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Thursday 7 July 2022

# Maps on Banach algebras which spectrally preserve algebraic operations

Speaker: Francois Schulz, University of Johannesburg
Time: 11:00
Venue: MathSci Room 1005

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.