Geodesic
Algebraic Inner Derivations on Operator Algebras
Canadian journal of mathematics, Tome 35 (1983) no. 4, pp. 710-723

Voir la notice de l'article provenant de la source Cambridge University Press

Let A be a C*-algebra, let p be a polynomial over C, and let a in M(A) (the multiplier algebra of A) be such that p(ad a) = 0. In this paper we study the following problem: when does there exist λ in Z(M(A)) (the centre of M(A)) such that p(a – λ) = 0? The first result of this type known to us is due to I. N. Herstein [7], who showed that for a simple ring with identity, such a λ always exists when p is of the form p(x) = xk for some positive integer k. Later, in [8], C. R. Miers showed that the result is true for any primitive unital C*-algebra and any polynomial whatever. It was also shown in [8] that if A is a unital C*-algebra acting on H and p is any polynomial, then such a λ exists in the larger algebra Z(A′′). In particular, the strict result holds for any von Neumann algebra, A. 
Miers, C. Robert; Phillips, John. Algebraic Inner Derivations on Operator Algebras. Canadian journal of mathematics, Tome 35 (1983) no. 4, pp. 710-723. doi: 10.4153/CJM-1983-040-6
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