## A note on a paper by Piszczek on Jordan decomposition in noncommutative Schwartz space

### Volume 148 / 2017

Colloquium Mathematicum 148 (2017), 225-230
MSC: Primary 46K05; Secondary 46H35.
DOI: 10.4064/cm6922-6-2016
Published online: 10 March 2017

#### Abstract

The positive cone in a closed $^*$-subalgebra of the noncommutative Schwartz algebra of smooth operators is normal, which immediately implies decomposition of a continuous self-adjoint functional as a difference of two positive functionals. A~decomposition as a difference of two representable positive functionals holds precisely for self-adjoint functionals continuous in the $C^*$-operator norm.