Weakly compact homomorphisms

Galé, J. E.; Ransford, T. J.; White, M. C.

doi:10.1090/S0002-9947-1992-1085942-9

Weakly compact homomorphisms
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by J. E. Galé, T. J. Ransford and M. C. White PDF

Trans. Amer. Math. Soc. 331 (1992), 815-824 Request permission

Abstract:

We study the structure of weakly compact homomorphisms between Banach algebras. In particular, it is shown that between many pairs of algebras, the only weakly compact homomorphisms are those of finite rank.

References

Bernard Aupetit, Propriétés spectrales des algèbres de Banach, Lecture Notes in Mathematics, vol. 735, Springer, Berlin, 1979 (French). MR 549769
Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029
Man Duen Choi and Edward G. Effros, Nuclear $C^*$-algebras and the approximation property, Amer. J. Math. 100 (1978), no. 1, 61–79. MR 482238, DOI 10.2307/2373876
P. C. Curtis Jr. and R. J. Loy, The structure of amenable Banach algebras, J. London Math. Soc. (2) 40 (1989), no. 1, 89–104. MR 1028916, DOI 10.1112/jlms/s2-40.1.89
W. J. Davis, T. Figiel, W. B. Johnson, and A. Pełczyński, Factoring weakly compact operators, J. Functional Analysis 17 (1974), 311–327. MR 0355536, DOI 10.1016/0022-1236(74)90044-5

Linear operators

José E. Galé, Weakly compact homomorphisms and semigroups in Banach algebras, J. London Math. Soc. (2) 45 (1992), no. 1, 113–125. MR 1157555, DOI 10.1112/jlms/s2-45.1.113

Weakly compact homomorphisms between Jordan-Banach algebras

F. Ghahramani, Compact homomorphisms of $C^*$-algebras, Proc. Amer. Math. Soc. 103 (1988), no. 2, 458–462. MR 943066, DOI 10.1090/S0002-9939-1988-0943066-7
Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
Herbert Kamowitz, Compact endomorphisms of Banach algebras, Pacific J. Math. 89 (1980), no. 2, 313–325. MR 599123
Herbert Kamowitz and Dennis Wortman, Endomorphisms of an extremal algebra, Proc. Amer. Math. Soc. 104 (1988), no. 3, 852–858. MR 931733, DOI 10.1090/S0002-9939-1988-0931733-0
Herbert Kamowitz, Stephen Scheinberg, and Dennis Wortman, Compact endomorphisms of Banach algebras. II, Proc. Amer. Math. Soc. 107 (1989), no. 2, 417–421. MR 984801, DOI 10.1090/S0002-9939-1989-0984801-2
A. A. Kirillov, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt. MR 0412321
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367
Lynn H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0054173
Martin Mathieu, Weakly compact homomorphisms from $C^*$-algebras are of finite rank, Proc. Amer. Math. Soc. 107 (1989), no. 3, 761–762. MR 998737, DOI 10.1090/S0002-9939-1989-0998737-4
Shûichi Ohno and Junzo Wada, Compact homomorphisms on function algebras, Tokyo J. Math. 4 (1981), no. 1, 105–112. MR 625122, DOI 10.3836/tjm/1270215742
Gilles Pisier, Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics, vol. 60, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986. MR 829919, DOI 10.1090/cbms/060
Gilles Pisier, Factorization of operators through $L_{p\infty }$ or $L_{p1}$ and noncommutative generalizations, Math. Ann. 276 (1986), no. 1, 105–136. MR 863711, DOI 10.1007/BF01450929
Shôichirô Sakai, Weakly compact operators on operator algebras, Pacific J. Math. 14 (1964), 659–664. MR 163185
Joseph L. Taylor, Homology and cohomology for topological algebras, Advances in Math. 9 (1972), 137–182. MR 328624, DOI 10.1016/0001-8708(72)90016-3