One-sided ideals of the non-commutative Schwartz space

Piszczek, Krzysztof

doi:10.1007/s00605-015-0758-z

Cited by 3 publications

(4 citation statements)

References 18 publications

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“…1.10 & Ex. 1.13] [16,17,18,19] and his forthcoming paper "The noncommutative Schwartz space is weakly amenable").…”

Section: Introductionmentioning

confidence: 99%

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Characterization of commutative algebras embedded into the algebra of smooth operators

Ciaś

2017

Bull. London Math. Soc.

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The paper deal with the noncommutative Fréchet * -algebra L(s ′ , s) of the so-called smooth opertors, i.e. linear and continuous operators acting from the space s ′ of slowly increasing sequences to the Fréchet space s of rapidly decreasing sequences. By a canonical identification, this algebra of smooth operators can be also seen as the algebra of the rapidly decreasing matrices. We give a full description of closed commutative * -subalgebras of this algebra and we show that every closed subspace of s with basis is isomorphic (as a Fréchet space) to some closed commutative * -subalgebra of L(s ′ , s). As a consequence, we give some equivalent formulation of the long-standing Quasi-equivalence Conjecture for closed subspaces of s.

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“…1.10 & Ex. 1.13] [16,17,18,19] and his forthcoming paper "The noncommutative Schwartz space is weakly amenable").…”

Section: Introductionmentioning

confidence: 99%

“…2.6]). Recently, Piszczek obtained several rusults concerning closed ideals, automatic continuity, amenability, Jordan decomposition and Grothendieck-type inequality in K ∞ (see Piszczek [16,17,18,19] and his forthcoming paper "The noncommutative Schwartz space is weakly amenable").…”

Section: Introductionmentioning

confidence: 99%

Characterization of commutative algebras embedded into the algebra of smooth operators

Ciaś

2017

Bull. London Math. Soc.

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“…The algebra L(s , s) is isomorphic as a Fréchet * -algebra to the algebra In these forms, the algebra L(s , s) usually appears and plays a significant role in K -theory of Fréchet algebras (see Bhatt [16][17][18][19] and his forthcoming paper "The noncommutative Schwartz space is weakly amenable"). Moreover, in the context of algebras of unbounded operators, the algebra L(s , s) appears in the book [20] as …”

Section: Introductionmentioning

confidence: 99%

Commutative subalgebras of the algebra of smooth operators

Ciaś

2016

Monatsh Math

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We consider the Fréchet * -algebra L(s , s) ⊆ L( 2 ) of the so-called smooth operators, i.e. continuous linear operators from the dual s of the space s of rapidly decreasing sequences to s. This algebra is a non-commutative analogue of the algebra s. We characterize closed * -subalgebras of 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.

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“…2.6]). Very recently, Piszczek obtained several rusults concerning closed ideals, automatic continuity (for positive functionals and derivations), amenability and Jordan decomposition in K ∞ (see Piszczek [16,15] and his forthcoming papers 'Automatic continuity and amenability in the non-commutative Schwartz space' and 'The noncommutative Schwartz space is weakly amenable'). Moreover, in the context of algebras of unbounded operators, the algebra L(s ′ , s) appears in the book [17] as The algebra of smooth operators can be seen as a noncommutative analogue of the commutative algebra s. The most important features of this algebra are the following:…”

Section: Introductionmentioning

confidence: 99%

Commutative subalgebras of the algebra of smooth operators

Ciaś¹

2015

Preprint

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We consider the Fréchet * -algebra L(s ′ , s) ⊆ L(ℓ2) of the so-called smooth operators, i.e. continuous linear operators from the dual s ′ of the space s of rapidly decreasing sequences into s. This algebra is a non-commutative analogue of the algebra s. We characterize all 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.

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One-sided ideals of the non-commutative Schwartz space

Abstract: We show that maximal one-sided ideals of the non-commutative Schwartz space are closed. We also characterize all closed one-sided ideals of this algebra. As a result, all maximal left ideals are fixed.

Cited by 3 publications

References 18 publications

Characterization of commutative algebras embedded into the algebra of smooth operators

Characterization of commutative algebras embedded into the algebra of smooth operators

Commutative subalgebras of the algebra of smooth operators

Commutative subalgebras of the algebra of smooth operators

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