Spectral preservers and approximate spectral preservers on operator algebras

Spectrally isometric elementary operators

“…Suppose S is a spectral isometry. By the results above, there exist two linearly independent subsets {a 1 , a 2 , a 3 } and {b 1 …”

“…(It is a fact that a unital surjective linear mapping is an isometry if and only if it is a selfadjoint spectral isometry.) As it stands, this conjecture is still open though there has been substantial progress towards it, see, e.g., [1], [8], [15], [18] and the references contained therein. The present paper aims to contribute to these studies but rather than putting additional conditions on the algebras involved, we investigate special spectral isometries on arbitrary semisimple Banach algebras, that is, we put the constraints on the operators.…”

Elementary operators - still not elementary?

“…Here we record a result that was obtained in a different way and hence is special to spectral isometries. There is also some evidence that Problem 3.1 could have an affirmative answer for all semisimple Banach algebras; see, e.g., [1] and [10]. In general, however, the question is wide open and one might therefore try to find a counterexample.…”

Spectral isometries into commutative Banach algebras

“…It has been observed, see in particular [10], that the behaviour on commutative subalgebras is vital for the conjecture to hold. Moreover, under additional hypotheses, the conjecture has even been verified for certain Banach algebras; see, e.g., [3] and [1]. This motivated us to re-visit the situation for commutative Banach algebras and to fill in some loose ends in the literature.…”