Atlas: The Role of the separating map in rings of continuous functions I and II by Edward Beckenstein

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Melvin Henriksen at 75: His research and coworkers
March 1-2, 2002
City College of CUNY
New York, NY, USA

Organizers
Ralph Kopperman, City College, CUNY, Prabudh Misra, Coll. of Staten Island, CUNY

The Role of the separating map in rings of continuous functions I and II
by
Edward Beckenstein
St. John's University, Staten Island, NY 10301, USA
Coauthors: Lawrence Narici (St. John's University, Jamaica, NY 11439)

The Role of the separating map in rings of continuous functions I and II

C(X) and C( Y) denote the rings of real- or complex-valued functions on the Tihonov spaces X and Y. For a subalgebra W of C(X), a map H:W --> C(Y) is separating if, for all f, g in W, fg=0 ===> HfHg=0. Such maps have played a prominent role in the characterization of linear isometries of spaces of continuous and integrable functions since the early days of analysis, in results such as the Banach-Stone theorem and Banach's proof that all linear isometries of L_p[0, 1], p =/= 2, onto itself are separating. They can also effect an improvement of : If H:C(X) --> C(Y) is a ring isomorphism then the realcompactification \upsilonX is homeomorphic to \upsilonY. The strength of ring isomorphism is not needed-H need only be biseparating (H and H^-1 separating)-to deduce the homeomorphism of X and Y. An advantage of the approach using separating maps is the ability to treat real- and complex-valued functions on a fairly equal footing their utility in dealing with functions other than continuous ones (e.g., integrable functions). In Part I, we define them, give some examples (differentiation is, integration is not), and indicate their use in Stone-Banach and Gelfand-Komolgorov-Hewitt theorems. In Part II we indicate the mechanism (the support map) from which their power derives, discuss some automatic continuity results and discuss the conjecture that linear separating bijections are biseparating.

Date received: February 12, 2002