Generalized derivations on some convolution algebras

Generalized derivations on some convolution algebras Let G be a locally compact abelian group, $$\omega $$ ω be a weighted function on $${\mathbb {R}}^+$$ R + , and let $$\mathfrak {D}$$ D be the Banach algebra $$L_0^\infty (G)^*$$ L 0 ∞ ( G ) ∗ or $$L_0^\infty (\omega )^*$$ L 0 ∞ ( ω ) ∗ . In this paper, we investigate generalized derivations on the noncommutative Banach algebra $$\mathfrak {D}$$ D . We characterize $$\textsf {k}$$ k -(skew) centralizing generalized derivations of $$\mathfrak {D}$$ D and show that the zero map is the only $$\textsf {k}$$ k -skew commuting generalized derivation of $$\mathfrak {D}$$ D . We also investigate the Singer–Wermer conjecture for generalized derivations of $$\mathfrak {D}$$ D and prove that the Singer–Wermer conjecture holds for a generalized derivation of $$\mathfrak {D}$$ D if and only if it is a derivation; or equivalently, it is nilpotent. Finally, we investigate the orthogonality of generalized derivations of $$L_0^\infty (\omega )^*$$ L 0 ∞ ( ω ) ∗ and give several necessary and sufficient conditions for orthogonal generalized derivations of $$L_0^\infty (\omega )^*$$ L 0 ∞ ( ω ) ∗ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png aequationes mathematicae Springer Journals

Generalized derivations on some convolution algebras

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis; Combinatorics
ISSN
0001-9054
eISSN
1420-8903
D.O.I.
10.1007/s00010-017-0531-6
Publisher site
See Article on Publisher Site

Abstract

Let G be a locally compact abelian group, $$\omega $$ ω be a weighted function on $${\mathbb {R}}^+$$ R + , and let $$\mathfrak {D}$$ D be the Banach algebra $$L_0^\infty (G)^*$$ L 0 ∞ ( G ) ∗ or $$L_0^\infty (\omega )^*$$ L 0 ∞ ( ω ) ∗ . In this paper, we investigate generalized derivations on the noncommutative Banach algebra $$\mathfrak {D}$$ D . We characterize $$\textsf {k}$$ k -(skew) centralizing generalized derivations of $$\mathfrak {D}$$ D and show that the zero map is the only $$\textsf {k}$$ k -skew commuting generalized derivation of $$\mathfrak {D}$$ D . We also investigate the Singer–Wermer conjecture for generalized derivations of $$\mathfrak {D}$$ D and prove that the Singer–Wermer conjecture holds for a generalized derivation of $$\mathfrak {D}$$ D if and only if it is a derivation; or equivalently, it is nilpotent. Finally, we investigate the orthogonality of generalized derivations of $$L_0^\infty (\omega )^*$$ L 0 ∞ ( ω ) ∗ and give several necessary and sufficient conditions for orthogonal generalized derivations of $$L_0^\infty (\omega )^*$$ L 0 ∞ ( ω ) ∗ .

Journal

aequationes mathematicaeSpringer Journals

Published: Jan 15, 2018

References

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