Automatic continuity of almost multiplicative linear functionals on Fréchet algebras
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A linear functional $T$ on a Fr$\acute{\mathbf{\text{e}}}$chet algebra $(A, (p_n))$ is called \textit{almost multiplicative} with respect to the sequence $(p_n)$, if there exists $\varepsilon\geq0$ such that $|Tab - Ta Tb|\leq \varepsilon p_n(a) p_n(b)$ for all $n \in \mathbb{N}$ and for every $a, b \in A$. We show that an almost multiplicative linear functional on a Fr$\acute{\mathbf{\text{e}}}$chet algebra is either multiplicative or it is continuous, and hence every almost multiplicative linear functional on a functionally continuous Fr$\acute{\mathbf{\text{e}}}$chet algebra is continuous.
Abstract
A linear functional $T$ on a Fr$\acute{\mathbf{\text{e}}}$chet algebra $(A, (p_n))$ is called \textit{almost multiplicative} with respect to the sequence $(p_n)$, if there exists $\varepsilon\geq0$ such that $|Tab - Ta Tb|\leq \varepsilon p_n(a) p_n(b)$ for all $n \in \mathbb{N}$ and for every $a, b \in A$. We show that an almost multiplicative linear functional on a Fr$\acute{\mathbf{\text{e}}}$chet algebra is either multiplicative or it is continuous, and hence every almost multiplicative linear functional on a functionally continuous Fr$\acute{\mathbf{\text{e}}}$chet algebra is continuous.
참고문헌 (20)
- Esmaeeil Ansari-Piri, "The nearly additive maps", 대한수학회보, 46(2), pp.199~207 (2009), 대한수학회
- J. Baker, "The stability of the cosine equation", Proc. Amer. Math. Soc, 80(3), pp.411~416 (1980)
- H. G. Dales, "Banach Algebras and Automatic Continuity" (2000), Clarendon Press
- M. Fragoulopoulou, "Uniqueness of topology for semisimple LFQ-algebras", Proc. Amer. Math. Soc, 117(4), pp.963~969 (1993)
- M. Fragoulopoulou, "Topological Algebras with Involution" (2005), Elsevier
- H. Goldmann, "Uniform Fr´echet Algebras" (1990), North Holland
- T. G. Honary, "Automatic continuity of homomorphisms between Banach algebras and Frechet algebras", Bull. Iranian Math. Soc, 32(2), pp.1~11 (2006)
- R. A. J. Howey, "Approximately multiplicative functionals on algebras of smooth func-tions", J. London Math. Soc. (2), 68(3), pp.739~752 (2003)
- T. Husain, "Multiplicative Functionals on Topological Algebras" (1983), Pitman Books Limited
- K. Jarosz, "Perturbations of Banach Algebras" (1985), Springer-Verlag
- K. Jarosz, "Almost multiplicative functionals", Studia Math, 124(1), pp.37~58 (1997)
- B. E. Johnson, "Approximately multiplicative functionals", J. London Math. Soc, 34(2), pp.489~510 (1986)
- B. E. Johnson, "Approximately multiplicative maps between Banach algebras", J. London Math. Soc, 37(2), pp.294~316 (1988)
- E. Kaniuth, "A Course in Commutative Banach Algebras", Graduate Texts in Mathematics, 246 (2000), Springer
- G. Krishna Kumar, "Linear maps preserving pseudospectrum and condition spectrum", Banach J. Math. Anal, 6(1), pp.45~60 (2012)
- S. H. Kulkarni, "Almost multiplicative functions on commutative Banach algebras", Studia Math, 197(1), pp.93~99 (2010)
- H. Lin, "Almost multiplicative morphisms and some applications", J. Operator Theory, 37(1), pp.121~154 (1997)
- A. Mallios, "Topological Algebras" (1986), North Holland
- E. A. Michael, "Locally multiplicatively convex topological algebras", Mem. Amer. Math. Soc, 1952(11), pp.1~82 (1952)
- P. Semrl, "Almost multiplicative functions and almost linear multiplicative functionals", Aequationes Math, 63(1-2), pp.180~192 (2002)