Let 1 p+. We show that the positive part of the closed unit ball of a non-commutative L p-space, as a metric space, is a complete Jordan-invariant for the underlying von Neumann algebra.

A not necessarily continuous, linear or multiplicative function from an algebra A into itself is called a local automorphism if agrees with an automorphism of A at each point in A. In this paper, we study the question when a local automorphism of a C*-algebra, or a W*-algebra, is an automorphism.

We study the control systems governed by abstract Volterra equations without uniqueness in a Banach space. By using the technique of the theory of condensing maps and multivalued analysis tools, we obtain the existence result, investigate the topological structure of the solution set, and prove the invariance of a reachability set of the control system under nonlinear perturbations. Examples concerning fractional order differential equations and first order evolution equations with multiple delays are proposed to demonstrate the applications to approximate controllability results.