Monatsh Math (2015) 178:599–610DOI 10.1007/s00605-015-0758-z

One-sided ideals of the non-commutative Schwartzspace

Krzysztof Piszczek1

Received: 16 July 2014 / Accepted: 9 March 2015 / Published online: 24 March 2015© The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract We show that maximal one-sided ideals of the non-commutative Schwartzspace are closed. We also characterize all closed one-sided ideals of this algebra. Asa result, all maximal left ideals are fixed.

Keywords Fréchet space · (Fréchet) LMC-algebra · (one-sided) Ideal · Fixed ideal

Mathematics Subject Classification Primary 46H40 · 46H30 · 46K05;Secondary 46H20

1 Introduction

The problemof characterizing ideals of a given topological algebra belongs to classicalones and there is vast literature devoted to it. The aim of this paper is to addressthis problem in the case of one particular lmc Fréchet ∗-algebra, the so-called non-commutative Schwartz space. This object has received reasonable attention recently,mostly due to Cias—see [3], Domanski—see [7] and the author—see [21,22]. Thispaper is a continuation of the previous work. The motivation for the investigation ofthe non-commutative Schwartz space—which we will denote from now on by S—comes from several directions. First, it has several, natural function (and sequence)space representations, for instance S is isomorphic (as a topological vector space)

Communicated by A. Constantin.

B Krzysztof Piszczekkpk@amu.edu.pl

1 Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznan,ul. Umultowska 87, 61-614 Poznan, Poland

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600 K. Piszczek

to the space C∞(M) of smooth functions on a compact smooth manifold M or tothe Schwartz space S(R) of test functions for tempered distributions (again as a tvs).This last isomorphism justifies also the name for S (the other name algebra of smoothoperators—used e.g. by Cias—comes from the first-mentioned isomorphism). Thislinks our object with the structure theory of Fréchet spaces, especially with questionsconcerning nuclearity or splitting of short exact sequences—see [17, Part IV]. Thisalgebra appears also in the context of K-theory—see [4,20] or in the context of cycliccohomology for crossed products—see [12,23]. Another interestingmotivation comesfrom the theory of operator spaces and its locally convex analogues—see [10,11]. Thenon-commutative Schwartz space plays also a role in quantum mechanics, where it iscalled the space of physical states and its dual is the so-called space of observables—see [8] for details. This algebra shares also some nice features with C∗-algebras, e.g.it admits a Hölder functional calculus—see [3, Th. 5.2] and all positive functionals aswell as all derivations are automatically continuous—see [21, Ths. 11, 13].On the otherhand it causes some technical difficulties, e.g. it has neither a bounded approximateidentity—see [21, Prop. 2] nor is it a locally C∗-algebra—see [13, Ths. 8.2, 8.3].

The paper is divided into four parts. The next section contains basic notation andterminology. The remaining two sections contain our main results. They are devotedonly to one-sided ideals since S is topologically simple, i.e. there are no non-trivialclosed two-sided ideals—see [21, Th. 4]. In the third Section we show that all maximalone-sided ideals of S are closed—see Theorem 5. In the last Section we characterizeall closed one-sided ideals of S—see Theorem 7. As a consequence, we obtain thata left ideal of the non-commutative Schwartz space is maximal if and only if it isfixed—see Corollary 8.

For unexplained details we refer the reader to [17] in the case of the structuretheory of Fréchet spaces and to [5] in the case of the ‘algebraic-in-flavour’ aspects ofthe paper.

The author is very indebted to Tomasz Kania for many valuable comments anddiscussions and to David Blecher for pointing out the reference [9].

2 Notation and terminology

Throughout the paper we denote N := {1, 2, 3, . . .} and N0 := N ∪ {0}. Recall thatby

s =⎧⎨

⎩ξ = (ξ j ) j∈N ⊂ C

N : |ξ |2k :=+∞∑

j=1

|ξ j |2 j2k < +∞ for all k ∈ N0

⎫⎬

⎭

we denote the space of rapidly decreasing sequences. This space becomes Fréchetwhen endowed with the above defined sequence (| · |k)k∈N0 of norms. The basis(Uk)k∈N0 of zero neighbourhoods of s is defined by Uk := {ξ ∈ s : |ξ |k ≤ 1}. Thisspace is topologically isomorphic to several spaces of functions, e.g. the spaceC∞(M)

of smooth functions on any smooth, compact manifold M or the Schwartz space ofrapidly decreasing functions S(R)—see [17, Example 29.5(2), (4)]. It is also present

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One-sided ideals of the non-commutative... 601

in the structure theory of (nuclear) Fréchet spaces as well as in the theory of splittingof short exact sequences of Fréchet spaces—see [17,25, Ch. 30]. The topological dualof s is

s′ =⎧⎨

⎩η = (η j ) j∈N ⊂ C

N : |η|′2k :=+∞∑

j=1

|η j |2 j−2k < +∞ for some k ∈ N0

⎫⎬

⎭.

If we now endow the space S := L(s′, s) of linear and continuous operators fromthe dual of s into s with the topology of uniform convergence on bounded sets then itbecomes a Fréchet space with the sequence (‖ · ‖n)n∈N0 of norms given by

‖x‖n := sup{|xξ |n : ξ ∈ U ◦n },

whereU ◦n = {ξ ∈ s′ : ∑

j |ξ j |2 j−2n ≤ 1}. It is fairly easy to observe that the identitymap ι : s↪→s′ is a continuous embedding which allows us to endow the space S withthe multiplication defined by

xy := x ◦ ι ◦ y, x, y ∈ S.

This multiplication is easily seen to be separately continuous therefore by [26, Th.1.5] it is also jointly continuous. If we define the duality bracket between s and s′ by

〈ξ, η〉 :=∑

j∈Nξ jη j , ξ ∈ s, η ∈ s′ (1)

then our algebra can also be given an involution map defined by

〈x∗ξ, η〉 := 〈ξ, xη〉, x ∈ S, ξ, η ∈ s′.

With these operations S becomes a locally multiplicatively convex (lmc for short)Fréchet ∗-algebra and this will be the main object of the paper. It has several repre-sentations as the lmc Fréchet ∗-algebra—see e.g. [7, Th. 1.1]. Of particular interestfor us will be the ∗-algebra isomorphism S � K∞, where

K∞ :=⎧⎨

⎩x = (xi, j )i, j∈N : ‖x‖2n :=

∞∑

i, j=1

|xi j |2(i j)2n < +∞ for all n ∈ N0

⎫⎬

⎭

is the lmc Fréchet ∗-algebra endowed with the topology defined by the sequence ofnorms (‖·‖n)n∈N0 , multiplication of matrices and the ‘star’ operation being the matrixconjugate transpose. It turns out that we also have the isomorphism S � s—this time,however, as Fréchet spaces. Note that any reasonable multiplication (e.g. pointwise)on the sequence space s makes it into a commutative algebra while S is highly non-commutative. Thus, as a Fréchet space, S can be thought of as the space of smoothfunctions or the Schwartz space of rapidly decreasing functions. On the other hand, if

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602 K. Piszczek

we endow s with the pointwise multiplication and involution being the coordinatewisecomplex conjugation then we get a ∗-isomorphism (onto its range) D : s → S sendinga sequence a = (a j ) into a diagonal operator Da := diag(a j ) j . The above-mentionedFréchet space isomorphismS � s justifies the name non-commutative Schwartz spacewhich we will be using throughout the paper.

The isomorphism S � K∞ shows that our algebra can be thought of as an algebraof matrices. Moreover the zeroth norm in S is just the C∗-algebra norm in B(�2) andso we obtain a continuous inclusion map S↪→B(�2). In fact we even have S↪→K (�2)

since every element of S is a compact operator on �2. The above inclusions haveseveral consequences. Among them we point out that the non-commutative Schwartzspace does not have a unit and so by S1 we denote its unitization. This unit is in factthe identity map on �2 therefore we still have S1 ⊂ B(�2). Furthermore by [7, Th. 2.3]an element x ∈ S1 is invertible in S1 if and only if it is invertible in B(�2). Recall nowthat by the natural inclusions s↪→�2 and �2↪→s′ we have SB(�2)S ⊂ S. Therefore,if we denote by σA(x) the spectrum of an element x in the algebra A, by [2, Prop.A.2.8] (cf. [7, Cor. 2.5]) we have

σS1(x) = σB(�2)(x), x ∈ S1.

This equality shows in particular that an element x ∈ S is positive in S if and only if itis positive as an operator on �2. Consequently, the order structure, multiplication andthe ‘star’ operation in S and S1 are inherited from the C∗-algebra B(�2). Furthermore,the space of self-adjoint elements of S, denoted by Ssa is a real Fréchet space and theset S+ of positive elements is a convex cone in S. Moreover by [7, Cor. 2.4] S1 is aQ-algebra and by [13, Ths. 8.2, 8.3] the topology of both S and S1 cannot be givenby a sequence of C∗-norms. It has also been shown recently in [3] that S as well asS1 admit a functional calculus for normal elements. Its description is a bit technicalnevertheless we can work with the Hölder functions which vanish at 0. In particularwe can take roots of positive elements. To obtain this functional calculus one has toapply a purely Fréchet space property (called (DN )) which holds trivially in Banachspaces—see [3, Th. 5.2] for details.

3 Closedness of maximal one-sided ideals

In this Section we prove that all maximal one-sided ideals of the non-commutativeSchwartz space are closed. Recall that one-sided ideals are in fact all ideals one canconsider since by [21, Th. 4] S is topologically simple, i.e. there are no non-trivialtwo-sided ideals. We will show in fact a stronger property, namely that null sequencesin S factor, i.e. for every sequence (xn)n∈N ⊂ S tending to 0 there is a single elementa ∈ S and another sequence (yn)n∈N ⊂ S also tending to zero such that xn = ayn

for all n ∈ N. This property may also be defined for algebra modules—see [5, Def.2.6.11]—however we will not need this general setting here. It turns out that if nullsequences in a Banach algebra factor then maximal right ideals are closed—see [14,Th. 3] (cf. also [5, Prop. 2.6.13]). However the proof of this result carries over onto

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One-sided ideals of the non-commutative... 603

Fréchet lmc algebras. If the algebra is involutive, this applies also to maximal leftideals. We state this result explicitly.

Theorem 1 If null sequences in a Fréchet lmc ∗-algebra factor then all maximalone-sided ideals are closed.

In order to prove that null sequences in S factor we cannot follow the Banach-algebraproof of Varopoulos, who proved in [24] that if a Banach algebra admits a boundedapproximate identity then null sequences in this algebra factor. Instead we will relyon [1] and [9]. Our proof will be divided into several steps. We start by defining α-additive ordered Fréchet spaces (we restrict ourselves to the Fréchet-space setting butthis definition can mutatis mutandis be extended onto arbitrary lcs). By an orderedlocally convex space we mean a real lcs partially ordered by a closed convex cone ofpositive elements. Our definition is slightly different from the one used e.g. by Asimovin the setting of Banach spaces—see [1, p. 118]. While the dual to a Banach space isstill a Banach space (with the dual order if the initial space was ordered), this is nolonger true for Fréchet spaces (these are called (DF)-spaceswhich are notmetrizable ingeneral). Therefore we think such a slight change is reasonable. Of course for Banachordered spaces (which are by definition also Fréchet spaces) we get two differentnotions of α-additiveness. The relation between these notions for an arbitrary orderedBanach space X is that it is α-additive in our sense if and only if the dual orderedBanach space X ′ (with the dual order) is α-additive in the sense used by Asimov.

Definition 2 Let α� 0. An ordered Fréchet space X is α-additive if for any p ∈ N

there exists q ∈ N such that for any n ∈ N and an arbitrary choice of n positivefunctionals φ1, . . . , φn ∈ X ′ we have that

n∑

k=1

‖φk‖∗q ≤ α

∥∥∥∥∥

n∑

k=1

φk

∥∥∥∥∥

∗

p

.

Proposition 3 Ssa is π2

6 -additive.

Proof We will be using in the proof the representation S � K∞ with the �∞-normsof the matrices that is

‖x‖k,∞ := sup{|xi j |(i j)k : i, j ∈ N}.Consequently, the dual norms in S ′ are given by the formulae

‖φ‖∗k,1 :=

+∞∑

i, j=1

|φi j |(i j)−k (φ ∈ S ′).

Take a single positive functional φ. Positivity then implies—see [18, Ex. 3.2(i), p. 39]or the proof of [21, Lemma 5]—that

|φi j |2 ≤ φi iφ j j for all i, j ∈ N.

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604 K. Piszczek

By Cauchy–Schwarz this leads to

‖φ‖∗k,1 =

+∞∑

i, j=1

|φi j |(i j)−k ≤⎛

⎝+∞∑

i, j=1

|φi j |2(i j)−2(k−1)

⎞

⎠

12⎛

⎝+∞∑

i, j=1

(i j)−2

⎞

⎠

12

≤ π2

6

⎛

⎝+∞∑

i, j=1

φi iφ j j (i j)−2(k−1)

⎞

⎠

12

= π2

6

+∞∑

j=1

φ j j j−2(k−1).

For arbitrary positive functionals φ1, . . . , φn this gives

n∑

m=1

‖φm‖∗k ≤ π2

6

+∞∑

j=1

(n∑

m=1

φmj j

)

j−2(k−1) ≤ π2

6

∥∥∥∥∥

n∑

m=1

φm

∥∥∥∥∥

∗

k−1,1

.

We have thus shown that for any k ∈ N0, n ∈ N and arbitrary choice of positivefunctionals φ1, . . . φn ∈ S ′

n∑

m=1

‖φm‖∗k+1,1 ≤ π2

6

∥∥∥∥∥

n∑

m=1

φm

∥∥∥∥∥

∗

k,1

.

This proves that Ssa is π2

6 -additive.

The next step is to show that Ssa is quasi-directed. An ordered locally convex spaceis (see [9]) quasi-directed if for any neighbourhood U of zero there is another neigh-bourhood V of zero such that every finite subset of V is majorized by some elementof U . We will now prove that the ordered self-adjoint part of the non-commutativeSchwartz space is quasi-directed. The idea of proof comes from the proof of [1, Th.1.1]. However at some point we will have to modify it in order to adapt it to theFréchet-space setting.

Proposition 4 Ssa is quasi-directed.

Proof Let α = π2

6 and fix n ∈ N. Following Asimov we denote by S(n)sa the n-fold

Cartesian product of Ssa with itself. The topology in this Fréchet space is given by thebasis (U(n)

k )k∈N0 of zero neighbourhoods, where

U(n)k = {

x(n) = (x1, . . . , xn) ∈ S(n)sa : ‖x(n)‖k := max{‖x j‖k : j = 1, . . . , n} ≤ 1

}.

With the closed positive cone P(n) := {p(n) : p j�0, j = 1, . . . , n}S(n)sa becomes an

ordered Fréchet space. We also define D(n)k := {x(n) = (x, . . . , x) : , ‖x‖k ≤ α} to be

the diagonal in S(n)sa and

A(n)k := D(n)

k − P(n) = {(x − p1, . . . , x − pn) : ‖x‖k ≤ α, p j�0}.

Following exactly the proof of [1, Th. 1.1] we obtain that

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One-sided ideals of the non-commutative... 605

U(n)k+1 ⊂ A(n)

k (k ∈ N0). (2)

Now we modify the proof of Asimov. Let x(n) ∈ U(n)k+1. By (2) there is z(n)

1 − p(n)1 ∈

A(n)k such that

‖x(n) − (z(n)1 − p(n)

1 )‖k+2 <1

2, ‖z(n)

1 ‖k ≤ α, p(n)1 � 0.

If y(n)1 := x(n) − (z(n)

1 − p(n)1 ) then x(n) = y(n)

1 + z(n)1 − p(n)

1 ≤ y(n)1 + z(n)

1 and

y(n)1 ∈ 1

2A(n)k+1. Therefore there is z

(n)2 − p(n)

2 ∈ 12A

(n)k+1 such that

‖y(n)1 − (z(n)

2 − p(n)2 )‖k+3 <

1

22, ‖z(n)

2 ‖k+1 ≤ α

2, p(n)

2 � 0.

If y(n)2 := y(n)

1 − (z(n)2 − p(n)

2 ) then x(n) ≤ z(n)1 + z(n)

2 + y(n)2 and y(n)

2 ∈ 122A(n)k+2.

Proceeding recursively we will find sequences (y(n)m )m∈N, (z(n)

m )m∈N ⊂ Ssa such that

‖y(n)m ‖k+1+m ≤ 1

2m, z(n)

m ∈ 1

2m−1D(n)k−1+m, x(n) ≤ z(n)

1 + · · · + z(n)m + y(n)

m (m ∈N).

It is now easy to show that y(n)m → 0 when m tends to infinity and the series

∑+∞m=1 z

(n)m

converges absolutely. Therefore there is z(n) := ∑+∞m=1 z

(n)m and x(n) ≤ z(n). Moreover

‖z(n)‖k ≤+∞∑

m=1

‖z(n)m ‖k−1+m ≤ 2α.

The above argument was independent of the choice of n ∈ N which means that forany k ∈ N0, n ∈ N and for arbitrary self-adjoint x1, . . . , xn ∈ Uk+1 there is z ∈ 2αUk

such that x j ≤ z for all j = 1, . . . , n. Since (Uk)k is a basis of zero neighbourhoodsthis implies that Ssa is quasi-directed. ��

Now we are ready to prove the following result.

Theorem 5 Null sequences in the non-commutative Schwartz space factor. Conse-quently, all maximal one-sided ideals of S are closed.

Proof Let (xn)n∈N be a null sequence in S. By [21, Lemmata 8 and 12] we have

xn = (xn x∗n )

14 yn, ‖yn‖k,∞ ≤ √

2‖xn‖4k,∞ (n ∈ N),

where ‖a‖k,∞ = sup{|ai j |(i j)k : i, j ∈ N} for a ∈ S. This implies that the sequences

(yn)n∈N ∈ S and (zn)n∈N ∈ S+, zn := (xn x∗n )

14 both tend to zero. By Proposition 4

and [9, Th. 1] there is a positive element z ∈ S which majorizes the sequence (zn)n ,

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606 K. Piszczek

i.e. zn ≤ z for all n ∈ N. Now we apply [21, Lemma 12] to the pairs (z, zn) and weobtain a decomposition

z12n = z

14 wn, ‖wn‖k,∞ ≤ √

2max

{∥∥∥z

14

∥∥∥2k,∞ ,

∥∥∥∥z

12n

∥∥∥∥

12

4k,∞

}

(n ∈ N). (3)

Consequently,

xn = zn yn = z14

(wnz

14 wn yn

):= z

14 tn (n ∈ N).

By (3) the sequence (wnz14 wn)n∈N is bounded and yn → 0 therefore the sequence

(tn)n∈N is a null sequence and the proof is complete. ��Remark The �∞-norms of the sequence (wn)n∈N in (3) can in fact be controlled in

a more convenient way. More precisely, since the squre root is an operator monotonefunction on [0,+∞)—see [19, Prop. 1.3.8], we can combine this fact with [21, Cor.

6, Lemma 8] in order to get ‖wn‖k,∞ ≤ √2‖z

12 ‖

124k,∞ for all n ∈ N.

4 Description of closed one-sided ideals

The description we are going to provide is analogous to the characterization of closedone-sided ideals of the algebra of compact (or nuclear) operators on a Hilbert space– see [16, Th. (III. 2.5)]. The proof of this result relies strongly on the fact thatevery closed subspace in a Hilbert space is complemented which is not the case forthe Schwartz space. Therefore we choose to adapt the proof of [15, Prop. 7.3]. Thegeneral idea is the same however in several points we have to appeal to the fact thatwe are dealing precisely with the Schwartz space instead of just a Banach space withthe approximation property (which is the case in [15]).

Recall for convenience that

s =⎧⎨

⎩ξ = (ξ j ) j∈N ⊂ C

N : |ξ |2k :=+∞∑

j=1

|ξ j |2 j2k < +∞ for all k ∈ N0

⎫⎬

⎭.

In particular the zeroth norm ‖·‖0 is just the �2-norm of a rapidly decreasing sequenceand we can compute the inner product 〈·, ·〉 of two vectors in s. In this case this innerproduct is in fact given by the duality (1). The definition below which we start withgoes back to Grønbæk’s paper. Let F be a closed subspace in s and let I be a closedleft ideal of S. We define

�(F) := lin{ξ ⊗ η : ξ ∈ s, η ∈ F},�(I ) := {η ∈ s : ∃ ξ �= 0 : ξ ⊗ η ∈ I }.

Remark It can be easily observed that �(F) is a closed left ideal of S. Moreover,�(I ) is a closed subspace of s. Indeed, take any non-zero ζ ∈ s and let η ∈ �(I ). By

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One-sided ideals of the non-commutative... 607

definition there is some non-zero element ξ ∈ s such that ξ ⊗ η ∈ I . Since I is a leftideal we get

ζ ⊗ η =(

ζ ⊗ 1

‖ξ‖20ξ

)

(ξ ⊗ η) ∈ I.

Therefore

�(I ) = {η ∈ s | ∀ ξ ∈ s : ξ ⊗ η ∈ I }.

Let now ηn → η, (ηn)n∈N ⊂ �(I ). For arbitrary ξ ∈ s we have

ξ ⊗ η = limn

ξ ⊗ ηn ∈ I

since I is closed. Consequently, η ∈ �(I ).

Proposition 6 Let F be a closed subspace in s and let I be a closed left ideal of S.Then

(i) I = �(�(I )),(ii) F = �(�(F)).

Proof (i) The inclusion ‘⊃’ is clear. To show the other one choose x ∈ I and for every

n ∈ N define un :=(

In 00 0

)

, where In is the n × n identity map. Then for all n ∈ N

we have un x ∈ I and dim im un x < +∞. We may suppose that

un x =mn∑

j=1

ξ j ⊗ η j , ξ j , η j ∈ s, j = 1, . . . , mn (4)

and the vectors (ξ j ) j are linearly independent. By the Gram–Schmidt orthogonaliza-tion we obtain orthogonal vectors e1, . . . , emn ∈ �2 satisfying

〈ξmn , emn 〉 �= 0 and 〈ξ j , emn 〉 = 0 ( j = 1, . . . , mn − 1). (5)

But e j ’s are linear combinations of ξ j ’s therefore (e j )mnj=1 ⊂ s. By (5) this leads to

ξmn ⊗ ηmn = 1

〈ξmn , emn 〉(ξmn ⊗ emn )un x ∈ I.

Reenumerating the indices in (4) if necessary,

ξ j ⊗ η j ∈ I ( j = 1, . . . , mn).

Therefore

η j ∈ �(I ) ( j = 1, . . . , mn).

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608 K. Piszczek

Consequently, un x ∈ �(�(I )) for all n ∈ N. By [21, Prop. 2] (un)n∈N is a sequentialapproximate identity therefore x ∈ �(�(I )) and the desired equality follows.

(ii) This time the inclusion ‘⊂’ is clear. Let now η ∈ �(�(F)) and take ξ �= 0such that ξ ⊗ η ∈ �(F). By definition

ξ ⊗ η = limn→+∞ xn, xn =

mn∑

j=1

ξnj ⊗ ηn

j , ξ ∈ s, ηnj ∈ F (n ∈ N, j = 1, . . . , mn).

Let ζ := 1‖ξ‖20

and define ηn := ∑mnj=1〈ξn

j , ζ 〉ηnj . Then each ηn ∈ F and (ξ ⊗ ζ )xn =

ξ ⊗ ηn . This implies that

ξ ⊗ η = limn→+∞ ξ ⊗ ηn .

Consequently, for every k ∈ N0 we have

‖η − ηn‖k = 1

‖ξ‖k‖ξ ⊗ η − ξ ⊗ ηn‖k → 0.

Therefore η ∈ F and the proof is thereby complete. ��Theorem 7 Let I be a closed one-sided ideal of the non-commutative Schwartz space.

(i) If I is a left ideal then there is a closed subspace E ⊂ s′ such that

I = {x ∈ S : x |E ≡ 0}.

(ii) If I is a right ideal then there is a closed subspace E ⊂ s′ such that

I = {x ∈ S : im x = E⊥}.

Proof (i) Let I be a closed left ideal of S and define E := ∩η∈�(I ) ker η ⊂ s′ andJ := {x ∈ S : x |E ≡ 0}. It is clear that �(�(I )) ⊂ J and so by Proposition 6(i)we get I ⊂ J . On the other hand J = �(�(J )) since it is a closed left ideal of S.Suppose now η ∈ �(J ). Then�(J )⊥ ⊂ (C·η)⊥ and by theBipolar Theorem [17,Th. 22.13] C · η ⊂ �(I ) (recall these are closed subspaces). Consequently,�(J ) ⊂ �(I ) and �(�(J )) ⊂ �(�(I )). Thus again by Proposition 6(i) J ⊂ Iand the equality follows.

(ii) If I is a closed right ideal then I ∗ := {x∗ : x ∈ I } is a closed left ideal. By (i)there is E ⊂ s′ such that I ∗ = {x∗ : x∗|E ≡ 0}. Using now the duality (1) it iseasy to obtain the desired conclusion.

Formaximal left ideals the above characterizationmay be rephrased in the languageof fixed ideals. Following [6] a fixed ideal is a left ideal I of the form

I = Iξ := {x ∈ S : x(ξ) = 0} (ξ ∈ s′\{0}).

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One-sided ideals of the non-commutative... 609

Corollary 8 A left ideal of the non-commutative Schwartz space is maximal if andonly if it is fixed.

Proof If I is a maximal left ideal then by Theorem 5 it is closed. Therefore by Theo-rem 7(i) I = {x ∈ S : x |E ≡ 0} for some closed subspace E ⊂ s′. Maximality of Igives now dim E = 1. On the other hand every fixed ideal is maximal. Indeed, let Iξbe a fixed ideal and take y ∈ S with 0 �= y(ξ) ∈ s. This means that 〈y(ξ), e j 〉 for somecoordinate functional e j : s → C, e j (η) := η j . But such a functional is easily seen tobe a rapidly decreasing sequence therefore there exists μ ∈ s such that 〈y(ξ), μ〉 = 1.For any n ∈ N denote ξn := (ξ1, . . . , ξn, 0, 0, . . .). Then (un − (ξn ⊗ μ)y)(ξ) = 0.Thus un − (ξn ⊗ μ)y ∈ Iξ and, consequently,

un = un − (ξn ⊗ μ)y + (ξn ⊗ μ)y ∈ 〈Iξ ∪ {y}〉 (n ∈ N).

Since (un)n∈N ⊂ S is an approximate identity, we arrive at S = 〈Iξ ∪ {y}〉 and theproof is complete. ��Acknowledgments The research of the author has been supported in the years 2014–2017 by the NationalCenter of Science, Poland, Grant no. DEC-2013/10/A/ST1/00091.

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