Super-derivations
Let \((A, \gamma)\) be a graded \(C^*\)-algebra, i.e., \(\gamma\) is a \(*\)- automorphism of the \(C^*\)-algebra \(A\) satisfying \(\gamma^ 2 = 1\). A linear map of a \(\gamma\)-invariant dense subalgebra \(D(d)\) of \(A\) into \(A\) is said to be a super-derivation if \(d(ab) = d(a)b + \gamma (a)d(b)\) for \(a\) and \(b\) in \(D(d)\) and if \(\gamma d = - d \gamma\). If \(\alpha\) is a one-parameter strongly continuous group of \(*\)-automorphisms of the graded \(C^*\)-algebra \((A, \gamma)\) with \(\alpha \gamma = \gamma \alpha\) and \(d\) is a super-derivation of \(A\) with \(d^ 2 \subset L\) where \(L\) is the generator of \(\alpha\), the authors show that the following conditions are equivalent: (i) \(d^ 2 = L\), (ii) \(d\) is everywhere defined (and hence bounded) and (iii) \(d\) is closed, \(D(d^ 2)\) is dense, \(d \alpha_ t = \alpha_ td\) for all \(t \in \mathbb{R}\), and \(A^ \alpha (-\varepsilon, \varepsilon) \subset D (\delta)\) for some \(\varepsilon > 0\). Here \(A^ \alpha (-\varepsilon, \varepsilon)\) is a spectral subspace of \(\alpha\). The authors remark super-derivations whose square extends to a generator of a one parameter automorphism group is an important ingredient in the definition of a quantum algebra in the work of \textit{A. Jaffe}, \textit{A. Lesniewski}, \textit{K. Osterwalder} [cf. Commun. Math. Phys. 118, No. 1, 1-14 (1988; Zbl 0656.58048)]. The authors also consider a perturbation of a super-derivation by an inner super-derivation. If \(q\) is an element of \(A\) with \(\gamma (q) = - q\), then the linear map \(\delta_ q (a) = qa - \gamma (a)q\) is a so- called inner super-derivation. If \(d\) is a super-derivation and \(\alpha\) is a one-parameter group of \(*\)-automorphisms of \(A\) with generator \(L\) satisfying \(\gamma \alpha_ t = \alpha_ t \gamma\) such that (i) \(d \alpha = \alpha d\), (ii) \(D(d^ 2)\) is a core for \(d\) and (iii) \(d^ 2 \subset L\), then the authors show that \(\overline {d^ 2} = L\), \(d_ q = d + \delta_ q\) is a super-derivation, \(L_ q = L + \delta_ \Omega\) is a generator, and \(d_ q\) and \(L_ q\) satisfy conditions i), ii), and iii). Here \(\Omega = d(q) + q^ 2\).
perturbation of a super-derivation by an inner super-derivation
scientific article; zbMATH DE number 509889