Continuous multiplicative maps of Toeplitz algebras

We characterize continuous, nondegenerate multiplicative maps on the algebra of upper-triangular Toeplitz matrices. As an application, we show that the continuous multiplicative isometries of (for any norm) are necessarily similarities or similarities composed with complex conjugation; for unitarily...

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Bibliographic Details
Published in:Linear & multilinear algebra 2017-10, Vol.65 (10), p.2011-2023
Main Author: Poon, Edward
Format: Article
Language:English
Subjects:
15A60
Algebra
Analogies
Conjugation
isometry
multiplicative map
Norms
preserver
Toeplitz algebra
unitarily invariant norm
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Summary:We characterize continuous, nondegenerate multiplicative maps on the algebra of upper-triangular Toeplitz matrices. As an application, we show that the continuous multiplicative isometries of (for any norm) are necessarily similarities or similarities composed with complex conjugation; for unitarily invariant norms, such isometries are precisely the unitary similarities, or unitary similarities composed with complex conjugation. We partially extend these results to algebras generated by non-derogatory matrices.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2017.1283389