Completely positive maps and*-isomorphism ofC *-algebras II

LetA, B be unitalC ^*-algebras,D _A ¹ the set of all completely positive maps ϕ fromA toM _n (C), with Tr ϕ(I)≤1(n≥3). If Ψ is an α-invariant affine homeomorphism betweenD _A ¹ andD _B ¹ with Ψ (0)=0, thenA is^*-isomorphic toB. Obtained results can be viewed as non-commutative Kadison-Shultz theorems. This work is supported by the National Natural Science Foundation of China.     

Weak-local derivations and homomorphisms on C*-algebras

We prove that every weak-local derivation on a C*-algebra is continuous, and the same conclusion remains valid for weak*-local derivations on von Neumann algebras. We further show that weak-local derivations on C*-algebras and weak*-local derivations on von Neumann algebras are derivations. We also study the connections between bilocal derivations and bilocal*-automorphism with our notions of extreme-strong-local derivations and automorphisms.     

Inverse closedness ofC *-algebras in Banach algebras

LetA denote a unital Banach algebra, and letB denote aC ^*-algebra which is contained as a unital subalgebra inA. We prove thatB is inverse closed inA if the norms ofA andB coincide. This generalizes well known result about inverse closedness ofC ^*-subalgebras inC ^*-algebras.     

Continuity of generalized derivations on JB*-algebras

We prove that every generalized Jordan derivation D from a JB?-algebra 𝒜 into itself or into its dual space is automatically continuous. In particular, we establish that every generalized Jordan derivation from a C?-algebra to a Jordan Banach module is continuous. As a consequence, every generalized derivation from a C?-algebra to a Banach bimodule is continuous.     

Inner ideals in C*-algebras

Research supported by a grant from the Schweizerische Nationalfonds/Fonds national suisse     

Derivations on ideals in commutative AW*-algebras

LetA be a commutativeAW*-algebra.We denote by S(A) the *-algebra of measurable operators that are affiliated with A. For an ideal I in A, let s(I) denote the support of I. Let Y be a solid linear subspace in S(A). We find necessary and sufficient conditions for existence of nonzero band preserving derivations from I to Y. We prove that no nonzero band preserving derivation from I to Y exists if either Y ? Aor Y is a quasi-normed solid space. We also show that a nonzero band preserving derivation from I to S(A) exists if and only if the boolean algebra of projections in the AW*-algebra s(I)A is not σ-distributive.