Maps on Banach algebras which spectrally preserve algebraic operations
Surjective linear maps between Banach algebras which preserve spectral proper- ties have been extensively studied. Another direction is where linearity is dropped but some algebraic aspects are preserved spectrally. For instance, for a surjec- tive map φ between unital complex Banach algebras A and B one can say that \(\varphi\) is “spectrally additive” if the spectrum of \(x + y\) is equal to the spectrum of \(\varphi(x) + \varphi(y)\) for each \(x, y \in A\). Notice that continuity, linearity or even additivity of \(\varphi\) are not requirements in the previous definition. Similarly, one may assume that \(\varphi\) is “spectrally multiplicative”. Do any of these restrictions on φ force it to be linear or perhaps even an isomorphism? By establishing a new characterization of rank one elements in a Banach algebra, one which only uses addition and invertibility, we show that any spectrally additive map acting on a semisimple domain preserves rank one elements in both directions. This settles an open question raised recently in [1]. Furthermore, by building on the techniques surrounding preservers and the spectral rank, trace and determinant which originated in the aforementioned paper, we prove that if \(A\) is semisimple and either \(A\) or \(B\) has an essential socle, then any spectrally additive map \(\varphi : A \to B\) is a continuous Jordan-isomorphism. If we also assume that either \(A\) or \(B\) is prime, we can conclude that \(\varphi\) is either a continuous algebra isomorphism or anti-isomorphism. This then shows that it is possible to derive a Kowalski-Slodkowski type theorem for spectrum-preserving maps. In fact, this extends many classical results on linear or additive spectrum-preserving maps (see for instance [2] or [3]). In a similar vein, we can also prove results for spectrally multiplicative maps. Here it is actually enough to assume that the multiplicative condition only holds for certain subsets of the spectrum; namely, the peripheral spectrum and the boundary of the spectrum, respectively. For further details on these results see [4].
This is joint work with Miles Askes (University of Johannesburg) and Rudi Brits (University of Johannesburg), and a part of this project is supported by the National Research Foundation of South Africa (Grant Number: 129692).
References
[1] M. Askes, R. Brits and F. Schulz, Spectrally additive group homomorphisms on Banach alge- bras, J. Math. Anal. Appl. 508 (2022), 125910.
[2] B. Aupetit and H. du T. Mouton, Spectrum preserving linear mappings in Banach algebras, Stud. Math. 109 (1994), 91–100.
[3] M. Omladiˇc and P. ˇSemrl, Spectrum-preserving additive maps, Linear Algebra Appl. 153 (1991) 67–72.
[4] F. Schulz, A note on peripherally multiplicative maps on Banach algebras, Ann. Funct. Anal. 10 (2019) 218–228.