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Mappings onto multiplicative subsets of function algebras and spectral properties of their products. (English) Zbl 1335.46044

Let \(A\) and \(B\) be function algebras on locally compact Hausdorff spaces \(X\) and \(Y\), \({\mathcal J}_i\) be arbitrary sets. Let \(S_i: {\mathcal J}_i \to {\mathcal S}_i\), \(T_i: {\mathcal J}_i \to {\mathcal T}_i\), \(i=1, 2\), be surjective (not necessarily linear) maps onto multiplicative subsets \({\mathcal S}_i \subset A\), \({\mathcal T}_i \subset B\). In this paper, the authors characterise mappings \(S_i\) and \(T_i\) under the conditions on the peripheral spectra of their products \[ \sigma_{\pi}(S_1(a)S_2(b))\subset \sigma_{\pi}(T_1(a)T_2(b)) \quad \text{and} \quad \sigma_{\pi}(S_1(a)S_2(b))\cap \sigma_{\pi}(T_1(a)T_2(b))\neq \emptyset \] for \(a\in {\mathcal J}_1\) and \(b\in {\mathcal J}_2\). Mainly, they show that the mappings in the first pair equal the mappings in the second one up to certain weighted composition operators on the corresponding Choquet boundaries. As a direct consequence, the authors obtain a large amount of previous results about mappings subject to various spectral conditions.

MSC:

46J10 Banach algebras of continuous functions, function algebras
47B48 Linear operators on Banach algebras
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[1] Gleason, A. M., A characterization of maximal ideals, J. Anal. Math., 19, 171-172, (1967) · Zbl 0148.37502
[2] Grigoryan, S. and Tonev, T., Shift-invariant Uniform Algebras on Groups, Monografie Matematyczne 68, Birkhäuser, Basel, 2006. · Zbl 1076.65032
[3] Hatori, O.; Hino, K.; Miura, T.; Takagi, H., Peripherally monomial-preserving maps between uniform algebras, Mediterr. J. Math., 6, 47-59, (2009) · Zbl 1050.81707
[4] Hatori, O.; Lambert, S.; Luttman, A.; Miura, T.; Tonev, T.; Yates, R., Spectral preservers in commutative Banach algebras, No. 547, 103-123, (2011), Providence, RI · Zbl 1239.46036
[5] Hatori, O.; Miura, T.; Shindo, R.; Takagi, H., Generalizations of spectrally multiplicative surjections between uniform algebras, Rend. Circ. Mat. Palermo, 59, 161-183, (2010) · Zbl 1209.46027
[6] Hatori, O.; Miura, T.; Takagi, H., Characterizations of isometric isomorphisms between uniform algebras via nonlinear range-preserving properties, Proc. Amer. Math. Soc., 134, 2923-2930, (2006) · Zbl 1102.46032
[7] Hatori, O.; Miura, T.; Takagi, H., Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative, J. Math. Anal. Appl., 326, 281-296, (2007) · Zbl 1113.46047
[8] Honma, D., Surjections on the algebras of continuous functions which preserve peripheral spectrum, No. 435, 199-205, (2007), Providence, RI · Zbl 1141.46324
[9] Hosseini, M.; Sady, F., Multiplicatively range-preserving maps between Banach function algebras, J. Math. Anal. Appl., 357, 314-322, (2009) · Zbl 1171.46021
[10] Jiménez-Vargas, A.; Lee, K.; Luttman, A.; Villegas-Vallecillos, M., Generalized weak peripheral multiplicativity in algebras of Lipschitz functions, Cent. Eur. J. Math., 11, 1197-1211, (2013) · Zbl 1298.46043
[11] Jiménez-Vargas, A.; Luttman, A.; Villegas-Vallecillos, M., Weakly peripherally multiplicative surjections of pointed Lipschitz algebras, Rocky Mountain J. Math., 40, 1903-1921, (2010) · Zbl 1220.46033
[12] Jiménez-Vargas, A.; Villegas-Vallecillos, M., Lipschitz algebras and peripherally-multiplicative maps, Acta Math. Sin. (Engl. Ser.), 24, 1233-1242, (2008) · Zbl 0254.58001
[13] Johnson, J., Peripherally-multiplicative spectral preservers between function algebras, Ph.D. Thesis, University of Montana, Missoula, MT, 2013.
[14] Johnson, J.; Tonev, T., Spectral conditions for composition operators on algebras of functions, Commun. Math. Appl., 3, 51-59, (2012)
[15] Kahane, J. P.; Żelazko, W., A characterization of maximal ideals in commutative Banach algebras, Studia Math., 29, 339-343, (1968) · Zbl 0883.32014
[16] Kowalski, S.; Słodkowski, Z., A characterization of multiplicative linear functionals in Banach algebras, Studia Math., 67, 215-223, (1980) · Zbl 0456.46041
[17] Lambert, S.; Luttman, A.; Tonev, T., Weakly peripherally-multiplicative operators between uniform algebras, No. 435, 265-281, (2007), Providence, RI · Zbl 1148.46030
[18] Lee, K.; Luttman, A., Generalizations of weakly peripherally multiplicative maps between uniform algebras, J. Math. Anal. Appl., 375, 108-117, (2011) · Zbl 0681.20029
[19] Leibowitz, G., Lectures on Complex Function Algebras, Scott, Foresman and Co., Glenview, IL, 1970. · Zbl 0219.46037
[20] Luttman, A.; Tonev, T., Uniform algebra isomorphisms and peripheral multiplicativity, Proc. Amer. Math. Soc., 135, 3589-3598, (2007) · Zbl 1134.46030
[21] Miura, T., Real-linear isometries between function algebras, Cent. Eur. J. Math., 9, 778-788, (2011) · Zbl 1243.46043
[22] Molnár, L., Some characterizations of the automorphisms of \(B\)(\(H\)) and \(C\)(\(X\)), Proc. Amer. Math. Soc., 130, 111-120, (2001) · Zbl 1205.82001
[23] Rao, N. V.; Roy, A. K., Multiplicatively spectrum-preserving maps of function algebras, Proc. Amer. Math. Soc., 133, 1135-1142, (2004) · Zbl 1068.46028
[24] Rao, N. V.; Roy, A. K., Multiplicatively spectrum-preserving maps of function algebras II, Proc. Edinb. Math. Soc., 48, 219-229, (2005) · Zbl 1124.32007
[25] Shindo, R., Weakly-peripherally multiplicative conditions and isomorphisms between uniform algebras, Publ. Math. Debrecen, 78, 675-685, (2011) · Zbl 1274.46104
[26] Tonev, T., Weak multiplicative operators on function algebras without units, No. 91, 411-421, (2010), Warsaw · Zbl 1217.46034
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