Elementary operators on prime \(C^*\)-algebras. I
The aim of this paper is to study properties of the elementary operator \(Sx=\sum ^{n}_{j=1}a_ jxb_ j\) on a prime \(C^ *\)-algebra A, where \(a_ j,b_ j\in M(A)\), the multiplier algebra of A. In doing this, the primeness of A (in the sense of non-commutative ring theory) is considered to be a fundamental device and exploited in several ways. It is shown that the extended centroid of a prime \(C^ *\)-algebra A is trivial and thus the canonical epimorphism between \(M(A)\otimes M(A)^{op}(M(A)^{op}\) denoting the opposite algebra of M(A)) and El(A), the algebra of all elementary operators on A, is an isomorphism. This extends results of \textit{C. K. Fong} and \textit{A. R. Sourour} [Can. J. Math. 31, 845-857 (1979; Zbl 0408.47028)] for the algebra B(H) and the Calkin algebra C(H); in particular, their use of Voiculescu's non- commutative Weyl-von Neumann theorem is replaced by much more elementary techniques. Applications are characterizations of hermitian-preserving and of completely positive elementary operators on prime \(C^ *\)- algebras as well as a description of their extreme points in the corresponding convex cone [extending results by \textit{M.-D. Choi}, Linear Alg. Appl. 10, 285-290 (1975; Zbl 0327.15018) and others]. Another section of the paper deals with the spectral theory of elementary operators where a complete description is given for the various parts \(\sigma _{\ell}(S)\), AP\(\sigma\) (S), etc. and for \(\sigma\) (S) in terms of the joint spectrum in the sense of R. E. Harte; ultraweak denseness of the range of S (in the case of a \(W^ *\)-algebra) is also discussed. These results generalize many of the results on the spectra of S known so far for B(H), C(H), or other special cases only, mainly established by Fialkow in a series of papers. Using the primeness, however, opens a much more direct and simplified approach. The second part of the paper which will appear in the Glasgow Math. J. 30 (1988) contains characterizations of compact and weakly compact elementary operators.
completely positive elementary operators on prime \(C^*\)-algebras
scientific article; zbMATH DE number 4058208
scientific article