Elementary Operators that are Spectrally Bounded

Spectrally isometric elementary operators

“…From Theorem 3.5 in [4], see also [6,Corollary 3.7], we can deduce the following characterisation for spectral boundedness of a unital elementary operator S ∈ E 2 (A). Note that the exceptional case pointed out in [6,Corollary 3.7] cannot occur if S1 = 1.…”

“…The recent papers [4]- [6] by Boudi and Mathieu contain necessary and sufficient conditions for an elementary operator S on A to be spectrally bounded; some restrictions on the length of S had to be imposed too. We shall recall some of these results below as we will need them in the discussion on spectral isometries in the next section.…”

“…Several newer investigations have been compiled in [10]. Elementary operators that are spectrally bounded , that is, r(Sx) ≤ M r(x) for some M ≥ 0 and all x ∈ A, are investigated in [4] and [6], extending earlier work in [9], for instance. General spectrally bounded operators do not allow for a detailed structure theory; for example, every bounded linear operator from a commutative C*-algebra is spectrally bounded.…”

Elementary operators - still not elementary?

“…The conditions in the above result in particular imply that each two-sided multiplication M aj ,bj in the representation S = n j=1 M aj ,bj is spectrally bounded and there is some "orthogonality" between their ranges. However, the conditions are not necessary: by [11], So far, necessary and sufficient conditions for S ∈ E (A) to be spectrally bounded in general are only known if (S) ≤ 2; see [6]. In the remainder of this article, we will discuss the results obtained in [27] for the case (S) = 3.…”

“…Since A is semisimple, we can piece the information from the quotients A/P together to obtain a global description. For the details, see [6][7][8].…”