Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Linear isometries between subspaces
of continuous functions

Authors: Jesús Araujo and Juan J. Font
Journal: Trans. Amer. Math. Soc. 349 (1997), 413-428
MSC (1991): Primary 46E15; Secondary 46E25
MathSciNet review: 1373627
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We say that a linear subspace $A$ of $C_0 (X)$ is strongly separating if given any pair of distinct points $x_1, x_2$ of the locally compact space $X$, then there exists $f \in A$ such that $ \left | f(x_1 ) \right | \neq \left | f(x_2 ) \right | $. In this paper we prove that a linear isometry $T$ of $A$ onto such a subspace $B$ of $C_0(Y)$ induces a homeomorphism $h$ between two certain singular subspaces of the Shilov boundaries of $B$ and $A$, sending the Choquet boundary of $B$ onto the Choquet boundary of $A$. We also provide an example which shows that the above result is no longer true if we do not assume $A$ to be strongly separating. Furthermore we obtain the following multiplicative representation of $T$: $(Tf)(y)=a(y)f(h(y))$ for all $y \in \partial B$ and all $f \in A$, where $a$ is a unimodular scalar-valued continuous function on $\partial B$. These results contain and extend some others by Amir and Arbel, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.

References [Enhancements On Off] (What's this?)

  • 1. D. Amir, On isomorphisms of continuous function spaces, Israel J. Math. 3 (1966), 205-210. MR 34:516
  • 2. J. Araujo and J. J. Font, On \v{S}ilov boundaries for subspaces of continuous functions. To appear in Topology Appl.
  • 3. D. Amir and B. Arbel, On injections and surjections of continuous function spaces, Israel J. Math. 15 (1973), 301-310. MR 53:1239
  • 4. E. Behrends, M-structure and the Banach-Stone theorem, Lecture Notes in Mathematics 736, Berlin-Heidelberg-New York, Springer (1979). MR 81b:46002
  • 5. Y. Benyamini, Small into isomorphisms between spaces of continuous functions, Proc. A.M.S. 83 (1981), 479-485. MR 82j:46033
  • 6. M. Cambern, On isomorphisms with small bound, Proc. A.M.S. 18 (1967), 1062-1066. MR 36:669
  • 7. M. Cambern and V.D. Pathak, Isometries of spaces of differentiable functions, Math. Japonica 26 (1981), 253-260. MR 82h:46043
  • 8. B. Cengiz, A generalization of the Banach-Stone theorem, Proc. A.M.S. 40 (1973), 426-430. MR 47:9258
  • 9. H. B. Cohen, A bound-two isomorphism between C(X) Banach spaces, Proc. A.M.S. 50 (1975), 215-217. MR 52:1279
  • 10. K. DeLeeuw, Banach spaces of Lipschitz functions, Studia Math. 21 (1961), 55-66. MR 25:4341
  • 11. K. DeLeeuw, W. Rudin and J. Wermer, The isometries of some functions spaces, Proc. A.M.S. 11 (1960), 694-698. MR 22:12380
  • 12. F. O. Farid and K. Varadarajan, Isometric shift operators on $C(X)$, Can. J. Math. 46 (1994), 532-542. MR 95d:47034
  • 13. A. Gutek, D. Hart, J. Jamison and M. Rajagopalan, Shift operators on Banach spaces, J. Funct. Anal. 101 (1991), 97-119. MR 92g:47046
  • 14. K. Geba and Z. Semadeni, Spaces of continuous functions (V), Studia Math. 19 (1960), 303-320. MR 22:8313
  • 15. H. Holszty\'{n}ski, Continuous mappings induced by isometries of spaces of continuous functions, Studia Math. 26 (1966), 133-136. MR 33:1711
  • 16. K. Jarosz, Into isomorphisms of spaces of continuous functions, Proc. Amer. Math. Soc. 90 (1984), 373-377. MR 85k:46024
  • 17. K. Jarosz, Perturbations of Banach algebras, Lecture Notes in Math. 1120, Springer-Verlag, (1985). MR 86k:46074
  • 18. K. Jarosz and V. D. Pathak, Isometries and small bound isomorphisms of function spaces, Lecture Notes in Pure and Appl. Math. 136, Dekker (1992), 241-271. MR 93b:47061
  • 19. A.A. Miljutin, Isomorphisms of spaces of continuous functions on compacta of the power of the continuum (Russian), Teor. Funkcii Funkcional Anal. i Prilozhen. 2 (1966), 150-156. MR 34:6513
  • 20. S. B. Myers, Banach spaces of continuous functions, Ann. of Math. 49 (1948), 132-140. MR 9:291c
  • 21. M. Nagasawa, Isomorphisms between commutative Banach algebras with application to rings of analytic functions, Kodai Math. Sem. Rep. 11 (1959), 182-188. MR 22:12379
  • 22. W. P. Novinger, Linear isometries of subspaces of continuous functions, Studia Math. 53 (1975), 273-276. MR 54:5818
  • 23. V.D. Pathak, Linear isometries of absolutely continuous functions, Can. J. Math. 34 (1982), 298-306. MR 83f:46023
  • 24. K. Sundaresan, Spaces of continuous functions into a Banach space, Studia Math. 48 (1973), 15-22. MR 48:9377

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 46E15, 46E25

Retrieve articles in all journals with MSC (1991): 46E15, 46E25

Additional Information

Jesús Araujo
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, Facultad de Ciencias, Avda. de los Castros, s. n., E-39071 Santander, Spain

Juan J. Font
Affiliation: Departamento de Matemáticas, Universitat Jaume I, Campus Penyeta Roja, E-12071 Castellón, Spain

Received by editor(s): October 16, 1995
Additional Notes: Research of the first author was supported in part by the Spanish Dirección General de Investigación Científica y Técnica (DGICYT, PS90-100).
Research of the second author was supported in part by Fundació Caixa Castelló, (A-39-MA)
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society