對於定義在歐氏空間開集上的光滑向量場空間的局部算子，Peetre 在1960年得到一個微分的表現式。我們期望將此結果由歐氏空間開集延伸到光滑巴拿赫流形上。作為更進一步的推廣和增加應用性，我們考慮抽象化的可微分函數範疇，這裡特別包含了由 n 階可微分向量場所構成的範疇 C^{n} ，其中 0≤n≤∞，以及李普希茨函數。因局部算子是保斥性算子的一個特例，在此論文中，主要的工作是在探討作用在巴拿赫流形上的向量叢之間的保斥性算子的結構問題。在最後的二章中，我們以 C^{n}及李普希茨函數空間當作例子去探討保斥性算子的結構，和由此所誘導出關於巴拿赫流形的微分結構等價關係。

Peetre characterized local operators defined on the smooth section space over an open subset of an Euclidean space as ``linear differential operators'. We look for an extension to such maps of smooth vector sections of smooth Banach bundles. Since local
operators are special disjointness preserving operators, it leads to the study of the disjointness structure of smooth Banach manifolds.

In this thesis, we take an abstract approach to define the``smooth functions', via the so-called S-category.
Especially, it covers the standard classes C^{n} and local Lipschitz functions, where 0≤n≤∞. We will study
the structure of disjointness preserving linear maps between S-smooth functions defined on separable Banach manifolds. In particular, we will give an extension of Peetre's theorem to characterize disjointness preserving linear mappings between C^n
or local Lipschitz functions defined on locally compact metric spaces.

Chapter 1: Introduction 1
Chapter 2: Local operators and Peetre’s Theorem 4
Chapter 3: S-categories 7
3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Banach-Stone type representation . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter 4: Algebras of differentiable functions 15
4.1 n-differentiable function spaces . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Smooth function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 5: Lipschitz algebras 22
5.1 A Banach-Stone type Theorem for Lipschitz manafolds . . . . . . . . . . . . 23
5.2 Disjointness preserving linear maps between little Lipschitz functions on locally
compact metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.3 Some results on automatic continuity . . . . . . . . . . . . . . . . . . . . . . 35
5.4 Disjointness preserving linear maps between little Lipschitz functions on compact
metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Appendix 43
Conclusions and open problems..............................43

[1] Y. Abramovich, A. I. Veksler and A. V. Koldunov, On operators preserving disjointness,
Soviet Math. Dokl., 248 (1979), 1033–1036.
[2] Y. A. Abramovich and A. K. Kitover, Inverses of disjointness preserving operators, Vol.
143, Mem. Amer. Math. Soc., 2000.
[3] J. Araujo, Separating maps and linear isometries between some spaces of continuous
functions, J. Math. Anal. Appl., 226 (1998), 23–39.
[4] J. Araujo, Realcompactness and spaces of vector-valued functions, Fund. Math., 172
(2002), 27–40.
[5] J. Araujo and K. Jarosz, Automatic continuity of biseparating maps, Studia Math., 155
(2003), 231–239.
[6] J. Araujo, Realcompactness and Banach–Stone theorems, Bull. Bel. Math. Soc. Simon
Stevin, 11 (2004), 247–258.
[7] J. Araujo, Linear biseparating maps between spaces of vector-valued differentiable functions
and automatic continuity, Advances in Math., 187 (2004), 488–520.
[8] W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J., 32 (1983),
199–215.
[9] S. Banach, Th´eorie des op´erations lin´eaires, New York, 1932.
[10] E. Beckenstein, L. Narici and A. Todd, Automatic continuity of linear maps on spaces of
continuous functions, Manuscripta Math., 62 (1988), 257–275.
[11] E. Behrends, M-structure and the Banach-Stone theorem, Vol. 736, Lecture Notes in
Math., Springer, Berlin, 1979.
[12] R. Bonic and J. Frampton, Smooth functions on Banach manifolds, J. of Math. Mech. 15
(1966), 877–898.
[13] F. Cabello S´anchez, J. Cabello S´anchez, Z. Ercan and S. Onal, Memorandum
on multiplicative bijections and order (2007), preprint available at
http://kolmogorov.unex.es/fcabello.
[14] L. Dubarbie, Separating maps between spaces of vector-valued absolutely continuous
functions, (to appear in Canad. Math. Bull.)
[15] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience,
New York, 1958.
[16] J. J. Font and S. Hern´andez, On separating maps between locally compact spaces, Arch.
Math. (Basel), 63 (1994), 158–165.
[17] M. Isabel Garrido, Jesu´s A. Jaramillo and A´ ngeles Prieto, Banach-Stone theorems for
Banach manifolds, Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp.), 94 (2000), 525–528.
[18] M. Isabel Garrido and Jes´us A. Jaramillo, Lipschitz-type functions on metric spaces, J.
Math. Anal. Appl., 340 (2008), 282–290.
[19] H.-L. Gau, J.-S. Jeang and N.-C. Wong, Biseparating linear maps between continuous
vector-valued function spaces, J. Aust. Math. Soc., 74 (2003), 101–109.
[20] J. M. Guti´errez and J. G. Llavona, Composition operators between algebras of differentiable
functions, Trans. Amer. Math. Soc., 338 (1993), 769–782.
[21] S. Hern´andez, E. Beckenstein and L. Narici, Banach–Stone theorems and separating
maps, Manuscripta Math., 86 (1995), 409–416.
[22] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, Berlin Heidelberg,
1965.
[23] K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull.,
33 (1990), 139–144.
[24] J.-S. Jeang and N.-C.Wong, Weighted composition operators of C0(X)0s, J. Math. Anal.
Appl., 201 (1996), 981–993.
[25] J.-S. Jeang and N.-C. Wong, On the Banach-Stone Problem, Studia Math, 155 (2003),
95–105.
[26] M. Jerison, The space of bounded maps into a Banach space, Ann. of Math. (2) , 52
(1950), 309–327.
[27] A. Jim´enez-Vargas, Linear bijections preserving the H¨o lder seminorm, Proc. Amer.
Math. Soc., 135 (2007), 2539–2547.
[28] A. Jim´enez-Vargas, Disjointness preserving operators between little Lipschitz algebras,
J. Math. Anal. Appl., 337 (2008), 984–993.
[29] A. Jim´enez-Vargas and M. Villegas-Vallecillos, Order isomorphisms of little Lipschitz
algebras, to appear in Houston J. Math.
[30] A. Jim´enez-Vargas, M. Villegas-Vallecillos and Y-S Wang, Banach-Stone theorems for
vector-valued little Lipschitz functions, Publ. Math. Debrecen, 74 (2009), 81–100.
[31] A. Jim´enez-Vargas and Y-SWang, Linear biseparating maps between vector-valued little
Lipschitz function spaces, to appear in Acta Math. Sinica (English Series).
[32] J. A. Johnson, Banach spaces of Lipschitz functions and vector-valued Lipschitz functions,
Trans. Amer. Math. Soc., 148 (1970), 147–169.
[33] R. Kantrowitz and M. M. Neumann, Disjointness preserving and local operators on
algebras of differentiable functions, Glasgow Math. J., 43 (2001), 295–309.
[34] J. Lamperti, On the isometries of certain function spaces, Pacific. J. Math., 8 (1958),
459–466.
[35] K. S. Lau, A representation theorem for isometries of C(X,E), Pacific. J. Math., 60
(1975), 229–233.
[36] K. de Leeuw, Banach spaces of Lipschitz functions, Studia. Math., 21 (1961/1962), 55–
66.
[37] R. Narasimhan, Analysis on real and complex manifolds, Amsterdam, New York : North-
Holland, 1973.
[38] L. Narici and E. Beckenstein, , The separating map: a survey, Rend. Circ. Mat. Palermo
(2) Suppl., 52 (1998), 637–648.
[39] B. Pavlovi´c, Automatic continuity of Lipschitz algebras, J. Funct. Anal., 131 (1995),
115–144.
[40] B. Pavlovi´c, Discontinuous maps from Lipschitz algebras, J. Funct. Anal., 155 (1998),
436–454.
[41] J. Peetre, Une caract´erisation abstraite des op´erateurs diff´erentiels, Math. Scand., 7
(1959), 211–218.
[42] J. Peetre, Rectifications ´a l’article Une caract´erisation abstraite des op´erateurs
diff´erentiels, Math. Scand., 8 (1960), 116–120.
[43] C. H. Scanlon, Rings of functions with certain Lipschitz properties, Pacific J. Math., 32
(1970), 197–201.
[44] D. Sherbert, Banach algebras of Lipschitz functions, Pacific. J. Math., 13 (1963), 1387–
1399.
[45] D. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz
function, Trans. Amer. Math. Soc., 111 (1964), 240–272.
[46] C. L. Terng, Natural vector bundles and natural differential operators, American J. of
Math., 100 (1978), 775–828.
[47] B. Z. Vulih, on linear multiplicative operators, Dokl. Akad. Nauk USSR, 41 (1943),
148–151.
[48] B. Z. Vulih, Multiplication in linear semi-ordered spaces and its application to the theory
of operationd, Mat. Sb, 22 (1948), 267–317.
[49] N. Weaver, Lipschitz algebras, World Scientific, Singapore, 1999.