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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: Suppose that $L^{1}(\omega)$ is a weighted convolution algebra on $\mathbf{R}^{+}=[0,\infty)$ with the weight $\omega (t)$ normalized so that the corresponding space $M(\omega)$ of measures is the dual space of the space $C_{0}(1/\omega)$ of continuous functions. Suppose that $\phi: L^{1}(\omega)\rightarrow \ensuremath{L^{1}(\omega')} $ is a continuous nonzero homomorphism, where \ensuremath{L^{1}(\omega')} is also a convolution algebra. If $L^{1}(\omega)\ast f$ is norm dense in $L^{1}(\omega)$, we show that $\ensuremath{L^{1}(\omega')}\ast\phi (f)$ is (relatively) weak$^{\ast}$ dense in \ensuremath{L^{1}(\omega')}, and we identify the norm closure of $\ensuremath{L^{1}(\omega')}\ast\phi (f)$ with the convergence set for a particular semigroup. When $\phi$ is weak$^{\ast}$ continuous it is enough for $L^{1}(\omega)\ast f$ to be weak$^{\ast}$ dense in $L^{1}(\omega)$. We also give sufficient conditions and characterizations of weak$^{\ast}$ continuity of $\phi$. In addition, we show that, for all nonzero $f$ in \ensuremath{L^{1}(\omega )}, the sequence $f^{n}/\vert\vert f^{n}\vert\vert$ converges weak$^{\ast}$ to 0. When $\omega$ is regulated, $f^{n+1}/\vert\vert f^{n}\vert\vert$ converges to 0 in norm.

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  • [A] G. R. Allan, An inequality involving product measures, in J. M. Bachar et al. (eds.), Radical Banach algebras and automatic continuity, 277-279, Lecture Notes in Math. #975, Springer-Verlag, New York, 1983. MR 84m:46062
  • [BD] W. G. Bade and H. G. Dales, Continuity of derivations from radical convolution algebras, Studia Math. 95 (1989), 59-91. MR 90k:46115
  • [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. MR 85j:46094
  • [D] H. G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monographs, 24, Clarendon Press, Oxford, 2000. MR 2002e:46001
  • [DS] N. Dunford and J. T. Schwartz, Linear operators, Part I, Wiley Interscience, New York, 1958. MR 22:8302
  • [Gh] F. Ghahramani, Isomorphisms between radical weighted convolution algebras, Proc. Edinburgh Math. Soc. (2) 26 (1983), 343-351. MR 85h:43002
  • [GG1] F. Ghahramani and S. Grabiner, Standard homomorphisms and convergent sequences in weighted convolution algebras, Illinois J. Math. 36 (1992), 505-527. MR 93d:46089
  • [GG2] F. Ghahramani and S. Grabiner, The $L^P$ theory of standard homomorphisms, Pacific J. Math. 168 (1995), 49-60. MR 96e:43004
  • [GG3] F. Ghahramani and S. Grabiner, Convergence factors and compactness in weighted convolution algebras, Canad. J. Math. 54 (2002), 303-323.
  • [GGM] F. Ghahramani, S. Grabiner, and J. P. McClure, Standard homomorphisms and regulated weights on weighted convolution algebras, J. Functional Anal. 91 (1990), 278-286. MR 91k:43007
  • [GhM] F. Ghahramani and J. P. McClure, Automorphisms and derivations of a Fréchet algebra of locally integrable functions, Studia Math. 103 (1992), 51-69. MR 93j:46055
  • [Gr1] S. Grabiner, Homomorphisms and semigroups in weighted convolution algebras, Indiana Univ. Math. J. 37 (1988), 589-615. MR 90f:43007
  • [Gr2] S. Grabiner, Semigroups and the structure of weighted convolution algebras, in Proceedings of the Conference on Automatic Continuity and Banach Algebras, R. J. Loy, ed., Proc. Centre Math. Anal., Australian National University, vol. 21 (1989), 155-169. MR 91c:43004
  • [Gr3] S. Grabiner, Weighted convolution algebras and their homomorphisms, in Functional Analysis and Operator Theory, Banach Center Publications 30 (1994), 175-190, Polish Acad. of Sci., Warsaw. MR 95e:43004
  • [HP] E. Hille and R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloquium Publ. 31, Providence, R.I., 1957. MR 19:664d
  • [LRRW] R. J. Loy, C. J. Read, V. Runde, and G. A. Willis, Amenable and weakly amenable Banach algebras with compact multiplication, J. Functional Analysis 171 (2000), 78-114. MR 2001h:46088
  • [S] M. Solovej, Norms of powers in the Volterra algebra, Bull. Austral. Math. Soc. 50 (1994), 55-57. MR 95g:46101
  • [W] G. A. Willis, The norms of powers of functions in the Volterra algebra, in J. M. Bachar et al. (eds.), Radical Banach algebras and automatic continuity, 345-349, Lecture Notes in Math. #975, Springer-Verlag, New York, 1983. MR 84m:46063

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Additional Information

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

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

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