N (cid:27) and for p = ∞ λ ∞ ( A ) := (cid:26) ξ = ( ξ j ) j ∈ N ∈ C N : | ξ | ∞ ,q := sup j ∈ N | ξ j | a j,q < ∞ for all q ∈ N (cid:27) with the locally convex topology given by the seminorms ( | · | p,q ) q ∈ N (see e.g. [12, Def. p. 326]).Sometimes, for simplicity, we will write λ ∞ ( a j,q ) (i.e. only the entries of the matix) insteadof λ ∞ ( A ).It is well-known (see [12, Lemma 27.1]) that the spaces λ p ( A ) are Fr´echet spaces and sometimesthey are Fr´echet ∗ -algebras with pointwise multiplication and conjugation (e.g. if a j,q ≥ j ∈ N and q ∈ N , see also [14, Prop. 3.1]); in that case they are called K¨othe algebras .Clearly, s is the K¨othe space λ ( A ) for A = ( j q ) j ∈ N ,q ∈ N and it is a Fr´echet ∗ -algebra. More-over, since the matrix A satisfies the so-called Grothendieck-Pietsch condition (see e.g. [12,Prop. 28.16(6)]), s is nuclear, and thus it has also other K¨othe space representations (see again[12, Prop. 28.16 & Ex. 29.4(1)]), i.e. for all 1 ≤ p ≤ ∞ , s = λ p ( A ) as Fr´echet spaces.We use ℓ -norms in the definition of s to clarify our ideas, for example we have | ξ | = || ξ || ℓ for ξ ∈ s and | η | ′ = || η || ℓ for η ∈ ℓ . However, in some situations the supremum norms | · | ∞ ,q (as they are relatively easy to compute) or the ℓ -norms will be more convenient. Proposition 4.2.
Let A = ( a j,q ) j ∈ N ,q ∈ N , B = ( b j,q ) j ∈ N ,q ∈ N be K¨othe matrices and for abijection σ : N → N let A σ := ( a σ ( j ) ,q ) j ∈ N ,q ∈ N . Assume that λ ∞ ( A ) and λ ∞ ( B ) are Fr´echet ∗ -algebras. Then the following assertions are equivalent: OMMUTATIVE SUBALGEBRAS OF THE ALGEBRA OF SMOOTH OPERATORS (i) λ ∞ ( A ) ∼ = λ ∞ ( B ) as Fr´echet ∗ -algebras; (ii) there is a bijection σ : N → N such that λ ∞ ( A σ ) = λ ∞ ( B ) as Fr´echet ∗ -algebras; (iii) there is a bijection σ : N → N such that λ ∞ ( A σ ) = λ ∞ ( B ) as sets; (iv) there is a bijection σ : N → N such that ( α ) ∀ q ∈ N ∃ r ∈ N ∃ C > ∀ j ∈ N a σ ( j ) ,q ≤ Cb j,r , ( β ) ∀ r ′ ∈ N ∃ q ′ ∈ N ∃ C ′ > ∀ j ∈ N b j,r ′ ≤ C ′ a σ ( j ) ,q ′ . Proof. (i) ⇒ (ii) Assume that there is an isomorphism Φ : λ ∞ ( A ) → λ ∞ ( B ) of Fr´echet ∗ -algebras. Clearly, if ξ = ξ , then Φ( ξ ) = Φ( ξ ) = (Φ( ξ )) , and the same is true for Φ − , i.e. Φmaps the idempotents of λ ∞ ( A ) onto the idempotents of λ ∞ ( B ). Hence for a fixed k ∈ N , thereis I ⊂ N such that Φ( e k ) = e I , where e I is a sequence which has 1 on an index set I ⊂ N and 0 otherwise. Suppose that | I | ≥ j ∈ I . Then e I = e j + e I \{ j } , where e j ∈ λ ∞ ( B ) and e I \{ j } = e I − e j ∈ λ ∞ ( B ). Therefore,there are nonempty subsets I j , I ′ j ⊂ N such that Φ( e I j ) = e j and Φ( e I ′ j ) = e I \{ j } . We have e I j e I ′ j = Φ − ( e j )Φ − ( e I \{ j } ) = Φ − ( e j e I \{ j } ) = 0 , and thus I j ∩ I ′ j = ∅ . Consequently,Φ( e k ) = e j + e I \{ j } = Φ( e I j ) + Φ( e I ′ j ) = Φ( e I j ∪ I ′ j ) , whence 1 = |{ k }| = | I j ∪ I ′ j | ≥
2, a contradiction. Hence Φ( e k ) = e n k for some n k ∈ N ,i.e. for the bijection σ : N → N defined by n σ ( k ) := k we have Φ( e σ ( k ) ) = e k . Therefore, aFr´echet ∗ -isomorphism Φ is given by ( ξ σ ( k ) ) k ∈ N ( ξ k ) k ∈ N for ( ξ σ ( k ) ) k ∈ N ∈ λ ∞ ( A ), and thus λ ∞ ( A σ ) = λ ∞ ( B ) as Fr´echet ∗ -algebras.(ii) ⇒ (iii) Obvious.(iii) ⇒ (iv) The proof follows from the observation that the identity map Id : λ ∞ ( A σ ) → λ ∞ ( B )is continuous (use the closed graph theorem).(iv) ⇒ (i) It is easy to see that Φ : λ ∞ ( A ) → λ ∞ ( B ) defined by e σ ( k ) e k is an isomorphismof Fr´echet ∗ -algebras. ✷ In the following proposition we characterize infinite-dimensional closed ∗ -subalgebras of nu-clear K¨othe algebras whose elements tends to zero (note that if a K¨othe space is contained in ℓ ∞ then it is a K¨othe algebra). Consequently, we obtain a characterization of closed ∗ -subalgebrasof s (Corollary 4.4). Proposition 4.3.
For
N ⊂ N let e N denote a sequence which has 1 on N and 0 otherwise. Let A = ( a j,q ) j ∈ N ,q ∈ N be a K¨othe matrix such that λ ∞ ( A ) is nuclear and λ ∞ ( A ) ⊂ c . Let E be aninfinite-dimensional closed ∗ -subalgebra of λ ∞ ( A ) . Then (i) there is a family {N k } k ∈ N of finite nonempty pairwise disjoint sets of natural numbers suchthat ( e N k ) k ∈ N is a Schauder basis of E ; (ii) E ∼ = λ ∞ (max j ∈N k a j,q ) as Fr´echet ∗ -algebras and the isomorphism is given by e N k e k for k ∈ N .Conversely, if {N k } k ∈ N is a family of finite nonempty pairwise disjoint sets of natural numbersand F is the closed ∗ -subalgebra of λ ∞ ( A ) generated by the set { e N k } k ∈ N , then (iii) ( e N k ) k ∈ N is a Schauder basis of F ; (iv) F ∼ = λ ∞ (max j ∈N k a j,q ) as Fr´echet ∗ -algebras and the isomorphism is given by e N k e k for k ∈ N . Proof.
In order to prove (i) and (ii) set N := { j ∈ N : ξ j = 0 for all ξ ∈ E } TOMASZ CIAŚ and define an equivalence relation ∼ on N \ N by i ∼ j ⇔ ξ i = ξ j for all ξ ∈ E. Since E is infinite-dimensional, our relation produces infinitely many equivalence classes N k , say N k := [min( N \ N ∪ . . . ∪ N k − )] / ∼ for k ∈ N .Fix κ ∈ N and take ξ ∈ E such that ξ j = 0 for j ∈ N κ . Denote η k := ξ j if j ∈ N k . Let M := { j ∈ N : | η j | = sup i ∈ N | η i |} . Assume we have already defined M , . . . , M l − . If there is j ∈ N \ {M ∪ . . . ∪ M l − } such that η j = 0 then we define M l := { j ∈ N : | η j | = sup {| η i | : i ∈ N \ M ∪ . . . ∪ M l − }} . Otherwise, denote I := { , . . . , l − } . If this procedure leads to infinite many sets M l then weset I := N . It is easily seen that for each l ∈ I there is I l ⊂ N such that M l = S k ∈I l N k . Byassumption ξ ∈ c , hence ( | η k | ) k ∈ N ∈ c as well, and thus each M l is a finite nonempty set.We first show that e M l ∈ E for l ∈ I . For l ∈ I fix m l ∈ M l . If I = { } , then ξ j = 0 for j / ∈ M , and e M = ξξ | η m | ∈ E . Let us consider the case |I| >
1. Since in nuclear Fr´echetspaces every basis is absolute (and thus unconditional), we have X l ∈I | η l | e M l = ∞ X j =1 | ξ j | e j = ξξ ∈ E, and, consequently, x n := X l ∈I (cid:18) | η l || η m | (cid:19) n e M l = (cid:18) ξξ | η m | (cid:19) n ∈ E for all n ∈ N . Then for q and n we get | x n − e M | ∞ ,q = (cid:12)(cid:12)(cid:12)(cid:12) ∞ X l ∈I (cid:18) | η l || η m | (cid:19) n e M l − e M (cid:12)(cid:12)(cid:12)(cid:12) ∞ ,q = (cid:12)(cid:12)(cid:12)(cid:12) X l ∈I\{ } (cid:18) | η l || η m | (cid:19) n e M l (cid:12)(cid:12)(cid:12)(cid:12) ∞ ,q ≤ X l ∈I\{ } (cid:18) | η l || η m | (cid:19) n | e M l | ∞ ,q ≤ | η m | (cid:18) | η m || η m | (cid:19) n − X l ∈I\{ } | η l | | e M l | ∞ ,q . Since ( e j ) j ∈ N is an absolute basis in λ ∞ ( A ), the above series is convergent. Note also that | η m | < | η m | . This shows that x n → e M in λ ∞ ( A ), and e M ∈ E . Assume that e M , . . . , e M l − ∈ E .If |I| = l − η m l = 0 and x ( l ) n := ξξ − ξξ P l − j =1 e M j | η m l | ! n ∈ E for n ∈ N . As above we show that x ( l ) n → e M l in λ ∞ ( A ), and thus e M l ∈ E . Proceeding byinduction, we prove that e M l ∈ E for l ∈ I .Now, we shall prove that ( e N k ) k ∈ N is a Schauder basis of E . Choose ι ∈ I such that κ ∈ I ι andfor k ∈ I ι let n k be an arbitrary element of N k . Then P k ∈I ι η n k e N k = ξe M ι ∈ E . Consequently,by [3, Lemma 4.1], e N κ ∈ E . Since κ was arbitrarily choosen, each e N k is in E and it is a simplematter to show that ( e N k ) k ∈ N is a Schauder basis of E .Moreover, | e N k | ∞ ,q = max j ∈N k a j,q hence, by [12, Cor. 28.13] and nuclearity, E is isomorphicas a Fr´echet space to λ ∞ (max j ∈N k a j,q ). The analysis of the proof of [12, Cor. 28.13] showsthat this isomorphism is given by e N k e k for k ∈ N , and thus it is also a Fr´echet ∗ -algebraisomorphism. OMMUTATIVE SUBALGEBRAS OF THE ALGEBRA OF SMOOTH OPERATORS Now, we prove (iii) and (iv). First note that every element of F is the limit of elements of theform P Mk =1 c k e N k , where M ∈ N and c , . . . , c M ∈ C . Therefore, if ξ ∈ F , then ξ i = ξ j for k ∈ N and i, j ∈ N k . This shows that each ξ ∈ F has the unique series representation ξ = P ∞ k =1 ξ n k e N k ,where ( n k ) k ∈ N is an arbitrarily choosen sequence such that n k ∈ N k for k ∈ N . Since the seriesis absolutely convergent, ( e N k ) k ∈ N is a Schauder basis of F . Statement (iv) follows by the samemethod as in (ii). ✷ Corollary 4.4.
Every infinite-dimensional closed ∗ -subalgebra of s is isomorphic as a Fr´echet ∗ -algebra to λ ∞ ( n qk ) for some strictly increasing sequence ( n k ) k ∈ N of natural numbers. Conversely,if ( n k ) k ∈ N is a strictly increasing sequence of natural numbers, then λ ∞ ( n qk ) is isomorphic as aFr´echet ∗ -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 K¨othe matrix ( j q ) j ∈ N ,q ∈ N . Let {N k } k ∈ N be a familyof finite nonempty pairwise disjoint sets of natural numbers. We have(1) max j ∈N k j q = (max { j : j ∈ N k } ) q for all q ∈ N and k ∈ N . Let σ : N → N be the bijection for which (max { j : j ∈ N σ ( k ) } ) k ∈ N is(strictly) increasing and let n k := max { j : j ∈ N σ ( k ) } for k ∈ N . Then, by Proposition 4.2, λ ∞ (cid:18) max j ∈N k j q (cid:19) ∼ = λ ∞ ( n qk )as Fr´echet ∗ -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 iscomplemented in s . Otherwise, by Proposition 4.3(i), E is a closed linear span of the set { e N k } k ∈ N for some family {N k } k ∈ N of finite nonempty pairwise disjoint sets of natural numbers.Define π : s → E by ( πx ) j := (cid:26) x n k for j ∈ N σ ( k ) n k ) k ∈ N and σ are as above. From (1) we have for every q ∈ N | πx | ∞ ,q = sup j ∈ N | ( πx ) j | j q ≤ sup k ∈ N | x n k | max j ∈N σ ( k ) j q = sup k ∈ N | x n k | (max { j : j ∈ N k } ) q = sup k ∈ N | x n k | n qk ≤ sup j ∈ N | x j | j q = | x | ∞ ,q , and thus π is well-defined and continuous. Since π is a projection, our proof is complete. ✷ Representations of closed commutative ∗ -subalgebras of L ( s ′ , s ) by K¨othealgebras The aim of this section is to describe all closed commutative ∗ -subalgebras of L ( s ′ , s ) as K¨othealgebras λ ∞ ( A ) for matrices A determined by orthonormal sequences whose elements belong tothe space s (Theorem 5.3 and Corollaries 5.4 and 5.5). For the convenience of the reader, wequote two results from [3] (with minor modifications which do not require extra arguments).For a subset Z of L ( s ′ , s ) we will denote by alg( Z ) (lin( Z ), resp.) the closed ∗ -subalgebra of L ( s ′ , s ) generated by Z (the closed linear span of Z , resp.).By [3, Lemma 4.4], every closed commutative ∗ -subalgebra E of L ( s ′ , s ) admits a specialSchauder 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 Schauderbasis of E . TOMASZ CIAŚ
Proposition 5.1. [3, Prop. 4.7]
Every sequence { P k } k ∈N ⊂ L ( s ′ , s ) of nonzero pairwise or-thogonal projections is the canonical Schauder basis of the algebra alg( { P k } k ∈N ) . In particu-lar, { P k } k ∈N is a basic sequence in L ( s ′ , s ) , i.e. it is a Schauder basis of the Fr´echet space lin( { P k } k ∈N ) . Theorem 5.2. [3, Th. 4.8]
Let E be a closed commutative infinite-dimensional ∗ -subalgebra of L ( s ′ , s ) and let { P k } k ∈ N be the canonical Schauder basis of E . Then E = alg( { P k } k ∈ N ) ∼ = λ ∞ ( || P k || q ) as Fr´echet ∗ -algebras and the isomorphism is given by P k e k for k ∈ N . Please note that a projection P ∈ L ( s ′ , s ) if and only if it is of the form P ξ = X k ∈ I h ξ, f k i f k for some finite set I and an orthonormal sequence ( f k ) k ∈ I ⊂ s .We will also use the identity(2) λ ∞ ( ||h· , f k i f k || q ) = λ ∞ ( | f k | q )which holds for every orthonormal sequence ( f k ) k ∈ N ⊂ 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 L ( s ′ , s ) is isomorphic as a Fr´echet ∗ -algebra to some closed ∗ -subalgebra of the algebra λ ∞ ( | f k | q ) for some orthonormal sequence ( f k ) k ∈ N ⊂ s . More precisely, if E is an infinite-dimensional closed commutative ∗ -subalgebraof L ( s ′ , s ) and ( P j ∈N k h· , f j i f j ) k ∈ N is its canonical Schauder basis for some family of finitepairwise disjoint subsets ( N k ) k ∈ N of natural numbers and an orthonormal sequence ( f j ) j ∈ N ⊂ s ,then E is isomorphic as a Fr´echet ∗ -algebra to the closed ∗ -subalgebra of λ ∞ ( | f k | q ) generated by { P j ∈N k e j } k ∈ N and the isomorphism is given by P j ∈N k h· , f j i f j P j ∈N k e j for k ∈ N .Conversely, if ( f k ) k ∈ N ⊂ s is an orthonormal sequence, then every closed ∗ -subalgebra of λ ∞ ( | f k | q ) is isomorphic as a Fr´echet ∗ -algebra to some closed commutative ∗ -subalgebra of L ( s ′ , s ) . Proof.
By Theorem 5.2, E = alg (cid:18)(cid:26) P j ∈N k h· , f j i f j (cid:27) k ∈ N (cid:19) for ( N k ) k ∈ N and ( f j ) j ∈ N ⊂ s asin the statement. Let F be the closed ∗ -subalgebra of λ ∞ ( | f k | q ) generated by { P j ∈N k e j } k ∈ N .Define Φ : alg( {h· , f k i f k } k ∈N ) → λ ∞ ( | f k | q )by h· , f k i f k e k , where N := S k ∈ N N k . By Proposition 5.1, {h· , f k i f k } k ∈N is the canonicalSchauder basis of alg( {h· , f k i f k } k ∈N ), and thus Theroem 5.2 and (2) imply that Φ is a Fr´echet ∗ -algebra isomorphism. Hence, ( P j ∈N k e j ) k ∈ N = (Φ( P j ∈N k h· , f j i f j )) k ∈ N is a Schauder basis ofΦ( E ) and Φ( E ) is a closed ∗ -subalgebra of λ ∞ ( | f k | q ). Therefore,Φ( E ) = lin (cid:18)(cid:26) X j ∈N k e j (cid:27) k ∈ N (cid:19) ⊂ F ⊂ Φ( E ) , whence Φ( E ) = F . In consequence Φ | E is a Fr´echet ∗ -algebra isomorphism of E and F , whichcompletes the proof of the first statement.If now ( f k ) k ∈ N ⊂ s is an arbitrary orthonormal sequence then, according to Proposition 5.1,Theorem 5.2 and indentity (2), λ ∞ ( | f k | q ) ∼ = alg( {h· , f k i f k } k ∈ N ) as Fr´echet ∗ -algebras. Conse-quently, every closed ∗ -subalgebra of λ ∞ ( | f k | q ) is isomorphic as a Fr´echet ∗ -algebra to someclosed ∗ -subalgebra of alg( {h· , f k i f k } k ∈ N ). ✷ The following characterization of infinite-dimensional closed commutative ∗ -subalgebras of L ( s ′ , s ) is a straightforward consequence of Proposition 4.3 and Theorem 5.3. OMMUTATIVE SUBALGEBRAS OF THE ALGEBRA OF SMOOTH OPERATORS Corollary 5.4.
Every infinite-dimensional closed commutative ∗ -subalgebra of L ( s ′ , s ) is iso-morphic as a Fr´echet ∗ -algebra to the algebra λ ∞ (max j ∈N k | f j | q ) for some orthonormal sequence ( f k ) k ∈ N ⊂ s and some family {N k } k ∈ N of finite nonempty pairwise disjoint sets of natural num-bers. In fact, if E is an infinite-dimensional closed commutative ∗ -subalgebra of L ( s ′ , s ) and ( P j ∈N k h· , f j i f j ) k ∈ N is its canonical Schauder basis, then E ∼ = λ ∞ (cid:18) max j ∈N k | f j | q (cid:19) as Fr´echet ∗ -algebras and the isomorphism is given by P j ∈N k h· , f j i f j e k for k ∈ N .Conversely, if ( f k ) k ∈ N ⊂ s is an orthonormal sequence and {N k } k ∈ N is a family of finitenonempty pairwise disjoint sets of natural numbers, then λ ∞ (max j ∈N k | f j | q ) is isomorphic as aFr´echet ∗ -algebra to some infinite-dimensional closed commutative ∗ -subalgebra of L ( s ′ , s ) . At the end of this section we consider the case of maximal commutative subalgebras of L ( s ′ , s ).A closed commutative ∗ -subalgebra of L ( s ′ , s ) is said to be maximal commutative if it is notproperly contained in any larger closed commutative ∗ -subalgebra of L ( s ′ , s ).We say that an orthonormal system ( f k ) k ∈ N of ℓ is s -complete , if every f k belongs to s andfor every ξ ∈ s the following implication holds: if h ξ, f k i = 0 for every k ∈ N , then ξ = 0.A sequence { P k } k ∈ N of nonzero pairwise orthogonal projections belonging to L ( s ′ , s ) is called L ( s ′ , s )- complete if there is no nonzero projection P belonging to L ( s ′ , s ) such that P k P = 0 forevery k ∈ N .One can easily show that an orthonormal system ( f k ) k ∈ N is s -complete if and only if thesequence of projections ( h· , f k i f k ) k ∈ N is L ( s ′ , s )-complete. Hence, by [3, Th. 4.10], closed com-mutative ∗ -subalgebra E of L ( s ′ , s ) is maximal commutative if and only if there is an s -completesequence ( f k ) k ∈ N such that ( h· , f k i f k ) k ∈ N is the canonical Schauder basis of E . Combining thiswith Corollary 5.4, we obtain the first statement of the following Corollary. Corollary 5.5.
Every closed maximal commutative ∗ -subalgebra of L ( s ′ , s ) is isomorphic as aFr´echet ∗ -algebra to the algebra λ ∞ ( | f k | q ) for some s -complete orthonormal sequence ( f k ) k ∈ N .More precisely, if E is a closed maximal commutative ∗ -subalgebra of L ( s ′ , s ) with the canonicalSchauder basis ( h· , f k i f k ) k ∈ N , then E ∼ = λ ∞ ( | f k | q ) as Fr´echet ∗ -algebras and the isomorphism is given by h· , f k i f k e k for k ∈ N .Conversely, if ( f k ) k ∈ N is an s -complete orthonormal sequence, then λ ∞ ( | f k | q ) is isomorphicas a Fr´echet ∗ -algebra to some closed maximal commutative ∗ -subalgebra of L ( s ′ , s ) . Proof.
In order to prove the second statement, take an arbitrary s -complete orthonormalsequence ( f k ) k ∈ N . By Proposition 5.1 and the remark above our Corollary, alg( {h· , f k i f k } k ∈ N ) ismaximal commutative and from the first statement it follows that it is isomorphic as a Fr´echet ∗ -algebra to λ ∞ ( | f k | q ). ✷ It is also worth pointing out the following result.
Proposition 5.6.
Every closed commutative ∗ -subalgebra of L ( s ′ , s ) is contained in some max-imal commutative ∗ -subalgebra of L ( s ′ , s ) . Proof.
Let E be a closed commutative ∗ -subalgebra of L ( s ′ , s ). Clearly, X := { e E : e E commutative ∗ -subalgebra of L ( s ′ , s ) and E ⊂ e E } with the inclusion relation is a partially ordered set. Consider a chain C in X and let E C := S F ∈C F . It is easy to check that E C ∈ X , and, of course, E C is an upper bound of C . Hence, bythe Kuratowski-Zorn lemma, X has a maximal element; let us call it M . By the continuity ofthe algebra operations, M L ( s ′ ,s ) is a closed commutative ∗ -subalgebra of L ( s ′ , s ), hence from the TOMASZ CIAŚ maximality of M , we have M = M L ( s ′ ,s ) , i.e. M is a (closed) maximal commutative ∗ -subalgebraof L ( s ′ , s ) containing E . ✷ Closed commutative ∗ -subalgebras of L ( s ′ , s ) with the property ( Ω ) In the present section we prove that a closed commutative ∗ -subalgebra of L ( s ′ , s ) is isomorphicas a Fr´echet ∗ -algebra to some closed ∗ -subalgebra of s if and only if it is isomorphic as a Fr´echetspace to some complemented subspace of s (Theorem 6.2), i.e. if it has the so-called property(Ω) (see Definition 6.1 below). We also give an example of a closed commutative ∗ -subalgebraof L ( s ′ , s ) which is not isomorphic to any closed ∗ -subalgebra of s (Theorem 6.10). Definition 6.1.
A Fr´echet space E with a fundamental sequence ( || · || q ) q ∈ N of seminorms hasthe property (Ω) if the following condition holds: ∀ p ∃ q ∀ r ∃ θ ∈ (0 , ∃ C > ∀ y ∈ E ′ || y || ′ q ≤ C || y || ′ − θp || y || ′ θr , where E ′ is the topological dual of E and || y || ′ p := sup {| y ( x ) | : || x || p ≤ } .The property (Ω) (together with the property (DN)) plays a crucial role in the theory ofnuclear Fr´echet spaces (for details, see [12, Ch. 29]).Recall that a subspace F of a Fr´echet space E is called complemented (in E ) if there isa continuous projection π : E → E with im π = F . Since every subspace of L ( s ′ , s ) has theproperty (DN) (and, by [3, Prop. 3.2], the norm || · || ℓ → ℓ is already a dominating norm),[12, Prop. 31.7] implies that a closed ∗ -subalgebra of L ( s ′ , s ) is isomorphic to a complementedsubspace of s if and only if it has the property (Ω). The class of complemented subspaces of s isstill 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 asimple description (see Corollary 4.4). The following theorem implies that, when restricting tothe family of closed commutative ∗ -subalgebras of L ( s ′ , s ), these two classes of Fr´echet spacescoincide. Theorem 6.2.
Let E be an infinite-dimensional closed commutative ∗ -subalgebra of L ( s ′ , s ) and let ( P j ∈N k h· , f j i f j ) k ∈ N be its canonical Schauder basis. Then the following assertions areequivalent: (i) E is isomorphic as a Fr´echet ∗ -algebra to some closed ∗ -subalgebra of s ; (ii) E is isomorphic as a Fr´echet space to some complemented subspace of s ; (iii) E has the property (Ω) ; (iv) ∃ p ∀ q ∃ r ∃ C > ∀ k max j ∈N k | f j | q ≤ C max j ∈N k | f j | rp . In order to prove Theorem 6.2, we will need Lemmas 6.3, 6.4 and Propositions 6.5, 6.7.The following result is a consequence of nuclearity of closed commutative ∗ -subalgebras of L ( s ′ , s ). Lemma 6.3.
Let ( f k ) k ∈ N ⊂ s be an orthonormal sequence and let ( N k ) k ∈ N be a family of finitepairwise disjoint subsets of natural numbers. For r ∈ N let σ r : N → N be a bijection such thatthe sequence (max j ∈N σr ( k ) | f j | r ) k ∈ N is non-decreasing. Then there is r ∈ N such that lim k →∞ k max j ∈N σr ( k ) | f j | r = 0 for all r ≥ r .Proof. By Corollary 5.4, λ ∞ (max j ∈N k | f j | q ) is a nuclear space. Hence, by the Grothendieck-Pietsch theorem (see e.g. [12, Th. 28.15]), for every q ∈ N there is r ∈ N such that ∞ X k =1 max j ∈N k | f j | q max j ∈N k | f j | r < ∞ . OMMUTATIVE SUBALGEBRAS OF THE ALGEBRA OF SMOOTH OPERATORS In particular (for q = 0), there is r such that for r ≥ r we have ∞ X k =1 j ∈N σr ( k ) | f j | r = ∞ X k =1 j ∈N k | f j | r < ∞ . Since the sequence (max j ∈N σr ( k ) | f j | r ) k ∈ N is non-decreasing, the conclusion follows from theelementary theory of number series. (cid:3) Lemma 6.4.
Let ( a k ) k ∈ N ⊂ [1 , ∞ ) be a non-decreasing sequence such that a k ≥ k for k bigenough. Then there exist a strictly increasing sequence ( b k ) k ∈ N of natural numbers and C > such that C a k ≤ b k ≤ Ca k for every k ∈ N . Proof.
Let k ∈ N be such that a k ≥ k for k > k and choose C ∈ N so that1 C a k ≤ k ≤ Ca k for k ∈ N := { , . . . , k } . Denote also N := { k ∈ N : a k = a k +1 } and, recursively, N j +1 := { k ∈ N : a k = a max N j +1 } . Clearly, N j are finite, pairwise disjoint, S j ∈ N N j = N and k < l for k ∈ N j , l ∈ N j +1 .Let b k := k for k ∈ N and let b m j + l − := C ⌈ max { a m j − , a m j }⌉ + l for j ∈ N and 1 ≤ l ≤ |N j | , where m j := min N j and ⌈ x ⌉ := min { n ∈ Z : n ≥ x } stands forthe ceiling of x ∈ R . We will show inductively that ( b k ) k ∈ N is a strictly increasing sequence ofnatural numbers such that(3) 1 C a k ≤ b k ≤ Ca k for every k ∈ N .Clearly, the condition (3) holds for k ∈ N . Assume that ( b k ) k ∈N ∪ ... ∪N j is a strictly increasingsequence of natural numbers for which the condition (3) holds. For simplicity, denote m :=min N j +1 . By the inductive assumption, we obtain b m − ≤ Ca m − , hence b m − b m − ≥ C ⌈ max { a m − , a m }⌉ + 1 − Ca m − ≥ Ca m − + 1 − Ca m − ≥ b m − < b m , and, clearly, b m < b m +1 < . . . < b max N j +1 .Fix 1 ≤ l ≤ | N j +1 | . We have b m + l − ≥ Ca m = Ca m + l − ≥ C a m + l − so the first inequality in (3) holds for k ∈ N j +1 . Next, by assumption, we get(4) a m + l − ≥ m + l − , whence(5) l ≤ a m − l +1 − m + 1 . Consider two cases. If a m ≥ a m − , then, from (5) b m − l +1 = C ⌈ a m ⌉ + l = C ⌈ a m + l − ⌉ + l ≤ Ca m + l − + a m + l − − m + 1 ≤ (2 C + 1) a m + l − ≤ Ca m + l − , where the last inequality holds because C ≥ a m − l +1 ≥ m + l − ≥ m ≥ k + 1) ≥ . TOMASZ CIAŚ
Finally, if a m − > a m , then, from (4), we obtain (note that, by the definition of N j and N j +1 ,we have a m − < a m ) b m − l +1 = C ⌈ a m − ⌉ + l ≤ C ⌈ ( a m − ⌉ + l = C ⌈ a m − a m + 1 ⌉ + l ≤ C ( a m − a m + 2) + l ≤ Ca m − Ca m + 2 C + Cl = Ca m + l − − C (2 a m + l − − − l ) ≤ Ca m + l − − C (4( m + l − − − l )= Ca m + l − − C (4 m + 3 l − ≤ Ca m + l − . Hence we have shown that the second inequality in (3) holds for k ∈ N j +1 , and the proof iscomplete. ✷ Proposition 6.5.
Let E be an infinite-dimensional closed commutative ∗ -subalgebra of L ( s ′ , s ) and let ( P j ∈N k h· , f j i f j ) k ∈ N be its canonical Schauder basis. Moreover, let ( n k ) k ∈ N be a strictlyincreasing sequence of natural numbers and let F be the closed ∗ -subalgebra of s generated by { e n k } k ∈ N . Then the following assertions are equivalent: (i) E is isomorphic to F as a Fr´echet ∗ -algebra; (ii) λ ∞ (max j ∈N k | f j | q ) ∼ = λ ∞ ( n qk ) as Fr´echet ∗ -algebras; (iii) there is a bijection σ : N → N such that λ ∞ (max j ∈N σ ( k ) | f j | q ) = λ ∞ ( n qk ) as Fr´echet ∗ -algebras; (iv) there is a bijection σ : N → N such that λ ∞ (max j ∈N σ ( k ) | f j | q ) = λ ∞ ( n qk ) as sets; (v) there is a bijection σ : N → N such that ( α ) ∀ q ∈ N ∃ r ∈ N ∃ C > ∀ k ∈ N max j ∈N σ ( k ) | f j | q ≤ Cn rk , ( β ) ∀ r ′ ∈ N ∃ q ′ ∈ N ∃ C ′ > ∀ k ∈ N n r ′ k ≤ C ′ max j ∈N σ ( k ) | f j | q ′ . Proof.
This is an immediate consequence of Proposition 4.2 and Corollary 5.4. ✷ Remark . In view of Corollary 4.4, every closed ∗ -subalgebra of s is isomorphic as a Fr´echet ∗ -algebra to λ ∞ ( n qk ) (i.e. the closed ∗ -subalgebra of s generated by { e n k } k ∈ N ) for some strictlyincreasing sequence ( n k ) k ∈ N ⊂ N , hence Proposition 6.5 characterizes closed commutative ∗ -subalgebras of L ( s ′ , s ) which are isomorphic as Fr´echet ∗ -algebras to some ∗ -subalgebra of s .The property (DN) for the space s gives us the following inequality. Proposition 6.7.
For every p, r ∈ N there is q ∈ N such that for all ξ ∈ s with || ξ || ℓ = 1 thefollowing inequality holds | ξ | rp ≤ | ξ | q . Proof.
Take p, r ∈ N and let j ∈ N be such that r ≤ j . Applying iteratively ( j -times) theinequality from Proposition 3.2 to ξ ∈ s with || ξ || ℓ = 1 we get | ξ | rp ≤ | ξ | j p ≤ | ξ | j p , and thus the required inequality holds for q = 2 j p . (cid:3) Now we are ready to prove Theorem 6.2.
Proof of Theorem 6.2. (i) ⇒ (ii): By Corollary 4.4, each closed ∗ -subalgebra of s is a comple-mented subspace of s . OMMUTATIVE SUBALGEBRAS OF THE ALGEBRA OF SMOOTH OPERATORS (ii) ⇔ (iii): See e.g. [12, Prop. 31.7].(iii) ⇒ (iv): By Corollary 5.4 and nuclearity (see e.g. [12, Prop. 28.16]), E ∼ = λ ∞ (cid:18) max j ∈N k | f j | q (cid:19) = λ (cid:18) max j ∈N k | f j | q (cid:19) as Fr´echet ∗ -algebras. Hence, by [18, Prop. 5.3], the property (Ω) yields ∀ l ∃ m ∀ n ∃ t ∃ C > ∀ k max j ∈N k | f j | tl max j ∈N k | f j | n ≤ C max j ∈N k | f j | t +1 m . In particular, taking l = 0, we get (iv).(iv) ⇒ (i): Take p from the condition (iv). By Lemma 6.3(ii), there is p ≥ p and a bijection σ : N → N such that (max j ∈N σ ( k ) | f j | p ) k ∈ N is non-decreasing and lim k →∞ k max j ∈N σ ( k ) | f j | p = 0.Consequently, for k big enough max j ∈N σ ( k ) | f j | p ≥ k, and therefore, by Lemma 6.4, there is a strictly increasing sequence ( n k ) k ∈ N ⊂ N and C > C max j ∈N σ ( k ) | f j | p ≤ n k ≤ C max j ∈N σ ( k ) | f j | p for every k ∈ N . Now, by the conditions (iv) and (6), we get that for all q there is r and C := CC r such that max j ∈N σ ( k ) | f j | q ≤ C max j ∈N σ ( k ) | f j | rp ≤ C n rk for all k ∈ N , so the condition ( α ) from Proposition 6.5(v) holds. Finally, by (6) and Proposition6.7 we obtain that for all r ′ there is q ′ and C := C r ′ such that n r ′ k ≤ C max j ∈N σ ( k ) | f j | r ′ p ≤ C max j ∈N σ ( k ) | f j | q ′ for every k ∈ N . Hence the condition ( β ) from Proposition 6.5(v) is satisfied, and therefore, byProposition 6.5, E is isomorphic as a Fr´echet ∗ -algebra to the closed ∗ -subalgebra of s generatedby { e n k } k ∈ N . ✷ Now, we shall give an example of some class of closed commutative ∗ -subalgebras of L ( s ′ , s )which are isomorphic to closed ∗ -subalgebras of s . Example 6.8.
Let H := [1]. We define recursively Hadamard matrices H n := (cid:20) H n − H n − H n − − H n − (cid:21) for n ∈ N . Then the matrices b H n := 2 − n H n are unitary, and thus their rows form an orthonor-mal system of 2 n vectors. Now fix an arbitrary sequence ( d n ) n ∈ N ⊂ N and define U := b H d . . . b H d . . . b H d ... ... . . . . Let f k denote the k -th row of the matrix U . Then ( f k ) k ∈ N is an orthonormal basis of ℓ andclearly each f k belongs to s . We will show that the closed (maximal) commutative ∗ -subalgebraalg( {h , · , f k i f k } k ∈ N ) of L ( s ′ , s ) is isomorphic to some closed ∗ -subalgebra of s . By Theorem 6.2,it is enough to prove that(7) ∃ p ∀ q ∃ r ∃ C > ∀ k | f k | ∞ ,q ≤ C | f k | r ∞ ,p . TOMASZ CIAŚ
Fix q ∈ N , k ∈ N and find n ∈ N such that 2 d + . . . + 2 d n − < k ≤ d + . . . + 2 d n . Then | f k | ∞ ,q | f k | q ∞ , = 2 − dn (2 d + . . . + 2 d n ) q − d n q (2 d + . . . + 2 d n ) q = 2 d n ( q − / (2 d + . . . + 2 d n ) − q ≤ p = C = 1 and r = 2 q .The next theorem solves in negative [3, Open Problem 4.13]. In contrast to the algebra s , whose all closed ∗ -subalgebras are complemented subspaces of s (Corollary 4.4), Theorems6.2 and 6.10 imply that there is a closed commutative ∗ -subalgebra of L ( s ′ , s ) which is notcomplemented in L ( s ′ , s ) (otherewise it would have the property (Ω), see [12, Prop. 31.7]). Inthe proof we will use the following identity. Lemma 6.9.
For every increasing sequence ( α j ) j ∈ N ⊂ (0 , ∞ ) and every p ∈ N we have sup j ∈ N α p − j +1 j · j − Y i =1 α i = p Y i =1 α i . Proof.
For j ≥ p + 1 we get α p − j +1 j · Q j − i =1 α i Q pi =1 α i = α p − j +1 j · j − Y i = p +1 α i = Q j − i = p +1 α i α j − p − j ≤ j ≤ p − α p − j +1 j · Q j − i =1 α i Q pi =1 α i = α p − j +1 j Q pi = j α i ≤ . Since α p − p +1 p · Q p − i =1 α i = Q pi =1 α i , the supremum is attained for j = p , and we are done. ✷ Theorem 6.10.
There is a closed commutative ∗ -subalgebra of L ( s ′ , s ) which is not isomorphicto any closed ∗ -subalgebra of s . Proof.
Let m k be the k -th prime number, N k, := m k , N k,j +1 := m N k,j k for j, k ∈ N . Denote a k, := c k and a k,j := c k Q j − i =1 N k,i N j − k,j for j ≥
2, where the sequence ( c k ) k ∈ N is choosen so that || ( a k,j ) j ∈ N || ℓ = 1, i.e. c k := (cid:18) ∞ X j =1 (cid:18) Q j − i =1 N k,i N j − k,j (cid:19) (cid:19) − / . The numbers c k are well-defined, because, by Lemma 6.9, ∞ X j =1 (cid:18) Q j − i =1 N k,i N j − k,j (cid:19) = ∞ X j =1 (cid:18) N − j +1 k,j · j − Y i =1 N k,i (cid:19) = ∞ X j =1 N k,j (cid:18) N − j +1 k,j · j − Y i =1 N k,i (cid:19) ≤ sup j ∈ N (cid:18) N − j +1 k,j · j − Y i =1 N k,i (cid:19) ∞ X j =1 N k,j = N k, ∞ X j =1 N k,j < N k, ∞ X j =1 j < ∞ . Finally, define an orthonormal sequence ( f k ) k ∈ N by f k := ∞ X j =1 a k,j e N k,j . OMMUTATIVE SUBALGEBRAS OF THE ALGEBRA OF SMOOTH OPERATORS We will show that alg( {h· , f k i f k } k ∈ N ) is a closed ∗ -subalgebra of L ( s ′ , s ) which is not isomorphicas an algebra to any closed ∗ -subalgebra of s . By Theorem 6.2 and nuclearity, it is enough toshow that each f k belongs to s and for every p, r ∈ N the following condition holdslim k →∞ | f k | ∞ ,p +1 | f k | r ∞ ,p = ∞ , where | ξ | ∞ ,q := sup j ∈ N | ξ j | j q .Note first that | f k | ∞ ,p = a k,p N pk,p . In fact, by Lemma 6.9, we get | f k | ∞ ,p = sup j ∈ N a k,j N pk,j = c k sup j ∈ N N pk,j · Q j − i =1 N k,i N j − k,j = c k sup j ∈ N N p − j +1 k,j · j − Y i =1 N k,i = c k p Y i =1 N k,i = c k N pk,p · Q p − i =1 N k,i N p − k,p = a k,p N pk,p . In particular, f k ∈ s for k ∈ N . Next, for j, k ∈ N , we have a k,j +1 N jk,j +1 a k,j = c k N jk,j +1 · Q ji =1 N k,i N jk,j +1 c k Q j − i =1 N k,i N j − k,j = Q ji =1 N k,i Q j − i =1 N k,i N j − k,j = N jk,j . Moreover, for every j, r ∈ N we get N k,j +1 N rk,j = m N k,j k N rk,j ≥ N k,j N rk,j −−−→ k →∞ ∞ , and clearly a k,j ≤ j, k ∈ N . Hence, for p, r ∈ N we obtain | f k | ∞ ,p +1 | f k | r ∞ ,p = a k,p +1 N p +1 k,p +1 a rk,p N prk,p = a k,p +1 N pk,p +1 a k,p · a r − k,p · N k,p +1 N prk,p = N pk,p · a r − k,p · N k,p +1 N prk,p ≥ N k,p +1 N prk,p −−−→ k →∞ ∞ , which is the desired conclusion. ✷ Acknowledgements.
I would like to thank Paweł Domański for his constant and generoussupport.
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