The Multiplier Algebra of the Noncommutative Schwartz Space

Home , Schwartz space

arXiv:2103.05352v1 [math.FA] 9 Mar 2021 smrhc(satopological a (as isomorphic with none prtr on operators unbounded h itreto”–waee ti losu oitouea introduce to us allows – consider is it we whatever – “intersection” The frpdydcesn sequences, decreasing rapidly of called e etscin and section) next see rs. ntespace the on (resp., tersection” sioopi a topological se a corresponding (as the isomorphic with is spaces distribution and function oad rn o DEC-2013/10/A/ST1/00091. no. grant Poland, h maximal the ieroeaoso h elkonShat space Schwartz well-known the on operators linear ocmuaieShat pc saspecific a is space Schwartz noncommutative onens rnil oko hssae–seT.45 Since 4.5. Th. see holds. – obviously space M Theorem this Open Hahn-Banach on Theorem, work Graph Principle Closed the Boundedness metrizable, nor Banach ther the called involution, space with algebra logical prtragba aual pern naayi,frexa for analysis, in appearing naturally algebras operator 1 stenm suggests, name the As 3 2 h algebra The pr topological and algebraic study to is paper this of aim The H UTPIRAGBAO H OCMUAIESCHWARTZ NONCOMMUTATIVE THE OF ALGEBRA MULTIPLIER THE Keywords: Acknowledgement. Classification: Subject Mathematics 2010 Λ( O n eoe by denoted and A ocmuaieShat space Schwartz noncommutative ∗ eso npriua hti snihra neither is it d rapidly that of particular space in Schwartz show classical We the with space Hilbert utpiragbacnb ena h largest the as seen be can algebra multiplier rv htcascltoso ucinlaayi,frexam principle, boundedness for uniform analysis, the functional or theorem of mapping tools open classical that prove Abstract. agba,adepcal 3,Pr .] oevr yProp by Moreover, I.2]. Part [33, especially and -algebras, := ) L L ( O ( S (Fréchet) x s MS ∗ ( ) ( = R agbawt domain with -algebra L ∩ )) edsrb h utpiragbao h ocmuaieSc noncommutative the of algebra multiplier the describe We x S h eerho ohatoshsbe upre yteNation the by supported been has authors both of research The L ∩ a edsrbdi aywy.Frisac,i a ese st as seen be can it instance, For ways. many in described be can i,j ( ′ s MS ( L m ) ′ R nta f o example, for of, instead ) i,j ( ( cne ler,(ocmuaie cwrzsae multip space, Schwartz (noncommutative) algebra, -convex s ftmee itiuin) tapasta ecnreplace can we that appears It distributions). tempered of ) S OAZCA N RYZO PISZCZEK KRZYSZTOF AND CIAŚ TOMASZ ℓ ∈ ), npriua ewl hwta,i pt ftefc that fact the of spite in that, show will we particular In . MS 2 ′ N ( R L ihdomain with ∗ ∈ agba otemti algebra matrix the to -algebra) ) where )), ( ∗ s C agba ote“intersection” the to -algebra) a eas iwda h ler f“utpir”frteso- the for “multipliers” of algebra the as viewed also be can ′ r h orsodn pcso otnoslna operato linear continuous of spaces corresponding the are ) N s 2 ′ : stesaeo lwyicesn eune frdefinitions (for sequences increasing slowly of space the is ∀ 1. N s S L n hsi a evee stelretagbaof algebra largest the as viewed be can it thus and , lokona the as known also , Introduction ( ∈ s S SPACE utpiragbao h ocmuaieSchwartz noncommutative the of algebra multiplier ( N rmr:4L0 61,4H5 eodr:4A3 46A11. 46A13, Secondary: 46H15. 46K10, 47L10, Primary: e h oko .Sh¨de 3]wihdeals which Schmüdgen [33] K. of book the see – Q R m 0 ∗ agbanor -algebra )(resp., )) agbao none prtr naseparable a on operators unbounded of -algebra 1 cne Fréchet-convex ∃ n L S ∈ ( ( s N R .Tealgebra The ). 0 fsot ail eraigfunctions decreasing rapidly smooth of ) sup : L pe otealgebra the to mple, l,tecoe rp hoe,the theorem, graph closed the ple, m ( cesn ucin stedomain. the as functions ecreasing unesae.Ti en that means This spaces. quence S i,j cne.O h te ad we hand, other the On -convex. ′ r tl available. still are ∈ ( aua nouinadti swhy is this and involution natural ler fsot operators smooth of algebra R N L ) steagbao continuous of algebra the is ))) | ∗ ( x ti oal ovxsaethe space convex locally a is it agbaioopi oseveral, to isomorphic -algebra s priso oeseictopo- specific some of operties pigTermadUniform and Theorem apping i,j ) L ∩ | max sto .,oragbais algebra our 4.3, osition MS ( wrzsae This space. hwartz s ′ ,where ), iragba PLS-space. algebra, lier lotrsott be to out turns also i j N n lCne fScience, of Center al L , j ( i N S n ′ s ( R stespace the is MS < ) h above the , ∞ S e“in- he ( R The . snei- is . )of )) MS rs. 2 TOMASZ CIAŚ AND KRZYSZTOF PISZCZEK continuous linear operators from ′(R) to (R), the algebra (s′,s) or to the algebra S S L N2 N N ∞ := (xi,j)i,j∈N C : N N0 sup xi,j i j < K { ∈ ∀ ∈ i,j∈N | | ∞} of rapidly decreasing matrices (see [11, Th. 1.1] for more representations). It is also isomorphic – topologically but not as an algebra – to the Fréchet space (R) (which explains its name). The noncommutative Schwartz space and the space (R) itselfS play a role in a number of fields, for example: structure theory of Fréchet spaces and splittS ing of short exact sequences (see [26, Part IV]); K-theory (see [9]); C∗-dynamical systems (see [18]); cyclic cohomology for crossed products (see [34]); operator analogues for locally convex spaces (see [16, 17]) and quantum mechanics, where it is called the space of physical states and its dual is the so-called space of observables (see [15]). Recently, some progress in the investigation of the noncommutative Schwartz space has been made. This contains: functional calculus (see [7]); description of closed, commutative, ∗-subalgebras (see [8]); automatic continuity (see [30]); amenability properties (see [29, 30]); Grothendieck inequality (see [25]). However, the significance of the algbra lies not only in the fact that it is a muliplier algebra MS ∗ of some well-known algebra of operators but also in its resemblance to the C -algebra (ℓ2) of ∗ B bounded operators on ℓ2 in the context of some class of Fréchet and topological -algebras. This can be already seen in the very definition of the maximal ∗-algebra on the domain s (which is, recall, isomorphic to ). It is also worth pointing outO that is considered as a Fréchet MS S analogon of the algebra (ℓ2) of compact operators on ℓ2 and (ℓ2) is the multiplier algebra of (ℓ ) whereas, as we haveK already noted, is the multiplierB algebra of – in fact, in order K 2 MS S to prove this, we apply methods used in the case of (ℓ2) and (ℓ2) (compare Theorem 3.9 with [6, Prop. 2.5] and [27, Example 3.1.2]). Moreover,K it seemsB that contains as closed ∗-subalgebras many important topological ∗-algebras, e.g. Fréchet algebrasMS C∞(M) of smooth functions on each compact smooth manifold M (the proof of this fact will be presented in a forthcoming paper). Finally, we find quite interesting to compare some commutative sequence ∗-algebras with the corresponding noncommutative operator ∗-algebras. This is done by the following diagram with the horizontal continuous embeddings of algebras:

s / ℓ / ℓ / c / ℓ / s′ O 1 2 0 ∞ O O OO OO OO OO O O O O O O O O O O O O O / (ℓ ) / (ℓ ) / (ℓ ) / (ℓ ) , S N 2 HS 2 K 2 B 2 MS where (ℓ ) and (ℓ ) is the algebra of nuclear and Hilbert-Schmidt operators, respectively. N 2 HS 2 The “vertical correspondences” from s up to c0 mean, for example, that every monotonical sequence of nonnegative numbers belonging to a commutative algebra from the first row is a sequence of singular numbers of some operator of its noncommutative analogues. Moreover, algebras from the first row are embedded in a canonical way, as the algebras of diagonal operators, into the corresponding algebras from the second row. It will be made clear below that (ℓ2) is not embedded in and vice versa. B This article isMS divided into four parts. Section 2 recalls basic notation and properties of the objects involved. In Section 3, we describe the multiplier algebra of the noncommutative Schwartz space. Section 4 deals with its topological properties and, finally, in Section 5 we consider properties of as a topological algebra. (For more information on functional analysis, see [20, 26], for more onMS Banach algebra theory, see [10]; and for non-Banach operator algebras, see [33].) THE MULTIPLIER ALGEBRA OF THE NONCOMMUTATIVE SCHWARTZ SPACE 3

2. Notation and Preliminaries In what follows, we set

N := 1, 2, 3,... , { } N := 0, 1, 2,... . 0 { } For locally convex spaces E and F , we denote by (E, F ) the space of all continuous linear operators from E to F , and we set (E) := (E, E).L These spaces will be considered with the L L topology τL(E,F ) of uniform convergence on bounded sets. By s we denote the space of rapidly decreasing sequences, that is, the Fréchet space ∞ 1/2 N 2 2n s := ξ = (ξ ) N C : ξ := ξ j < for all n N j j∈ ∈ | |n | j| ∞ ∈ 0 jX=1 with the topology given by the system ( ) N of norms. We will denote by s the Hilbert | · |n n∈ 0 n space corresponding to the norm n. By [26, Prop. 27.13] we may identify| · | the strong dual of s, that is, the space of all continuous linear functionals on s with the topology of uniform convergence on bounded subsets of s, with the space of slowly increasing sequences ∞ 1/2 ′ N 2 −2n s := ξ = (ξ ) N C : ξ := ξ j < for some n N j j∈ ∈ | |−n | j| ∞ ∈ 0 jX=1 equipped with the inductive limit topology for the sequence (s−n)n∈N0 , where s−n is the Hilbert ′ space corresponding to the norm −n. In other words, the locally convex topology on s is ′ | · | ′ given by the family B B∈B of norms , ξ B := supη∈B η,ξ , where denotes the class of all bounded subsets{| of ·s |. } | | |h i| B ′ The space (s ,s) is a Fréchet space, whose topology is described by the sequence ( n)n∈N0 of norms, L k · k ◦ x n := sup xξ n : ξ Un , ◦ ′ k k {| | ∈ } where Un = ξ s : ξ −n 6 1 are polars of the zeroneighbourhood basis (Un)n∈N0 in s. One can show that{ ∈is in fact| | isomorphic} (as a Fréchet space) to the space s (see [22, §41, 7.(5)] and S ′ ′ x ,(Lemma 31.1]). Since (s ,s) ֒ (ℓ2) (continuous embedding given by ℓ2 ֒ s s ֒ ℓ2 ,26] we can endow our space withL multiplication,→B involution and order structure of→ the →C∗-algebra→ ′ ∗ (ℓ2). With these operations (s ,s) becomes an m-convex Fréchet -algebra and is called the Bnoncommutative Schwartz spaceL or the algebra of smooth operators. We devote a large part of the present article to considering the spaces (s), (s′) and their “intersection” L L (s) (s′) := x (s) : x = x for some (and hence unique) x (s′) L ∩L { ∈L |s ∈L } equipped with the topology τL(s)∩L(s′) := τL(s) τL(s′); τL(s)∩L(s′) is therefore determined by the e ∩ e family q N of seminorms, where { n,B}n∈ 0,B∈B

′ (1) qn,B(x) := max sup xξ n, sup xξ B | | ◦ | | ξ∈B ξ∈Un and is the class of all bounded subsets of s. It is easy to show that (s) and (s′) are topologicalB algebras and, as we will show in Proposition 3.8, the same is trueL for (s)L (s′). The spaces (s) and (s′) can be seen as the completed tensor products s′Ls and∩Ls s′, respectively – seeL [22, §43,L 3.(7)]. Since s and s′ are nuclear (see [26, Prop. 28.16]),⊗ the injective⊗ and the projective tensor product topologies coincide (see [20, 21.2, Th. 1]). Consequentlyb (seeb 4 TOMASZ CIAŚ AND KRZYSZTOF PISZCZEK

[20, 15.4, Th. 2 & 15.5, Cor. 4]), (s) and (s′) admit natural PLS-topologies (see below) L L coinciding with τL(s) and τL(s′), that is,

(s) = proj N ind N (s ,s ), L N∈ 0 n∈ 0 L n N (2) ′ ′ ′ (s ) = proj N ind N (s ,s ). L N∈ 0 n∈ 0 L N n Remark 2.1. To be precise, the above two representations of (s) and (s′) are not of PLS- type. But this can be easily overcome following the first part ofL Remark 4.11.L As for the class of PLS-spaces (the best reference for which is [12] and the references therein) we will need more information. Recall that by a PLB-space we mean a locally convex space X whose topology is given by N X = projN∈N0 indn∈ 0 XN,n, where all the X ’s are Banach spaces and all the linking maps ιN,n+1 : X ֒ X are N,n N,n N,n → N,n+1 linear and continuous inclusions. If all the XN,n’s are Hilbert spaces then we call X a PLH- N,n+1 space. If the linking maps (ιN,n )N,n∈N0 are compact (nuclear) then X is called a PLS-space (PLN-space). Of particular importance for us will be the so-called Köthe-type PLB-spaces. Recall that a Köthe PLB-matrix is a matrix := (c ) N N of nonnegative scalars satisfying: C j,N,n j∈ ,N,n∈ 0 (i) j N N N0 n N0 : cj,N,n > 0, (ii) ∀ j ∈ N,N,n∃ ∈ N ∀: c∈ 6 c 6 c . ∀ ∈ ∈ 0 j,N,n+1 j,N,n j,N+1,n We define Λp( ) := x = (x ) N N n N : x < + (1 6 p< ), C { j j| ∀ ∈ 0 ∃ ∈ 0 || ||N,n,p ∞} ∞ where ∞ 1/p x := ( x c )p . || ||N,n,p | j | j,N,n jX=1 For p = we use the respective ‘sup’ norms. Then Λp( ) can be identified with the space ∞ C N p N projN∈N0 indn∈ 0 ℓ (cj,N,n) and it is called a Köthe-type PLB-space. If for all N,n 0 we have cj,N,n+1 p ∈ lim = 0 (compact linking maps) then Λ ( ) is a PLS-space and if for all N,n N0 we j cj,N,n C ∈ have cj,N,n+1 < (nuclear linking maps) then Λp( ) is a PLN-space. If Λp( ) = Λq( ), as j cj,N,n ∞ C C C sets, forP all 1 6 p,q 6 then we simply write Λ( ). For later use, we distinguish∞ specific Köthe PLB-matricesC on N N, defined as follows: × iN = (b ) N N , b := , B ij;N,n i,j∈ ,N,n∈ 0 ij;N,n jn (3) N ′ ′ ′ j = (b ) N N , b := B ij;N,n i,j∈ ,N,n∈ 0 ij;N,n in and

N N ′ i j (4) = (a ) N N , a := max b , b = max , . ij;N,n i,j∈ ,N,n∈ 0 ij;N,n ij;N,n ij;N,n jn in A { } Proposition 2.2. Let be any of the K¨othe-type PLB-matrices given by (3) or (4). Then Λp( ) = Λq( ) topologically,M for all 1 6 p,q 6 . M M ∞ 2 Proof. Since mij;N+2,n = (ij) mij;N,n+2 for all indices i, j N,N,n N0, we have the inequalities ∈ ∈ 6 and 6 C , k · kN,n,∞ k · kN,n,1 k · kN,n+2,1 k · kN+2,n,∞ THE MULTIPLIER ALGEBRA OF THE NONCOMMUTATIVE SCHWARTZ SPACE 5

4 which hold for all N,n N with C = ∞ (ij)−2 = π . ✷ ∈ 0 i,j=1 36 Using the notion of K¨othe-type PLB-spacesP we get from (2) and Proposition 2.2 topological isomorphisms:

(5) (s) = Λ( ) and (s′) = Λ( ′). L ∼ B L ∼ B In both cases the isomorphism is given by x ( xe , e ) N. 7→ h j ii i,j∈

3. Representations of the multiplier algebra In this section we want to describe the so-called multiplier algebra of which is, in some sense, the largest algebra of operators acting on . The algebra (s) (s′)S seems to be a good candidate, because if x and y (s) (s′),S then clearly xy,yxL ∩L . Now, using heuristic arguments, we may show∈ that S the algebra∈L ∩L(s) (s′) is optimal. Assume∈ S that y (E, F ) for some locally convex spaces E, F . If xy L for∩L every x then, in particular,∈L ( ,ξ ξ)y for all ξ s, and therefore y(η),ξ has∈ to S be well-defined∈ S for every ξ s and ηh· si′, which∈ S shows that∈ y : s′ s′. Similarly,h wei show that if yx for every x ∈then y : s ∈ s. Hence, y (s) (s′).→ ∈ S ∈ S → ∈LAnother,∩L more abstract, approach to multipliers goes through the so-called double centralizers (see Definition 3.1) and is due to Johnson [21]. Theorem 3.9 below shows that this approach leads again to (s) (s′). In Corollary 4.4, we show that the multiplier algebra of has other representationsL – also∩L important in further investigation. S The theory of double centralizers of C∗-algebras was developed by Busby (see [6] and also [27, pp. 38–39, 81–83]). Our exposition for the noncommutative Schwartz space will follow that of C∗-algebras. Definition 3.1. Let A be a ∗-algebra over C. A pair (L, R) of maps from A to A (neither linearity nor continuity is required) such that xL(y) = R(x)y for x,y A is called a double centralizer on A. We denote the set of all double centralizers on A by ∈ (A). Moreover, for a map T : A A, we define T ∗ : A A by T ∗(x) := (T (x∗))∗. DC Now, let→ (L , R ), (L , R ) →(A), λ C. We define: 1 1 2 2 ∈ DC ∈ (i) (L1, R1) + (L2, R2) := (L1 + L2, R1 + R2); (ii) λ(L1, R1) := (λL1, λR1); (iii) (L1, R1) (L2, R2) := (L1L2, R2R1); ∗· ∗ ∗ (iv) (L1, R1) := (R1,L1). A straightforward computation shows that (A) with the operations defined above is a ∗- algebra. The elements of A correspond to the elementsDC of (A) via the map, called the double representation of A (see [21, p. 301]), DC

(6) ̺: A (A), ̺(x) := (L , R ), → DC x x where Lx(y) := xy and Rx(y) := yx are the left and right multiplication maps, respectively. One can easily show that ̺ is a homomorphism of ∗-algebras. Deﬁnition 3.2. Let A be an algebra over C and let I be an ideal in A. (i) We say that A is faithful if for every x A we have: xA = 0 implies x = 0 and Ax = 0 implies x = 0. ∈ { } (ii) An ideal{ }I is called essential in A if for every x A the following implications hold: if xI = 0 then x =0 and if Ix = 0 then x = 0.∈ { } { } 6 TOMASZ CIAŚ AND KRZYSZTOF PISZCZEK

It is well-known that every C∗-algebra is an essential ideal in its multiplier algebra [27, p. 82]), so it is faithful (see also [6, Cor. 2.4]). For example, the C∗-algebra of compact operators on ℓ is an essential ideal in (ℓ ). The analogue of this result holds for and (s) (s′). 2 L 2 S L ∩L Proposition 3.3. An algebra is an essential ideal in (s) (s′). In particular, is faithful. S L ∩L S Proof. Clearly, is an ideal in (s) (s′). Assume that xz = 0 for all z . Then, in particular, for ξ Ss and z := ,ξLξ, we∩L get ,ξ x(ξ) = 0. Thus x(ξ) = 0 for∈ all S ξ s, i.e x = 0. This gives∈ x = 0 h· x i= 0. Applyingh· i now the involution we get the implication∈ x = 0 x = 0. S { } ⇒ ✷ S { }⇒ The following result can be deduced from [21, Th. 7 & Th. 14]. For the convenience of the reader we present a more direct proof. We follow the proof of [6, Prop. 2.5] (the case of C∗-algebras). Proposition 3.4. Let A be a faithful m-convex Fréchet algebra and let (L, R) (A). Then ∈ DC (i) L and R are linear continuous maps on A; (ii) L(xy)= L(x)y for every x,y A; (iii) R(xy)= xR(y) for every x,y ∈ A. ∈ Proof. (i) Let x,y,z A, α, β C. Then ∈ ∈ zL(αx + βy)= R(z)(αx + βy)= αR(z)x + βR(z)y = z(αL(x)+ βL(y)), and since A is faithful, L(αx + βy)= αL(x)+ βL(y). Now, let (x ) N A and assume that x 0 and L(x ) y (convergence in the topology j j∈ ⊂ j → j → of A). Let ( ) N be a fundamental system of submultiplicative seminorms on A. Then || · ||q q∈ 0 zy 6 zy zL(x ) + zL(x ) = z(y L(x )) + R(z)x || ||q || − j ||q || j ||q || − j ||q || j ||q 6 z y L(x ) + R(z) x 0, || ||q · || − j ||q || ||q · || j||q → as j , so zy q = 0 for every q N0, and therefore zy = 0. Hence, by the assumption on A, y →= 0.∞ Now,|| by|| the Closed Graph∈ Theorem for Fréchet spaces (see e.g. [26, Th. 24.31]), L is continuous. Analogous arguments work for the map R. (ii) Let x,y,z A. Then ∈ zL(xy)= R(z)xy = (R(z)x)y = (zL(x))y = z(L(x)y), and therefore, L(xy)= L(x)y. (iii) Analogously as in (ii). ✷ Now, we need to prove that elements of (s) (s′) can be seen as some unbounded operators L ∩L on ℓ2, namely as the elements of the class ∗(s) := x: s s : x is linear,s (x∗) and x∗(s) s . L { → ⊂ D ⊂ } Here (x∗) := η ℓ : ζ ℓ ξ s xξ,η = ξ,ζ D { ∈ 2 ∃ ∈ 2 ∀ ∈ h i h i} and x∗η := ζ for η (x∗) (one can show that such a vector ζ is unique). This ∗-operation defines a natural involution∈ D on ∗(s). ∗(s) is known as the maximal ∗-algebra with domain s L ∗ L O and it can be viewed as the largest -algebra of unbounded operators on ℓ2 with domain s (see [33, p. I.2.1] for details). Let denote the set of all bounded subsets of s. We endow ∗(s) with B L the topology τ ∗ given by the family p N of seminorms, L (s) { n,B}n∈ 0,B∈B

∗ (7) pn,B(x) := max sup xξ n, sup x ξ n . {ξ∈B | | ξ∈B | | } It is well-known that ∗(s) is a complete locally convex space and a topological ∗-algebra (see [33, Prop. 3.3.15] andL Remark 3.5 below). THE MULTIPLIER ALGEBRA OF THE NONCOMMUTATIVE SCHWARTZ SPACE 7

Remark 3.5. The so-called graph topology on s of the ∗-algebra ∗(s) is given by the system O L of seminorms ( ) ∗ , ξ := aξ ([33, Def. 2.1.1]). It is easy to see that the || · ||a a∈L (s) || ||a || ||ℓ2 usual Fréchet space topology on s is equal to the graph topology (consider the diagonal map s (ξ ) N (jnξ ) N s, n N ), and therefore the topology τ ∗ on ∗(s) coincides with ∋ j j∈ 7→ j j∈ ∈ ∈ 0 L (s) L the topology τ ∗ (see [33, pp. 81–82]) defined by the seminorms pa,B ∗ , { }a∈L (s),B∈B a,B ∗ p (x) := max sup axξ ℓ2 , sup ax ξ ℓ2 . ξ∈B || || ξ∈B || || n o Lemma 3.6. We have ∗(s) (s) and every operator x ∗(s) can be extended to an operator x (s′). Moreover, L ⊂L ∈L ∈L x∗η,ξ = η, xξ h i h i fore all ξ s′ and η s. ∈ ∈ e ∗ Proof. Take x (s). Let (ξj)j∈N s and assume that ξj 0 and xξj η as j . Then, for every ζ∈ Ls, we have ⊂ → → → ∞ ∈ xξ ,ζ = ξ ,x∗ζ 0 h j i h j i→ and, on the other hand, xξ ,ζ η,ζ . h j i → h i Hence η,ζ = 0 for every ζ s, and therefore η = 0. By the Closed Graph Theorem for Fréchet spaces,hx: si s is continuous,∈ and thus ∗(s) (s). →∗ ′ L ⊂L ∗ Fix x (s), ξ s , and define a linear functional ϕξ : s C, ϕξ(η) := x η,ξ . From the ∈ L ∗ ∈ → h i continuity of x : s s, it follows that for every q N0 there is r N0 and C > 0 such that x∗η 6 C η for all→η in s. Hence, with the same quantifiers,∈ we get∈ | |q | |r (8) ϕ (η) = x∗η,ξ 6 x∗η ξ 6 C η ξ , | ξ | |h i| | |q · | |−q | |r · | |−q and thus ϕ is continuous. Consequently, for each ξ s′ we can find a unique ζ s′ such that ξ ∈ ∈ η,ζ = ϕ (η)= x∗η,ξ h i ξ h i for all η s and we may define x: s′ s′ by xξ := ζ. Clearly, x is a linear extension of x, and moreover∈x is continuous. Indeed, by→ (8), for every q N there is r N and C > 0 such that ∈ 0 ∈ 0 e e ∗ e xξ −r = sup η, xξ = sup x η,ξ 6 C ξ −q e | | |η|r61 |h i| |η|r61 |h i| | | for all ξ s′. This showse the continuity ofe x, and the proof is complete. ✷ ∈ The following result follows also from [23, Prop. 2.2]. e Proposition 3.7. ∗(s)= (s) (s′) as sets. L L ∩L Proof. The inclusion ∗(s) (s) (s′) follows directly from Lemma 3.6. ′ L ⊂L ∩L ′ Let x (s) (s ). For each η s we define a linear functional ψη : s C, ψη(ξ) := xξ,η , where x:∈Ls′ s∩L′ is the continuous∈ extension of x. By the continuity of the→ operator x andh thei → Grothendieck Factorization Theorem [26, p. 24.33], it follows that for every r N theree is ∈ 0 q N eand C > 0 such that xξ 6 C ξ for ξ s′. Hence, for ξ s′, we have e ∈ 0 | |−q | |−r ∈ ∈ ψη(ξ) = xξ,η 6 xξ −q η q 6 C η q ξ −r. | e| |h i| | | · | | | | · | | This shows that ψη is continuous, and therefore there exists ζ s such that ψη( ) = ,ζ . Consequently, xξ,η = ξ,ζ for ξ e s, hencees (x∗) and x∗(s∈) s, that is, x · ∗(s).h· ✷i h i h i ∈ ⊂ D ⊂ ∈L If we endow (s) (s′) with the involution on ∗(s), we obtain that: L ∩L L Proposition 3.8. ∗(s) = (s) (s′) as locally convex spaces and ∗-algebras. Consequently, (s) (s′) is a completeL locallyL ∩L convex space and a topological ∗-algebra. L ∩L 8 TOMASZ CIAŚ AND KRZYSZTOF PISZCZEK

Proof. By deﬁnition and Proposition 3.7, ∗(s)= (s) (s′) as ∗-algebras. Let us compare L L ∩L ∗ fundamental systems pn,B n∈N0,B∈B (7) and qn,B n∈N0,B∈B (1) of seminorms on (s) and (s) (s′), respectively.{ } { } L L Take∩Lx ∗(s) with its unique extension x (s′). Let B be a bounded subset of s and n N . Then,∈ L by Lemma 3.6, ∈ L ∈ 0 ∗ ∗ e ′ (9) sup x η n = sup sup x η,ξ = sup sup η, xξ = sup xξ . ◦ ◦ ◦ B η∈B | | η∈B ξ∈Un |h i| η∈B ξ∈Un |h i| ξ∈Un | | e e This shows that pn,B(x)= qn,B(x). Since ∗(s) is a complete locally convex space and a topological ∗-algebra (see [33, Prop. 3.3.15] andL Remark 3.5), the result follows. ✷

By Propositions 3.3 and 3.4, ( ) ( ) ( ), and thus we may endow ( ) with the DC S ⊂L S ×L S DC S corresponding subspace topology, denoted by τDC(S). Note that a typical continuous seminorm of an element (L, R) of ( ) ( ) is given by the formula max sup L(x) , sup R(x) , L S ×L S { x∈M || ||n x∈M || ||n} where M is a bounded subset of and n N0. ′ S ∈ For u (s) (s ) we deﬁne the left and right multiplication maps Lu, Ru : , Lu(x) := ∈L ∩L ′ ′ S → S ′ ux, Ru(x) := xu, where u: s s is the extension of u according to the deﬁnition of (s) (s ). Theorem 3.9 below states→ that the double representation (6) of can be extendedL ∩L to an S isomorphism ofe topologicale ∗-algebras (s) (s′) and ( ), and thus (s) (s′) can be seen as the multiplier algebra of (compareL with∩L [27, Th. 3.1.8DC S & ExampleL 3.1.2]).∩L S ′ Theorem 3.9. The map ̺: (s) (s ) ( ), u (Lu, Ru) is an isomorphism of locally convex spaces and ∗-algebras.L ∩L Consequently,→ DC S ( ) 7→is a complete locally convex space and DC S topological ∗-algebra. e

Proof. Throughout the proof, for ξ, η s, ξ η denotes the one-dimensional operator , η ξ : s′ s. ∈ ⊗ h· i → ′ Clearly, for u (s) (s ), the left and right multiplication maps Lu, Ru : are well ∈ L ∩L S → S ′ defined. Moreover, it is easy to see that xLu(y) = Ru(x)y for x,y and u (s) (s ). ′ ∈ S ∈ L ∩L Hence, (Lu, Ru) ( ) for every u (s) (s ), that is, ̺ is well defined. The proof of the∈ DC factS that ̺ is a ∗-algebra∈L homomorphism∩L is straightforward and ̺ is injective, because is an essential ideal in (s) (s′) (Proposition 3.3).e We will show that ̺ is surjective. S L ∩L Let (L, R) ( ) and fixe e s with e = 1. We define a linear continuouse map (use ∈ DC S ∈ || ||ℓ2 Propositions 3.3 and 3.4) u: s s by e → uξ := L(ξ e)(e). ⊗ For ξ, η s we have ∈ uξ, η = L(ξ e)(e), η = L(ξ e)(e), (η e)(e) = (e η)[L(ξ e)(e)], e h i h ⊗ i h ⊗ ⊗ i h ⊗ ⊗ i (10) = [(e η)L(ξ e)](e), e = [R(e η)(ξ e)](e), e = R(e η)[(ξ e)(e)], e h ⊗ ⊗ i h ⊗ ⊗ i h ⊗ ⊗ i = R(e η)(ξ), e = ξ, (R(e η))∗(e) . h ⊗ i h ⊗ i This means that u∗η = (R(e η))∗(e) s for η s. Hence, s (u∗) and u∗(s) s. Consequently, u ∗(s), and thus,⊗ by Proposition∈ 3.7,∈u has the continuous⊂ D extension u: s′ ⊂ s′. By Propositions∈L 3.3 and 3.4, for ζ s we obtain → ∈ e L (ξ η)(ζ) = (uξ η)(ζ) = [L(ξ e)(e) η](ζ)= ζ, η L(ξ e)(e) u ⊗ ⊗ ⊗ ⊗ h i ⊗ = L(ξ e)( ζ, η e)= L(ξ e)[(e η)(ζ)] = [L(ξ e)(e η)](ζ) ⊗ h i ⊗ ⊗ ⊗ ⊗ = L((ξ e)(e η))(ζ)= L(ξ η)(ζ), ⊗ ⊗ ⊗ THE MULTIPLIER ALGEBRA OF THE NONCOMMUTATIVE SCHWARTZ SPACE 9 hence Lu(ξ η)= L(ξ η). Since ξ η : ξ, η s is linearly dense in , it follows that Lu = L. Likewise, (10)⊗ implies⊗ that, for ζ { s,⊗ ∈ } S ∈ R (ξ η)(ζ) = [(ξ η)u](ζ)= uζ, η ξ = R(e η)(ζ), e ξ = (ξ e)((R(e η)(ζ)) = u ⊗ ⊗ h i h ⊗ i ⊗ ⊗ = [(ξ e)R(e η)](ζ)= R((ξ e)(e η))(ζ)= R(ξ η)(ζ), ⊗ e ⊗ ⊗ ⊗ ⊗ and therefore Ru = R. Hence ̺(u) = (Lu, Ru) = (L, R), and thus ̺ is surjective. Next, we shall prove that ̺ is continuous. Let M be a bounded subset of and let n N . S ∈ 0 Since the involution on is continuouse (see [7, p. 148]), the set Me ∗ is bounded and there are S C > 0, k > n such that y∗ e 6 C y for all y . Define || ||n || ||k ∈ S B := xξ : x M,ξ s′, ξ 6 1 , 1 { ∈ ∈ | |−n } B := x∗ξ : x M,ξ s′, ξ 6 1 . 2 { ∈ ∈ | |−k } Then, for all m > k, we have

sup η m : η B1 6 sup x m < , {| | ∈ } x∈M || || ∞ sup η m : η B2 6 sup x m < , {| | ∈ } x∈M ∗ || || ∞ and therefore B1 and B2 are bounded subsets of s. Now,

sup Lu(x) n = sup ux n = sup sup u(xξ) n = sup u(η) n 6 qn,B1 (u). x∈M || || x∈M || || x∈M |ξ|−n61 | | η∈B1 | |

Moreover, by Lemma 3.6, (xu)∗ = u∗x∗ for x and u (s) (s′), and thus ∈ S ∈L ∩L ∗ ∗ ∗ ∗ ∗ sup Ru(x) n = sup xue n 6 C sup (xu) k = C sup u x k = sup sup u (x ξ) k x∈M || || x∈M || || x∈M || || x∈M || || x∈M |ξ|−k61 | | ∗ = sup u eη k 6 pk,B2 (u)=eqk,B2 (u), η∈B2 | | where the last identity follows from (9). Consequently,

max sup Lu(x) n, sup Ru(x) n 6 max qn,B1 (u),qk,B2 (u) 6 qk,B1∪B2 (u), x∈M || || x∈M || || { } and thus ̺ is continuous. Finally, we show that the inverse of ̺ is continuous. Let us take a bounded subset B of s and n N . Deﬁnee ∈ 0 e M := x : ξ B 0 0 , { ξ ∈ \ { }} ∪ { } −1 where xξ := ξ ξ ξ and 0 is the zero operator in . For all m N0, we have || ||ℓ2 ⊗ S ∈ sup x = sup ξ −1 ξ 2 6 sup ξ < , m ℓ2 m 2m x∈M || || ξ∈B\{0} || || | | ξ∈B\{0} | | ∞ where the ﬁrst inequality follows from the Cauchy-Schwartz inequality and the last one is a consequence of the boundedness of the set B. Hence M is a bounded subset of . Let S B′ := xξ : x M,ξ s′, ξ 6 1 . { ∈ ∈ | |−n } 10 TOMASZ CIAŚ AND KRZYSZTOF PISZCZEK

Clearly, 0 B′. If ξ B 0 , then ξ = x ( ξ −1ξ) and ξ −1ξ 6 1, hence ξ B′. ξ ℓ2 ℓ2 −n ∈ ′ ∈ \ { } || || || || ′ ∈ Consequently, B B . Again by identity (9), we get, for all u (s) (s ), ⊂ ∈L ∩L ∗ ∗ qn,B(u)= pn,B(u) = max sup uη n, sup u η n 6 max sup uη n, sup u η n ′ ′ η∈B | | η∈B | | η∈B | | η∈B | | } = max sup sup u(xξ) , sup sup u∗(xξ) = max sup ux , sup u∗x∗ | |n | |n || ||n || ||n x∈M |ξ|−n61 x∈M |ξ|−n61 x∈M x∈M

max sup ux n, sup xu n = max sup Lu(x) n, sup Ru(x) n , x∈M || || x∈M || || x∈M || || x∈M || || and therefore ̺−1 is continuous. e ✷

e 4. Topological properties of the multiplier algebra We start by showing how the multiplier algebra of can be realized as a matrix algebra. Before that we make rather easy but very efficient observationS . Proposition 4.1. The space (s) (s′) is isomorphic as a locally convex space to a complemented subspace of (s) (sL′). ∩L L ×L ′ Proof. We use matrix representations (5). If x = (xij)i,j∈N Λ( ) and y = (yij)i,j∈N Λ( ) N N ∈ B ∈ B then we denote M(x,y) C × , ∈ xij, i 6 j [M(x,y)]ij := (yij, i > j, and define a map P : Λ( ) Λ( ′) Λ( ) Λ( ′) by B × B → B × B P (x,y) := (M(x,y), M(x,y)). It is easily seen that P is a projection with im P = ∆(Λ( ) Λ( ′)) := (x,x) Λ( ) Λ( ′): x Λ( ) Λ( ′) . B × B { ∈ B × B ∈ B ∩ B } To get continuity, observe first that Λ( ) and Λ( ′) are webbed by [26, Lemma 24.28]. More- ′ ′ B B ′ over, (s) ∼= (s ) ∼= s s by [22, §43, 3.(7)], and therefore (s) and (s ) are ultrabornological by [19,L Ch. II,L Prop.⊗ 15 & Cor. 2] (see also [24] for a homologicL alL proof of this fact). Since ′ ′ ′ Λ( ) ∼= (s) and Λ( ) ∼= (s ), Λ( ) and Λ( ) are ultrabornological, as well. This implies thatB Λ( L) Λ( ′) isB ultrabornologicalL B by [32,B Ch. II, 8.2, Cor. 1] and has a web by [22, §35, 4.(6)]. ContinuityB × B of P follows now by the Closed Graph Theorem [26, Th. 24.31]. Now, let us consider the map Φ: (s) (s′) Λ( ) Λ( ′), Φ(x,y) := (Φ (x), Φ (y)), L ×L → B × B 1 2 ′ ′ where Φ1 : (s) Λ( ), Φ1 : (s ) Λ( ) are isomorphisms given by x ( xej , ei )i,j∈N. Clearly Φ isL an isomorphism→ B ofL locally→ convexB spaces and 7→ h i Φ(∆( (s) (s′))) = ∆(Λ( ) Λ( ′)), L ×L B × B where ∆( (s) (s′)) := (x, x) (s) (s′): x (s) (s′) . L ×L { ∈L ×L ∈L ∩L } This shows that the map Φ−1P Φ is a continuous projection onto ∆( (s) (s′)). Finally, by comparing fundamental systems of seminormse on (s) (s′) and ∆(L (s)×L(s′)), we see that the map L ∩L L ×L Ψ: (s) (s′) ∆( (s) (s′)), Ψ(x) := (x, x) L ∩L → L ×L is an isomorphisms, and thus (s) (s′) is isomorphic to ∆( (s) (s′)), a complemented subspace of (s) (s′). L ∩L L ×Le ✷ L ×L THE MULTIPLIER ALGEBRA OF THE NONCOMMUTATIVE SCHWARTZ SPACE 11

Corollary 4.2. The space (s) (s′) is a nuclear, ultrabornological PLS-space. L ∩L Proof. (s) and (s′) are nuclear, ultrabornological PLS-spaces, so is (s) (s′) as their product (seeL [26, Prop.L 28.7], [28, Cor. 6.2.14]). The desired propertiesL are×L inherited by complemented subspaces (see [26, Prop. 28.6], [32, Ch. II, 8.2, Cor. 1], [14, Prop. 1.2]), and thus, by Proposition 4.1, the proof is complete. ✷ Let be as in (4), that is, A iN jN := (a ) N N , a := max , . ij,N,n i,j∈ ,N,n∈ 0 ij,N,n jn in A Proposition 4.3. We have (s) (s′) = Λ( ) as topological ∗-algebras. L ∩L ∼ A Proof. The map T : Λ( ) (s) (s′), (Tx)e , e := x A →L ∩L h j ii ij (is a ∗-algebra isomorphism. To see it is continuous, observe that the embeddings Λ( ) ֒ (s and Λ( ) ֒ (s′) are continuous. Continuity of T −1 follows from the Open MappingA Theorem→L [26, p.A 24.30]→L since Λ( ) – as a PLS-space – has a web and (s) (s′) is ultrabornological by Corollary 4.2. A L ∩L ✷ Combining Proposition 3.8, Theorem 3.9 and Proposition 4.3 we obtain the following. Corollary 4.4. We have ( ) = ∗(s)= (s) (s′) = Λ( ) as topological ∗-algebras. DC S ∼ L L ∩L ∼ A From now on, by (s′,s) we denote any topological ∗-algebra isomorphic to ( ) and we call it the multiplierML algebra of the noncommutative Schwartz space. DC S We have just shown that (s′,s) is webbed and ultrabornological. By [35, Th. 4.2] this last property is equivalent toML barrelledness. Therefore by [26, Ths. 24.30, 24.31] and [28, Prop. 4.1.3] all the classical functional analytic tools are available for (s′,s). For convenience we state this result separately. ML Theorem 4.5. Let X = (s′,s) be the multiplier algebra of the noncommutative Schwartz space and let E,F,G be aML locally convex spaces with E webbed and F ultrabornological. The following hold: (1) Uniform Boundedness Principle: every pointwise bounded set B (X, G) is equicontinu- ous. ⊂ L (2) Closed Graph Theorem: every linear map T : X E and S : F X with closed graph is continuous. → → (3) Open Mapping Theorem: every continuous linear surjection T : E X and S : X F is open. → → We will now need a characterization of those Fréchet spaces which are PLN-spaces. This characterization seems to be known for specialists however we were not able to find any reference to that result. For convenience of the reader we state it explicitly below. Proposition 4.6. A Fréchet space is a PLN-space if and only if it is strongly nuclear. Proof. Let X be a strongly nuclear Fréchet space. By [20, 21.8, Th. 8], X is a topological subspace of (s′)I for a countable set I. Since s′ is an LN-space, the product (s′)I is a PLN-space. By [14, Prop. 1.2], X is a PLN-space. Conversely, every PLN-space is a topological subspace of a countable product of LN-spaces and these are strongly nuclear by [20, 21.8, Th. 6]. Consequently, by [20, Props. 21.1.3 and 21.1.5], X is strongly nuclear. ✷

Corollary 4.7. The multiplier algebra of the noncommutative Schwartz space is not a PLN- space. 12 TOMASZ CIAŚ AND KRZYSZTOF PISZCZEK

Proof. Suppose that the multiplier algebra is a PLN-space. Then so is s as its closed subspace. In fact, s is even complemented in Λ( ) – consider the projection in Λ( ) which cancels all but ﬁrst row-entries. By Proposition 4.6, As is strongly nuclear which leads toA a contradiction by [20, 21.8, Ex. 3]. ✷ The multiplier algebra of the noncommutative Schwartz space satisﬁes another useful property. Recall from [5, p. 433] that a PLS-space X is said to have the dual interpolation estimate for big θ if

N M K n m θ (0, 1) θ > θ k,C > 0 x′ X′ : ∀ ∃ ∀ ∃ ∀ ∃ 0 ∈ ∀ 0 ∃ ∀ ∈ N x′ ιM ∗ 6 C( x′ ιK ∗ )1−θ( x′ ∗ )θ. || ◦ N ||M,m || ◦ N ||K,k || ||N,n If we take θ 6 θ0 then X has the dual interpolation estimate for small θ and if we take θ (0, 1) then X has the dual interpolation estimate for all θ. For Köthe-type PLS-spaces Λp∈(B) it is enough – see the proof of [4, Th. 4.3] – to check the above condition for evaluation functionals ϕj(x) := xj, j N. Examples of PLS-spaces with this property can be found in [3, 5, 4]. Dual interpolation estimate∈ plays an important role in partial differential equations, e.g. surjectivity of operators, existence of linear right inverses, parameter dependence of solutions – see [13] for more details. Proposition 4.8. The multiplier algebra has the dual interpolation estimate for big θ but not for small θ. MS Proof. For any N N take M := N + 1 and for any K N take θ (0, 1) so that ∈ 0 ∈ 0 0 ∈ (1 θ )K + Nθ 6 M. − 0 0 In a similar fashion, for n = 1, any m N and θ > θ take k N so that ∈ 0 0 ∈ 0 (1 θ)k + nθ > m. − Then (with all the quantifiers in front) 1−θ θ jK jN jM 6 C . ik ! in ! im Exchanging indices i, j in the above inequality we obtain by (4) 1−θ θ (aij;K,k) (aij:N,n) 6 Caij;M,m. Since ∼= Λ( ), the dual interpolation estimate for big θ follows. If it had the condition for smallMSθ then,A by [3, Props. 5.3(b) & 5.4(b)] and [26, Cor. 29.22], the space s of rapidly decreasing sequences would be a Banach space – a contradiction. ✷ Remark 4.9. The above result together with [5, Prop. 1.1 & Cor. 1.2(c)] gives another proof of the fact that the multiplier algebra of the noncommutative Schwartz space is ultrabornological. We end this section with a technical lemma which characterizes when an arbitrary PLB-space is already a PLS-space; the proof uses interpolation theory and follows the idea of [31, Lemma 7]. As a consequence – see Remark 4.11(ii), we obtain another proof of the fact that is a PLS-space. MS

Lemma 4.10. Let (XM,m)M,m∈N be Banach spaces so that X := projM∈Nindm∈NXM,m is a PLB-space. The following conditions are equivalent: (i) X is a PLS-space, .ii) M L := L l m := m : X ֒ X is a compact inclusion) ∀ ∃ M ∀ ∃ l L,l → M,m Proof. We only need to show the implication (ii) (i). There is no loss of generality in assuming ⇒ that LM = M +1 and ml = l. Consider the commutative diagram THE MULTIPLIER ALGEBRA OF THE NONCOMMUTATIVE SCHWARTZ SPACE 13

j1 X2,1 X1,1

2 2 ι1 κ1 j2 X2,2 X1,2

2 2 where the inclusions ι1, κ1 are the respective linking maps and the inclusions j1, j2 are compact. Applying the real interpolation method with parameters θ1, 1 (0 <θ1 < 1) to the Banach couples Y1 := (X2,1, X1,1), Y2 := (X2,2, X1,2) we obtain, by [1, Th. 3.11.8], a continuous map (Y ) (Y ). Jθ1,1 1 →Jθ1,1 2 between the interpolation spaces (Y ) and (Y ). By [1, Cor. 3.8.2], we get for 0 <θ < Jθ1,1 1 Jθ1,1 2 1 θ2 < 1 the compact inclusion ( Y ) ֒ (Y) Jθ1,1 2 →Jθ2,1 2 therefore the map j2 : (Y ) (Y ) 1 Jθ1,1 1 →Jθ2,1 2 is also compact. We apply the same procedure to the commutative diagram

j2 X2,2 X1,2

3 3 ι2 κ2 j3 X2,3 X1,3 and obtain a compact operator j3 : (Y ) (Y ), 2 Jθ2,1 2 →Jθ3,1 3 where Y3 := (X2,3, X1,3) and θ2 < θ3 < 1. Proceeding this way we obtain a countable inductive system (jn+1 : (Y ) Y ), where Y := jn+1( (Y )). Let us observe that n Jθn,1 n → 1 1 n n Jθn,1 n the inductive topology of this system exists. Indeed, let x Y1 be a non-zero element. Since S ∈ ∗ Y1 X1 = indnX1,n, there exists, by [26, Lemma 24.6], a linear functional ϕ (indnX1,n) such that⊂ϕ(x) = 0 and ϕ κ X′ for all n N (by (κ : X X ) we denote∈ the imbedding 6 ◦ n ∈ 1,n ∈ n 1,n → 1 spectrum of X1). Recall that we distinguish here the space of linear functionals – denoted by ∗ ′ ∗ ( ) and the space of linear and continuous functionals – denoted by ( ) . Therefore ϕ Y1 . Moreover,· for every n N we have the commutative diagram · ∈ ∈ ϕ κn (Y ) X ◦ K Jθn,1 n 1,n n+1 jn ∼= ϕ κn+1 (Y ) X ◦ K Jθn+1,1 n+1 1,n+1 n+1 ′ N therefore ϕ jn θn,1(Yn) for every n . Again, by [26, Lemma 24.6], this implies that the ◦ ∈J n+1 ∈ inductive topology of (jn : θn,1(Yn) Y1) exists. Now, by [26, Lemma 24.34], we conclude n+1 J → n+1 that (jn : θn,1(Yn) Y1) is an LB-space and compactness of the linking maps (jn )n implies that it is evenJ an LS-space.→ It follows that we have linear and continuous maps

(11) X Y X . 2 → 1 → 1 14 TOMASZ CIAŚ AND KRZYSZTOF PISZCZEK

N Indeed, since for every n , θn,1(Yn) is an interpolation space for the couple Yn = (X2,n, X1,n) with the compact inclusion∈ j J: X ֒ X , we get compact inclusions n 2,n → 1,n X ֒ (Y ) ֒ X 2,n →Jθn,1 n → 1,n n+1 (observe that we loose injectivity in (11) because the linking maps (jn )n are not, in general, injective). The above argument works for all LB-spaces XM and XM+1 therefore

X = projM∈NYM where all YM ’s are LS-spaces. Consequently, X is a PLS-space. ✷ Remark 4.11. Here we give two new proofs of the fact that is a PLS-space. By Proposi- ∗ MS tions 2.2 & 4.3, we may use the topological -algebra isomorphism ∼= Λ( ) (recall that the PLB-matrix is given by (4)). MS A A (i) We do a slight perturbation of . Let := (dij,N,n)i,j∈N,N,n∈N0 be a 4-indexed K¨othe PLB-matrix given by A D N+ 1 N+ 1 j n i n dij;N,n := max , . i,j∈N in jn One can easily show that for all i, j N,N,n N we have ∈ ∈ 0 aij;N,n 6 dij;N,n 6 aij;N+1,n. This implies the topological isomorphism = Λ( ). Since MS ∼ D dij;N,n 1 = min i, j max i, j n(n+1) , dij;N,n+1 { } { } we get that for all N,n N0 ∈ d lim ij;N,n+1 = 0. i,j→+∞ dij;N,n Consequently, Λ( ), and therefore also , is a PLS-space. (ii) Since aij,N,nD+2 = (ij)−2 for all i,j,N,nMS , the inclusion map aij,N+2,n

ℓ (a ) N ֒ ℓ (a ) N 2 ij,N+2,n i,j∈ → 2 ij,N,n+2 i,j∈ is compact. By Lemma 4.10, the result follows. ✷

5. Algebraic properties of the multiplier algebra We say that a subalgebra B of an algebra A is spectral invariant in A if, for every x B, x is invertible in A if and only if it is invertible in B. We show that contains a spectral∈ invariant copy of the algebra s′, which implies that it is neither a -algebraMS nor m-convex. Q Proposition 5.1. Let be a K¨othe PLB-matrix given by (4) and let ∆( ) be the algebra of all diagonal matrices belongingA to Λ( ). Then A A (i) ∆( ) is a complemented subspace of Λ( ); (ii) ∆(A) is a closed commutative ∗-subalgebraA of Λ( ); A ′ ∗ A (iii) ∆( ) ∼= s as topological -algebras; (iv) ∆(A) is spectral invariant in Λ( ). A A Proof. (i) Deﬁne π : Λ( ) Λ( ) by A → A ∞ πx := ejjxejj, jX=1 THE MULTIPLIER ALGEBRA OF THE NONCOMMUTATIVE SCHWARTZ SPACE 15 where (eij)i,j∈N is a sequence of matrix units. Clearly, π is a projection. Note that continuity of a linear operator T on Λ( ) follows from to the condition A N M m n,C > 0 x Λ( ) Tx 6 C x . ∀ ∃ ∀ ∃ ∀ ∈ A || ||N,n || ||M,m But ∞ ∞ πx = x a 6 x a = x , || ||N,n | jj| jj,N,n | ij| ij,N,n || ||N,n jX=1 i,jX=1 so π is continuous, and thus ∆( ) is complemented in Λ( ). (ii) It is clear that ∆( ) is aA commutative ∗-subalgebraA of Λ( ), and by (i) it is closed in Λ( ). A A A(iii) Since ∞ N N ∆( )= x C × : x = 0 for i = j and N n x jN−n < A { ∈ ij 6 ∀ ∃ | jj| ∞} jX=1 N×N ′ = x C : x = 0 for i = j and (x ) N s , { ∈ ij 6 jj j∈ ∈ } the operator ∞ ϕ: s′ ∆( ), ϕξ := ξ e → A j jj jX=1 ∗ ′ is a -algebra isomorphism. Moreover, for all N,n N0 and all ξ s , we have ∞ ∈ ∈ ϕξ = ξ jN−n. || ||N,n | j| jX=1 Now, by Cauchy-Schwarz we get ∞ −m π ξ −m 6 ξj j 6 ξ −m+1 | | | | √6| | jX=1 N ′ for all m 0. Therefore ϕ is an isomorphism of locally convex spaces. Consequently, ∆( ) ∼= s as topological∈ ∗-algebras. A (iv) Let us take x ∆( ) which is invertible in Λ( ) and let y be its inverse. Then ∈ A A ∞ 1 for i = j, x y = x y = ik kj ii ij 0 otherwise. kX=1 Consequently, xii = 0, and thus yij = 0 for i = j. This shows that y ∆( ), and the proof is complete. 6 6 ∈ A Proposition 5.2. The following statements hold: (i) is not a -algebra; (ii) MS is not mQ-convex. MS Proof. By Proposition 5.1, (s′,s) contains a closed, spectral invariant ∗-subalgebra M isomorphic to s′. ML (i) Let F, G be the sets of invertible elements in M and (s′,s), respectively. Then F = G M. This shows that G is not open, because otherwise FMLwould be open, which contradicts [2,∩ Th. 2.8]. Hence (s′,s) is not a -algebra. (ii) Suppose thatML there is a basis ofQ zero neighborhoods in (s′,s) such that V 2 V for all V . Then V M is a basisV of zero neighborhoodsML in M and ⊆ ∈V { ∩ }V ∈V (V M)2 V 2 M V M, ∩ ⊆ ∩ ⊆ ∩ so M is m-convex, a contradiction (again apply [2, Th. 2.8]). Hence, (s′,s) is not m-convex. ML ✷ 16 TOMASZ CIAŚ AND KRZYSZTOF PISZCZEK

References

[1] J. Bergh and J. Löfström. Interpolation spaces. An introduction. Grundlehren der Mathe- matischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, 1976, pp. x+207. [2] J. Bonet and P. Domański. “Köthe coechelon spaces as locally convex algebras”. In: Studia Math. 199.3 (2010), pp. 241–265. [3] J. Bonet and P. Domański. “Parameter dependence of solutions of differential equations on spaces of distributions and the splitting of short exact sequences”. In: J. Funct. Anal. 230.2 (2006), pp. 329–381. [4] J. Bonet and P. Domański. “The splitting of exact sequences of PLS-spaces and smooth dependence of solutions of linear partial differential equations”. In: Adv. Math. 217.2 (2008), pp. 561–585. [5] J. Bonet and P. Domański. “The structure of spaces of quasianalytic functions of Roumieu type”. In: Arch. Math. (Basel) 89.5 (2007), pp. 430–441. [6] R.C. Busby. “Double centralizers and extensions of C∗-algebras”. In: Trans. Amer. Math. Soc. 132 (1968), pp. 79–99. [7] T. Ciaś. “On the algebra of smooth operators”. In: Studia Math. 218.2 (2013), pp. 145–166. [8] Tomasz Ciaś. “Commutative subalgebras of the algebra of smooth operators”. In: Monatsh. Math. 181.2 (2016), pp. 301–323. [9] J. Cuntz. “Cyclic theory and the bivariant Chern-Connes character”. In: Noncommutative geometry. Vol. 1831. Lecture Notes in Math. Berlin: Springer, 2004, pp. 73–135. [10] H.G. Dales. Banach algebras and automatic continuity. Vol. 24. London Mathematical Society Monographs. New Series. Oxford Science Publications. New York: The Clarendon Press Oxford University Press, 2000, pp. xviii+907. [11] P. Domański. “Algebra of smooth operators”. Unpublished manuscript. [12] P. Domański. “Classical PLS-spaces: spaces of distributions, real analytic functions and their relatives”. In: Orlicz centenary volume. Vol. 64. Banach Center Publ. Polish Acad. Sci., Warsaw, 2004, pp. 51–70. [13] P. Domański. “Real analytic parameter dependence of solutions of differential equations”. In: Rev. Mat. Iberoam. 26.1 (2010), pp. 175–238. [14] P. Domański and D. Vogt. “A splitting theory for the space of distributions”. In: Studia Math. 140.1 (2000), pp. 57–77. [15] D.A. Dubin and M.A. Hennings. Quantum mechanics, algebras and distributions. Vol. 238. Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Har- low; copublished in the United States with John Wiley & Sons, Inc., New York, 1990, pp. viii+238. [16] E.G. Effros and C. Webster. “Operator analogues of locally convex spaces”. In: Operator algebras and applications (Samos, 1996). Vol. 495. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Kluwer Acad. Publ., Dordrecht, 1997, pp. 163–207. [17] E.G. Effros and S. Winkler. “Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems”. In: J. Funct. Anal. 144.1 (1997), pp. 117–152. [18] G.A. Elliott, T. Natsume, and R. Nest. “Cyclic cohomology for one-parameter smooth crossed products”. In: Acta Math. 160.3-4 (1988), pp. 285–305. [19] A. Grothendieck. “Produits tensoriels topologiques et espaces nucléaires”. In: Mem. Amer. Math. Soc. 1955.16 (1955), 140 pp. [20] H. Jarchow. Locally convex spaces. Mathematische Leitfäden. Stuttgart: B. G. Teubner, 1981, p. 548. [21] B.E. Johnson. “An introduction to the theory of centralizers”. In: Proc. London Math. Soc. (3) 14 (1964), pp. 299–320. REFERENCES 17

[22] G. Köthe. Topological vector spaces. II. Vol. 237. Grundlehren der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Science]. Springer-Verlag, New York- Berlin, 1979, pp. xii+331. [23] K.-D. Kürsten. “The completion of the maximal Op∗-algebra on a Fréchet domain”. In: Publ. Res. Inst. Math. Sci. 22.1 (1986), pp. 151–175. [24] J. Larcher and J. Wengenroth. “A new proof for the bornologicity of the space of slowly increasing functions”. In: Bull. Belg. Math. Soc. Simon Stevin 21.5 (2014), pp. 887–894. [25] R.H. Levene and K. Piszczek. “Grothendieck’s Inequality in the noncommutative Schwartz space”. To appear in Studia Math. [26] R. Meise and D. Vogt. Introduction to functional analysis. Vol. 2. Oxford Graduate Texts in Mathematics. Translated from the German by M. S. Ramanujan and revised by the authors. New York: The Clarendon Press Oxford University Press, 1997, pp. x+437. [27] G.J. Murphy. C∗-algebras and operator theory. Academic Press, Inc., Boston, MA, 1990, pp. x+286. [28] Pedro Pérez Carreras and JoséBonet. Barrelled locally convex spaces. Vol. 131. North- Holland Mathematics Studies. Notas de Matemática [Mathematical Notes], 113. North- Holland Publishing Co., Amsterdam, 1987, pp. xvi+512. [29] A.Yu. Pirkovskii. “Flat cyclic Fréchet modules, amenable Fréchet algebras, and approxi- mate identities”. In: Homology, Homotopy Appl. 11.1 (2009), pp. 81–114. [30] K. Piszczek. “Automatic continuity and amenability in the non-commutative Schwartz space”. In: J. Math. Anal. Appl. 432.2 (2015), pp. 954–964. [31] K. Piszczek. “Tame (PLS)-spaces”. In: Funct. Approx. Comment. Math. 38.part 1 (2008), pp. 67–80. [32] H.H. Schaefer. Topological vector spaces. Third printing corrected, Graduate Texts in Math- ematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971, pp. xi+294. [33] K. Schmüdgen. Unbounded operator algebras and representation theory. Vol. 37. Operator Theory: Advances and Applications. Birkhäuser Verlag, Basel, 1990, p. 380. [34] L.B. Schweitzer. “Spectral invariance of dense subalgebras of operator algebras”. In: In- ternat. J. Math. 4.2 (1993), pp. 289–317. [35] Dietmar Vogt. “Topics on projective spectra of (LB)-spaces”. In: Advances in the theory of Fréchet spaces (Istanbul, 1988). Vol. 287. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Kluwer Acad. Publ., Dordrecht, 1989, pp. 11–27.

Tomasz Ciaś Faculty of Mathematics and Computer Science Adam Mickiewicz University, Poznań ul. Uniwersytetu Poznańskiego 4 61-614 Poznań, Poland e-mail: [email protected]

Krzysztof Piszczek Faculty of Mathematics and Computer Science Adam Mickiewicz University, Poznań ul. Uniwersytetu Poznańskiego 4 61-614 Poznań, Poland e-mail: [email protected]