Module homomorphisms from Frechet algebras

Document Type: Research Paper

Author

Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Shahrekord‎, ‎P.O‎. ‎Box 88186-34141‎, ‎Shahrekord‎, ‎Iran.

Abstract

We first study some properties‎ ‎of $A$-module homomorphisms $\theta:X\rightarrow Y$‎, ‎where $X$‎ ‎and $Y$ are Fréchet $A$-modules and $A$ is a unital‎ ‎Fréchet algebra‎. ‎Then we show that if there exists a‎ ‎continued bisection of the identity for $A$‎, ‎then $\theta$ is‎ ‎automatically continuous under certain condition on $X$‎. ‎In‎ ‎particular‎, ‎every homomorphism from $A$ into certain‎ ‎Fréchet algebras (including Banach algebra) is automatically‎ ‎continuous‎. ‎Finally‎, ‎we show that every unital Fréchet‎ ‎algebra with a continued bisection of the identity‎, ‎is‎ ‎functionally continuous‎.

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W.G‎. ‎Bade and P.C‎. ‎Curtis Jr.‎, ‎Homomorphisms of commutative Banach algebras‎, ‎ Amer‎. ‎J‎. ‎Math. 82 (1960) 589--608‎.

S‎. ‎Banach‎, ‎Théorie des Opérations Linéaires‎, ‎Monografje Matematyczne 1‎, ‎Polish Scientific Publishers‎, ‎Warsaw‎, 1932, ‎Reprinted Chelsea‎, ‎New York‎, ‎1955‎.

H.G‎. ‎Dales‎, ‎Banach Algebras and Automatic Continuity‎, ‎London Math‎. ‎Soc‎. ‎Monogr‎. ‎Ser‎. ‎24‎, ‎Clarendon Press‎, ‎Oxford‎, ‎2000‎.

T‎. ‎Figiel‎, ‎An example of infinite dimensional reflexive Banach space non-isomorphic to its Cartesian square‎, ‎ Studia Math. 42 (1972) 295--306‎.

B.E‎. ‎Johnson‎, ‎Continuity of homomorphisms of algebras of operators‎, ‎ J‎. ‎Lond‎. ‎Math‎. ‎Soc. 42 (1967) 537--541‎.

B.E‎. ‎Johnson‎, ‎Continuity of homomorphisms of Banach $G$-modules‎, ‎ Pacific J‎. ‎Math. 120 (1985)‎, ‎no‎. ‎1‎, ‎111--121‎.

K.W‎. ‎Jun‎, ‎Y.W‎. ‎Lee and D.W‎. ‎Park‎, ‎The continuity of derivations and module homomorphisms‎, ‎ Bull‎. ‎Korean Math‎. ‎Soc. 27 (1990)‎, ‎no‎. ‎2‎, ‎197--206‎.

N.J‎. ‎Laustsen‎, ‎On ring-theoretic (in)finiteness of Banach algebras of operators on Banach spaces, Glasg‎. ‎Math‎. ‎J. 45 (2003)‎, ‎no‎. ‎1‎, ‎11--19‎.

‎R.J‎. ‎Loy and G.A‎. ‎Willis‎, ‎Continuity of derivations on $B(E)$ for certain Banach spaces $E$‎, ‎ J‎. ‎Lond‎. ‎Math‎. ‎Soc‎. ‎(2) 40 (1989)‎, ‎no‎. ‎2‎, ‎327--346‎.

C.P‎. ‎Ogden‎, ‎Homomorphisms from $mathfrakB(ell (omega_eta))$‎, ‎ J‎. ‎Lond‎. ‎Math‎. ‎Soc‎. ‎(2) 54 (1996)‎, ‎no‎. ‎2‎, ‎346--358‎.

W‎. ‎Rudin‎, ‎Functional Analysis‎, ‎McGraw-Hill‎, ‎2nd edition‎, ‎New York‎, ‎1991‎.

A.M‎. ‎Sinclair‎, ‎Homomorphisms from $C^ast$-algebras‎, ‎ Proc‎. ‎Lond‎. ‎Math‎. ‎Soc‎. ‎(3) 29 (1974)‎, ‎no‎. ‎3‎, ‎435--452‎.