Multiplicative bijections of $\mathcal {C(X},I\mathcal {)}$
HTML articles powered by AMS MathViewer
- by Janko Marovt PDF
- Proc. Amer. Math. Soc. 134 (2006), 1065-1075 Request permission
Abstract:
Let $\mathcal {X}$ be a compact Hausdorff space which satisfies the first axiom of countability, let $I=\left [ 0,1\right ]$ and let $\mathcal {C}(\mathcal {X}$,$I)$ be the set of all continuous functions from $\mathcal {X}$ to $I.$ If $\varphi :\mathcal {C}(\mathcal {X}$,$I)\rightarrow \mathcal {C}(\mathcal {X}$,$I)$ is a bijective multiplicative map, then there exist a homeomorphism $\mu : \mathcal {X\rightarrow X}$ and a continuous map $k:\mathcal {X} \rightarrow \left ( 0,\infty \right ) ,$ such that $\varphi (f)(x)=f(\mu (x))^{k(x)}$ for all $x\in \mathcal {X}$ and for all $f\in \mathcal {C}(\mathcal {X},I).$References
- J. Aczél and J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its Applications, vol. 31, Cambridge University Press, Cambridge, 1989. With applications to mathematics, information theory and to the natural and social sciences. MR 1004465, DOI 10.1017/CBO9781139086578
- Paul Busch, Pekka J. Lahti, and Peter Mittelstaedt, The quantum theory of measurement, Lecture Notes in Physics. New Series m: Monographs, vol. 2, Springer-Verlag, Berlin, 1991. MR 1176754, DOI 10.1007/978-3-662-13844-1
- A. DvureÄenskij, S. PulmannovĂĄ, Recent Trends in Quantum Structures, Kluwer Academic Publisher, 2000.
- Stan Gudder and Richard Greechie, Sequential products on effect algebras, Rep. Math. Phys. 49 (2002), no. 1, 87â111. MR 1899078, DOI 10.1016/S0034-4877(02)80007-6
- Stan Gudder and Gabriel Nagy, Sequential quantum measurements, J. Math. Phys. 42 (2001), no. 11, 5212â5222. MR 1861337, DOI 10.1063/1.1407837
- Stan Gudder and Gabriel Nagy, Sequentially independent effects, Proc. Amer. Math. Soc. 130 (2002), no. 4, 1125â1130. MR 1873787, DOI 10.1090/S0002-9939-01-06194-9
- Lajos MolnĂĄr, Sequential isomorphisms between the sets of von Neumann algebra effects, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4, 755â772. MR 2034206
Additional Information
- Janko Marovt
- Affiliation: EPF-University of Maribor, Razlagova 14, 2000 Maribor, Slovenia
- Email: janko.marovt@uni-mb.si
- Received by editor(s): September 10, 2004
- Received by editor(s) in revised form: October 27, 2004
- Published electronically: July 20, 2005
- Communicated by: Joseph A. Ball
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1065-1075
- MSC (2000): Primary 46J10; Secondary 46E05
- DOI: https://doi.org/10.1090/S0002-9939-05-08069-X
- MathSciNet review: 2196040