## Banach Journal of Mathematical Analysis

### The multiplier algebra of the noncommutative Schwartz space

#### Abstract

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 $\mathcal{Q}$-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

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

Dates
Accepted: 20 September 2016
First available in Project Euclid: 6 May 2017

https://projecteuclid.org/euclid.bjma/1494036021

Digital Object Identifier
doi:10.1215/17358787-2017-0007

Mathematical Reviews number (MathSciNet)
MR3679898

Zentralblatt MATH identifier
06754305

#### Citation

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. https://projecteuclid.org/euclid.bjma/1494036021

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