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On interpolation of Banach algebras and factorization of weakly compact homomorphisms. (English) Zbl 1119.46024
The family \((\cdot,\cdot)_\Gamma\) of interpolation methods was introduced by J. Peetre [“A theory of interpolation of normed spaces” (Notas Mat. No. 39) (1968; Zbl 0162.44502)]. A special class of the family \(\Gamma\) was used in [A. Blanco, S. Kaijser and T. J. Ransford, J. Funct. Anal. 217, No. 1, 126–141 (2004; Zbl 1078.46050)] to prove that weakly compact homomorphisms between Banach algebras factor through a reflexive Banach algebra. It is crucial in this factorization that the interpolation method preserves the Banach algebra structure. In the paper under review, a necessary and sufficient condition on \(\Gamma\) for the interpolation method \((\cdot,\cdot)_\Gamma\) to preserve the Banach algebra structure is found. Applying this general result, it turns out that the classical real interpolation method \((\cdot,\cdot)_{\theta,q}\) preserves Banach algebras only if \(q=1\). Also, some results on the factorization of weakly compact homomorphisms between Banach algebras are derived as applications.

46B70 Interpolation between normed linear spaces
46H05 General theory of topological algebras
46M35 Abstract interpolation of topological vector spaces
Full Text: DOI
[1] Bergh, J.; Löfström, J., Interpolation spaces. an introduction, (1976), Springer Berlin · Zbl 0344.46071
[2] Bishop, E.A., Holomorphic completion, analytic continuation, and the interpolation of seminorms, Ann. math., 78, 468-500, (1963) · Zbl 0131.30901
[3] Blanco, A.; Kaijser, S.; Ransford, T.J., Real interpolation of Banach algebras and factorization of weakly compact homomorphisms, J. funct. anal., 217, 126-141, (2004) · Zbl 1078.46050
[4] Brudnyıˇ, Y.; Krugljak, N., Interpolation functors and interpolation spaces, vol. 1, (1991), North-Holland Amsterdam · Zbl 0743.46082
[5] Calderón, A.P., Intermediate spaces and interpolation, the complex method, Studia math., 24, 113-190, (1964) · Zbl 0204.13703
[6] Cobos, F., Some spaces in which martingale difference sequences are unconditional, Bull. Polish acad. sci. math., 34, 695-703, (1986) · Zbl 0617.46079
[7] Cobos, F.; Fernández-Cabrera, L.M.; Manzano, A.; Martínez, A., Real interpolation and closed operator ideals, J. math. pures appl., 83, 417-432, (2004) · Zbl 1051.46013
[8] Cobos, F.; Fernández-Cabrera, L.M.; Manzano, A.; Martínez, A., On interpolation of asplund operators, Math. Z., 250, 267-277, (2005) · Zbl 1078.46015
[9] Cobos, F.; Peetre, J.; Persson, L.E., On the connection between real and complex interpolation of quasi-Banach spaces, Bull. sci. math., 122, 17-37, (1998) · Zbl 0953.46037
[10] Conway, J.B., A course in functional analysis, (1990), Springer New York · Zbl 0706.46003
[11] Cwikel, M.; Peetre, J., Abstract K and J spaces, J. math. pures appl., 60, 1-50, (1981) · Zbl 0415.46052
[12] Davis, W.J.; Figiel, T.; Johnson, W.B.; Pelczyński, A., Factoring weakly compact operators, J. funct. anal., 17, 311-327, (1974) · Zbl 0306.46020
[13] Galé, J.E.; Ransford, T.J.; White, M.C., Weakly compact homomorphisms, Trans. amer. math. soc., 331, 815-824, (1992) · Zbl 0761.46037
[14] Kaijser, S., Interpolation of Banach algebras and open sets, Integral equations oper. theory, 41, 189-222, (2001) · Zbl 1022.46046
[15] Levy, M., L’espace d’interpolation réel \((A_0, A_1)_{\theta, p}\) contient \(\ell^p\), C. R. acad. sci. Paris Sér. A, 289, 675-677, (1979) · Zbl 0421.46028
[16] Lindenstrauss, J.; Tzafriri, L., Classical Banach spaces, I and II, (1979), Springer New York
[17] Naimark, M.A., Normed rings, (1964), P. Noordhoff N.V. Groningen · Zbl 0137.31703
[18] Nikolskii, N.K., Spectral synthesis for a shift operator and zeros in certain classes of analytic functions smooth up to the boundary, Dokl. akad. nauk SSSR, 190, 780-783, (1970)
[19] Nilsson, P., Reiteration theorems for real interpolation and approximation spaces, Ann. mat. pura appl., 132, 291-330, (1982) · Zbl 0514.46049
[20] Nilsson, P., Interpolation of Calderón and ovchinnikov pairs, Ann. mat. pura appl., 134, 201-332, (1983)
[21] Peetre, J., A theory of interpolation of normed spaces, Notes mat., 39, 1-86, (1968), Lecture Notes, Brasilia, 1963 · Zbl 0162.44502
[22] Triebel, H., Interpolation theory, function spaces, differential operators, (1978), North-Holland Amsterdam · Zbl 0387.46033
[23] Zafran, M., The dichotomy problem for homogeneous Banach algebras, Ann. math., 108, 97-105, (1978) · Zbl 0388.46036
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