[Settfa] Tuesday 10/12, <h2>Krzysztof Piszczek</h2>

[kubis at math.cas.cz]

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 Tuesday 10th December, 10:00am 
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 Speaker:Krzysztof Piszczek
, Pozna?
 Title: Jordan decomposition in the non-commutative Schwartz space

 Abstract  

 If $A,B$ are $^*$-algebras with a cone of positive elements $A_+,\,\text{resp.}\,B_+$, then it is classical to consider whether continuous self-adjoint mappings $\phi\colon A\to B$ decompose into a difference of two positive ones. This question has been considered in many different frameworks (e.g. $C^*$-algebra setting). We will consider this problem in a specific Fr\'echet algebra $\mathcal{S}:=L(s',s)$ of operators acting from the dual of $s$ into $s$ itself, where
 
\[s=\big\{\xi=(\xi_j)_{j\in\mathbb{N}}\subset\mathbb{C}^{\mathbb{N}}\colon\,\,|\xi|_k^2:=\sum_{j=1}^{+\infty}|\xi_j|^2j^{2k}<+\infty\,\,\text{for all}\,\,k\in\mathbb{N}\big\}\]
 
is the so-called \textit{space of rapidly decreasing sequences}. The algebra $\mathcal{S}$ shares some common features with the $C^*$-algebra of compact operators on the separable Hilbert space $\ell_2$ but there are some essential differences as well. During the talk we will show that there are indecomposable self-adjoint mappings $\phi\colon\mathcal{S}\to\mathcal{S}$. On the other hand, if we add a unit to the range space then there is a constructive way to decompose any self-adjoint $\phi\colon\mathcal{S}\to\mathcal{S}_1$.

 For more information see the seminar web page at 
 https://calendar.math.cas.cz/set-theory-and-analysis-actual .