Skip to main content
SpringerLink
Log in

Automatic continuity of linear maps on spaces of continuous functions

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

Let C(S)and C(T) denote the sup-normed Banach spaces of real- or complex-valued continuous functions on the compact Hausdorff spaces S and T, respectively. A linear map A∶C(T)→C(S) is calledseparating if when two functions x and y from C(T) have disjoint cozero sets then so do Ax and Ay. In the spirit of [3] and [4], we show that separating maps are automatically continuous in some important cases (Theorems 2.4 and 2.5). If a separating map is continuous, then it must be a continuous multiple of a composition map (Theorem 2.2). If A is injective, separating and detaching (Def. 2.4) then S and T are homeomorphic (Theorem 2.1).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. ABRAMOVIC, Y: Multiplicative representations of disjointness preserving operators, Indag. Math.45, 265–279 (1983)

    Google Scholar 

  2. ABRAMOVIC, Y., VEKSLER, A., and KOLDUNOV, V.: On operators preserving disjointness, Soviet Math. Dokl.20, 1089–1093 (1979)

    Google Scholar 

  3. ALBRECHT, E. and NEUMANN, M.: Automatic continuity of generalized local linear operators, Manuscripta Math.32, 263–294 (1980)

    Google Scholar 

  4. ALBRECHT, E. and NEUMANN, M.: Automatic continuity for operators of local type. Lecture Notes in Mathematics 975, 342–355, Berlin-Heidelberg-New York: Springer 1985

    Google Scholar 

  5. AMIR, D.: On isomorphisms of continuous function spaces, Israel J. Math.3, 205–210, (1965)

    Google Scholar 

  6. ARENDT, W.: Spectral properties of Lamperti operators, Indiana Univ. Math. J.32, 199–215 (1983)

    Google Scholar 

  7. BACHMAN, G.: Elements of abstract harmonic analysis. New York: Academic Press 1964

    Google Scholar 

  8. BECKENSTEIN, E. and NARICI, L.: A nonarchimedean Stone-Banach theorem, Proc. A. M. S.100, 242–246 (1987)

    Google Scholar 

  9. BECKENSTEIN, E. and NARICI, L.: Automatic continuity of certain linear isomorphisms, Acad. Roy. Belg. Bull. cl. sci. (5) Bruxelles73, 191–200 (1987)

    Google Scholar 

  10. CAMBERN, M.: On some isomorphisms with small bound, Proc. A. M. S.18, 1062–1066 (1967)

    Google Scholar 

  11. COHEN, H. B.: A bound-two isomorphism between C(X) Banach spaces, Proc. A. M. S.50, 215–217 (1975)

    Google Scholar 

  12. DE PAGTER, B.: A note on disjointness preserving operators, Proc. A. M. S.90, 543–549 (1984)

    Google Scholar 

  13. JAROSZ, K.: Automatic continuity of separating linear isomorphisms, to appear

Download references

Author information

Authors and Affiliations

  1. St. John's University, 10301, Staten Island, New York, USA

    Edward Beckenstein

  2. St. John's University, 11439, Jamaica, New York, USA

    Lawrence Narici

  3. Baruch college, 17 Lexington Avenue, 10010, New York, New York, USA

    Aaron R. Todd

Authors
  1. Edward Beckenstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lawrence Narici
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aaron R. Todd
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beckenstein, E., Narici, L. & Todd, A.R. Automatic continuity of linear maps on spaces of continuous functions. Manuscripta Math 62, 257–275 (1988). https://doi.org/10.1007/BF01246833

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01246833

Keywords

Search

Navigation