Elementary Operators and Completely Bounded Mappings

Abstract

Among the classes of bounded linear operators on C*-algebras that allow a fairly detailed analysis are the elementary operators, which are of the form

$$ S:x \mapsto \sum\limits_{j = 1}^n {a_j xb_j (x \in A)} $$

for given n-tuples a = (a_l,…,a_n), b = (b_l,…,b_n) ∈ M(A) ⁿ, A a C*algebra. Elementary operators, in the context of matrix algebras, were first studied by Stéphanos and Sylvester around the turn of the 19th century. In the last decades of the 20th century, both spectral and structural properties of elementary operators were thoroughly investigated, and this class of operators has found manifold applications, not only in operator theory, but also in non-commutative algebraic geometry [34] and soliton physics [81], among others. Elementary operators serve as building blocks for more general types of operators, and they comprise both inner derivations and inner automorphisms, which are the topic of the previous chapter.