Banach Journal of Mathematical Analysis

The multiplier algebra of the noncommutative Schwartz space

Tomasz Ciaś and Krzysztof Piszczek

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We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest -algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a Q-algebra nor m-convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem, or the uniform boundedness principle, are still available.

Article information

Banach J. Math. Anal., Volume 11, Number 3 (2017), 615-635.

Received: 19 July 2016
Accepted: 20 September 2016
First available in Project Euclid: 6 May 2017

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Zentralblatt MATH identifier

Primary: 47L10: Algebras of operators on Banach spaces and other topological linear spaces
Secondary: 46K10: Representations of topological algebras with involution 46H15: Representations of topological algebras 46A13: Spaces defined by inductive or projective limits (LB, LF, etc.) [See also 46M40] 46A11: Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)

(Fréchet) $m$-convex algebra (noncommutative) Schwartz space multiplier algebra $\mathrm{PLS}$-space


Ciaś, Tomasz; Piszczek, Krzysztof. The multiplier algebra of the noncommutative Schwartz space. Banach J. Math. Anal. 11 (2017), no. 3, 615--635. doi:10.1215/17358787-2017-0007.

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