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ON MULTIPLICATIVE MAPS INTO THE SPECTRUM
Keywords:
$C^*$-algebra, von Neumann algebra, positive operator, spectrum.DOI:
https://doi.org/10.17654/0972087124008Abstract
For a fixed integer $n \geq 2$, denote by $\mathcal{M}_n(\mathbb{C})$ the algebra of all complex matrices. Consider a positive matrix $P$ so that $P^2=P$. Then we characterize multiplicative maps $\phi: P \mathcal{M}_n(\mathbb{C}) P \rightarrow \mathcal{M}_n(\mathbb{C})$ satisfying $\sigma\left(\phi(P A P), \mathcal{M}_n(\mathbb{C})\right) \backslash\{0\}=\sigma(P A P, \mathfrak{A}) \backslash\{0\}$ for any $A \in$ $\mathcal{M}_n(\mathbb{C})$.
Received: March 7, 2024
Accepted: April 13, 2024
References
C. E. Rickart, General theory of Banach algebras, University Series in Higher Mathematics, van Nostrand, 1960.
R. Brits, M. Mabrouk and C. Toure, A multiplicative Gleason-Kahane-Zelazko theorem for -algebras, J. Math. Anal. Appl. 500(1) (2021), 125089.
S. H. Hochwald, Multiplicative maps on matrices that preserve the spectrum, Linear Algebra Appl. 212/213 (1994), 339-351.
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