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Group Algebra Modules. I

  • S. L. Gulick (a1), T. S. Liu (a1) and A. C. M. Van Rooij (a1)

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Some time ago, J. G. Wendel proved that the operators on the group algebra L1(G) which commute with convolution correspond in a natural way to the measure algebra M(G) (13). One might ask if Wendel's theorem can be restated in a more general setting. It is this question that is the point of departure for our present paper. Let K be a Banach module over L1(G). Our interest is in operators from L1(G) into K, and from K into L(G), which commute with the module composition (where L(G) is thought of as a module over L1(G) also). Such operators we call (L1(G), K)- and (K, L(G))-homomorphisms, respectively. Investigations of various other kinds of module homomorphisms occur in A. Figà-Talamanca (6) and B. E. Johnson (9; 10).

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References

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1. Arens, R. F., Operations in function classes, Monatsh. Math., 55 (1951), 119.
2. Arens, R. F., The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2 (1951), 839848.
3. Civin, P., Ideals in the second conjugate algebra of a group algebra, Math. Scand., 11 (1962), 161174.
4. Civin, P. and Yood, B., The second conjugate space of a Banach algebra as an algebra, Pacific J. Math., 11 (1961), 847870.
5. Cohen, P. J., Factorization in group algebras, Duke Math. J., 26 (1959), 199205.
6. Figà-Talamanca, A., Multipliers of p-integrable functions, Bull. Math. Soc., 70 (1964), 666669.
7. Gulick, S. L., Liu, T. S., and van Rooij, A. C. M., Group algebra modules, II (to appear).
8. Hewitt, E. and Ross, K. A., Abstract harmonic analysis, Part I (Berlin, 1963).
9. Johnson, B. E., An introduction to the theory of centralizers, Proc. London Math. Soc., 14 (1964), 299320.
10. Johnson, B. E., Centralisers on certain topological algebras, J. London Math. Soc., 39 (1964), 603614.
11. Kelley, J. L., Namioka, I., and co-authors, Linear topological spaces (New York, 1963).
12. Rudin, W., Fourier analysis in groups (New York, 1962).
13. Wendel, J. G., Left centralizers and isomorphisms of group algebras, Pacific J. Math., 2 (1952), 251261.
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