Derivations on the algebra of operators in Hilbert \(C^*\)-modules
A derivation of an algebra \(A\) is a linear mapping \(\delta: A \to A\) such that \(\delta(ab)=\delta(a)b+a\delta(b)\). A derivation of the form \(\delta(a)=ac_0-bc_0\) for some fixed \(c_0\in A\) is called an inner derivation. One of the interesting problems in the theory of derivations is to identify those algebras on which all the derivations are inner. It is known that every derivation of a \(W^*\)-algebra is inner; see \textit{S. Sakai} [Ann. Math. (2) 83, 273--279 (1966; Zbl 0139.30601)]. In the paper under review, the authors treat this problem in the setting of Hilbert \(C^*\)-modules. Let \(M\) be a full Hilbert \(C^*\)-module over a \(C^*\)-algebra \(A\), and let \(\mathrm{End}_A^*(M)\) be the algebra of all adjointable operators on \(M\). The authors prove that if \(A\) is unital and commutative, then every derivation of \(\mathrm{End}_A^*(M)\) is an inner derivation, and that if \(A\) is \(\sigma\)-unital and commutative, then innerness of derivations on compact operators completely decides innerness of derivations on \(\mathrm{End}_A^*(M)\). If \(A\) is unital (no commutativity is assumed) such that every derivation of \(A\) is inner, then they show that every derivation of \(\mathrm{End}_A^*(L_n(A))\) is also inner, where \(L_n(A)\) denotes the direct sum of \(n\) copies of \(A\).
scientific article; zbMATH DE number 6102781
scientific article