## Banach Journal of Mathematical Analysis

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

### The multiplier algebra of the noncommutative Schwartz space

Tomasz Ciaś and Krzysztof Piszczek

#### Abstract

We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest ${}^{\ast}$-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**

Received: 19 July 2016

Accepted: 20 September 2016

First available in Project Euclid: 6 May 2017

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/17358787-2017-0007

**Mathematical Reviews number (MathSciNet)**

MR3679898

**Zentralblatt MATH identifier**

06754305

**Subjects**

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.)

**Keywords**

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

#### 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