Superderivations of \(C^*\)-algebras implemented by symmetric operators (Q1317420)

Let \({\mathfrak A}\) be a \(C^*\)-algebra and let \(\rho\) be a representation of \({\mathfrak A}\) on the Hilbert space \(H\). A closed linear operator \(\delta\) defined on the dense \(*\)-subalgebra \(D(\delta)\) of \({\mathfrak A}\) into \(B({\mathfrak H})\) is said to be a \(*\)-superderivation of \(A\) relative to the pair \((\rho,\varphi)\), where \(\varphi\) is a \(*\)-automorphism of \(D (\delta) \), if \(\delta (AB) = \delta (A) \rho (B) + \rho (\varphi (A)) \delta (B)\) and if \(\delta (\varphi (A)^*) = \delta (A)^*\) for \(A\) and \(B\) in \({\mathfrak A}\). A pair \((S,G)\), consisting of a densely defined closed operator \(S\) on \(H\) and an invertible operator in \(B(H)\), is said to implement \(\delta\) if \(\delta (A) |_{D(S)} = i(S \rho (A)-G^{- 1} \rho (A)GS) |_{D(S)}\). The domains of the unbounded maps are assumed to match up. If \(S\) is symmetric and \(G\) is selfadjoint, then \((S,G)\) is called a symmetric implementation. If the pair \((T,G)\) also implements \(\delta\) and \(T\) extends \(S\), then \((T,G)\) is said to be a \(\delta\)-extension of \((S,G)\). The author shows that there is a one-one correspondence between symmetric \(\delta\)-extensions of the symmetric implementation \((S,G)\) of the \(*\)-superderivation \(\delta\) and certain subspaces \(L\) of \(N(S)\). To get an idea of what the certain subspaces \(L\) are, let \(N_ \pm (S^*) = \{x \in D (S^*) | S^*x = \pm ix\}\) be the deficiency subspaces of \(S\). Then the Hilbert space \(D(S^*)\) with the inner product \(\langle x,y \rangle = (x,y) + (S^*x, S^*y)\) can be written \(D(S^*) = D(S) \oplus N_ + (S) \oplus N_ - (S)\) and \(N(S)\) is defined to be \(N(S) = N_ + (S) \oplus N_ - (S)\). The space \(D(S^*)\) carries the indefinite forms \(\{\cdot, \cdot\} = \langle J \cdot, \cdot \rangle\), where \(J\) is the involution of \(N(S)\) given by \(x \oplus y \to x \oplus (-y)\), and \(\{\cdot, \cdot\}_ 1 = \{QGQ \cdot, \cdot\}\), where \(Q\) is the projection of \(D(S^*)\) onto \(N(S)\). The subspaces \(L\) are exactly those which are neutral with respect to \(\{\cdot, \cdot\}_ 1\) (i.e., \(\{x,y\}_ 1 = 0\) for \(x\) and \(y\) in \(L)\) together with some additional conditions. Using the subspaces \(L\), the author proves the existence of a maximal symmetric extension of a symmetric implementation. Part of the additional conditions on the subspaces \(L\) in the previous paragraph involve the representation \(\pi^ S_ \delta (A)x = Q \rho (A)x\) of \(N(S)\) on \(D(S^*)\). In case that the deficiency indices are finite and \((S,G)\) is a maximal symmetric implementation of \(\delta\), the author shows that the representation \(\pi^ S_ \delta\) extends to a bounded representation of \({\mathfrak A}\) on \(N(S)\).