Spectrally bounded generalized inner derivations

Abstract

Abstract. We characterize the generalized inner derivations on a unital Banach algebra which are spectrally bounded. In particular, a simplified argument for the recent result due to Bresar that every spectrally bounded inner derivation maps into the radical is given.

Spectrally isometric elementary operators

“…To begin with, the simple identity r(M a,b x) = r(bax) = r(M ba,1 x) together with Pták's description of spectrally bounded one-sided multiplications ( [21], see also [9] for an alternative proof) tells us that M a,b is spectrally bounded if and only if ba ∈ Z(A). Now if ab = M a,b 1 = 1 then ba = baab = abab = 1 too.…”

“…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?

“…Then xa − ax ∈ rad(A), the Jacobson radical of A, for all x ∈ A if and only if r(ax) ≤ r(a) r(x) for all x ∈ A; the latter condition is easily seen to be equivalent to spectral boundedness of L a and, since r(ax) = r(xa) for all x, equally equivalent to spectral boundedness of R a . It follows, see also [11], that the inner derivation R a − L a maps A into its radical rad(A) if and only if it is spectrally bounded. In fact, it was shown in [9] that an arbitrary (not necessarily continuous) derivation d on A maps into rad(A) if and only if d is spectrally bounded.…”

“…Clearly this condition is not sufficient. For example, L a − R a is spectrally bounded on a semisimple Banach algebra A if and only if L a − R a = 0, equivalently, a ∈ Z(A); see the discussion in Section 2 and [11]. However a − a ∈ Z(A) for every a ∈ A.…”

“…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.…”