Weak properties of weighted convolution algebras

Author:
Sandy Grabiner

Translated by:

Journal:
Proc. Amer. Math. Soc. **132** (2004), 1675-1684

MSC (2000):
Primary 43A10, 43A20, 43A22, 46J45

Published electronically:
January 12, 2004

MathSciNet review:
2051128

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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that is a weighted convolution algebra on with the weight normalized so that the corresponding space of measures is the dual space of the space of continuous functions. Suppose that is a continuous nonzero homomorphism, where is also a convolution algebra. If is norm dense in , we show that is (relatively) weak dense in , and we identify the norm closure of with the convergence set for a particular semigroup. When is weak continuous it is enough for to be weak dense in . We also give sufficient conditions and characterizations of weak continuity of . In addition, we show that, for all nonzero in , the sequence converges weak to 0. When is regulated, converges to 0 in norm.

**[A]**G. R. Allan,*An inequality involving product measures*, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), Lecture Notes in Math., vol. 975, Springer, Berlin-New York, 1983, pp. 277–279. MR**697588****[BD]**W. G. Bade and H. G. Dales,*Continuity of derivations from radical convolution algebras*, Studia Math.**95**(1989), no. 1, 59–91. MR**1024275****[BDL]**W. G. Bade, H. G. Dales, and K. B. Laursen,*Multipliers of radical Banach algebras of power series*, Mem. Amer. Math. Soc.**49**(1984), no. 303, v+84. MR**743548**, 10.1090/memo/0303**[D]**H. G. Dales,*Banach algebras and automatic continuity*, London Mathematical Society Monographs. New Series, vol. 24, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications. MR**1816726****[DS]**Nelson Dunford and Jacob T. Schwartz,*Linear Operators. I. General Theory*, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR**0117523****[Gh]**F. Ghahramani,*Isomorphisms between radical weighted convolution algebras*, Proc. Edinburgh Math. Soc. (2)**26**(1983), no. 3, 343–351. MR**722565**, 10.1017/S0013091500004417**[GG1]**F. Ghahramani and S. Grabiner,*Standard homomorphisms and convergent sequences in weighted convolution algebras*, Illinois J. Math.**36**(1992), no. 3, 505–527. MR**1161980****[GG2]**F. Ghahramani and S. Grabiner,*The 𝐿^{𝑝} theory of standard homomorphisms*, Pacific J. Math.**168**(1995), no. 1, 49–60. MR**1331994****[GG3]**F. Ghahramani and S. Grabiner,*Convergence factors and compactness in weighted convolution algebras,*Canad. J. Math.**54**(2002), 303-323.**[GGM]**F. Ghahramani, J. P. McClure, and S. Grabiner,*Standard homomorphisms and regulated weights on weighted convolution algebras*, J. Funct. Anal.**91**(1990), no. 2, 278–286. MR**1058973**, 10.1016/0022-1236(90)90145-B**[GhM]**F. Ghahramani and J. P. McClure,*Automorphisms and derivations of a Fréchet algebra of locally integrable functions*, Studia Math.**103**(1992), no. 1, 51–69. MR**1184102****[Gr1]**Sandy Grabiner,*Homomorphisms and semigroups in weighted convolution algebras*, Indiana Univ. Math. J.**37**(1988), no. 3, 589–615. MR**962925**, 10.1512/iumj.1988.37.37029**[Gr2]**Sandy Grabiner,*Semigroups and the structure of weighted convolution algebras*, Conference on Automatic Continuity and Banach Algebras (Canberra, 1989), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 21, Austral. Nat. Univ., Canberra, 1989, pp. 155–169. MR**1022002****[Gr3]**Sandy Grabiner,*Weighted convolution algebras and their homomorphisms*, Functional analysis and operator theory (Warsaw, 1992) Banach Center Publ., vol. 30, Polish Acad. Sci., Warsaw, 1994, pp. 175–190. MR**1285606****[HP]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[LRRW]**R. J. Loy, C. J. Read, V. Runde, and G. A. Willis,*Amenable and weakly amenable Banach algebras with compact multiplication*, J. Funct. Anal.**171**(2000), no. 1, 78–114. MR**1742859**, 10.1006/jfan.1999.3533**[S]**Michel Solovej,*Norms of powers in the Volterra algebra*, Bull. Austral. Math. Soc.**50**(1994), no. 1, 55–57. MR**1285659**, 10.1017/S0004972700009564**[W]**G. A. Willis,*The norms of powers of functions in the Volterra algebra*, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981), Lecture Notes in Math., vol. 975, Springer, Berlin-New York, 1983, pp. 280–281. MR**697589**

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Additional Information

**Sandy Grabiner**

Affiliation:
Department of Mathematics, Pomona College, Claremont, California 91711

DOI:
https://doi.org/10.1090/S0002-9939-04-07385-X

Received by editor(s):
October 9, 2002

Published electronically:
January 12, 2004

Additional Notes:
The research for this paper was done while the author enjoyed the gracious hospitality of the Australian National University in Canberra

Communicated by:
N. Tomczak-Jaegermann

Article copyright:
© Copyright 2004
American Mathematical Society