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Automatic continuity of certain isomorphisms between regular Banach function algebras. (English) Zbl 0901.46042

If \(A\) and \(B\) are regular commutative semisimple Banach algebras then a linear map \(T:A\to B\) is said to be separating (or disjointness preserving) if \(fg=0\) implies \(TfTg=0\). Here it is shown that if \(A\) satisfies Ditkin’s condition then a separating bijection is necessarily continuous and its inverse is separating. If \(B\) also satisfies Ditkin’s condition then the structure spaces of the two algebras are homeomorphic. In particular, it is shown that linear isometries between regular uniform algebras are separating; classical results, like the Banach-Stone theorem, follow.

MSC:

46H40 Automatic continuity
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[1] Reiter, Classical harmonic analvsis and locally compact groups (1968) · Zbl 0165.15601
[2] Font, Arch. Math. (Basel) 63 pp 158– (1994) · Zbl 0805.46049 · doi:10.1007/BF01189890
[3] DOI: 10.1007/BF01246833 · Zbl 0666.46018 · doi:10.1007/BF01246833
[4] Banach, Théorie des opérations linéaires (1932)
[5] DOI: 10.1016/0022-1236(86)90003-0 · Zbl 0611.47030 · doi:10.1016/0022-1236(86)90003-0
[6] DOI: 10.1512/iumj.1983.32.32018 · Zbl 0488.47016 · doi:10.1512/iumj.1983.32.32018
[7] DOI: 10.1016/S0166-8641(96)00132-0 · Zbl 0870.54018 · doi:10.1016/S0166-8641(96)00132-0
[8] DOI: 10.1090/S0002-9947-97-01713-3 · Zbl 0869.46014 · doi:10.1090/S0002-9947-97-01713-3
[9] DOI: 10.1006/jmaa.1995.1170 · Zbl 0828.47024 · doi:10.1006/jmaa.1995.1170
[10] Abramovich, Indag. Math. 45 pp 265– (1983) · doi:10.1016/1385-7258(83)90062-8
[11] DOI: 10.2996/kmj/1138844205 · Zbl 0166.40002 · doi:10.2996/kmj/1138844205
[12] Lamperti, Pacific J. Math. 8 pp 459– (1958) · Zbl 0085.09702 · doi:10.2140/pjm.1958.8.459
[13] DOI: 10.2307/2160705 · Zbl 0827.47051 · doi:10.2307/2160705
[14] Jarosz, Canad. Math. Bull. 33 pp 139– (1990) · Zbl 0714.46040 · doi:10.4153/CMB-1990-024-2
[15] DOI: 10.1090/S0002-9904-1967-11735-X · Zbl 0172.41004 · doi:10.1090/S0002-9904-1967-11735-X
[16] DOI: 10.1017/S0013091500018745 · Zbl 0807.47024 · doi:10.1017/S0013091500018745
[17] Hewitt, Abstract harmonic analysis II (1963)
[18] DOI: 10.1007/BF02568002 · Zbl 0827.46032 · doi:10.1007/BF02568002
[19] Yap, Studia Math. 40 pp 235– (1971)
[20] DOI: 10.1016/0019-3577(96)81755-1 · Zbl 0840.43006 · doi:10.1016/0019-3577(96)81755-1
[21] Dales, Banach algebras and automatic continuity
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