Determining a space by order only

14 Sep 2015 NUS mathematicians clarified the role of order in determining the linear and/or topological structures of function spaces.

A mathematical object may carry a number of different structures. For example, the set of real numbers has an algebraic structure (addition and multiplication are defined), an order structure (a notion of ``a < b”) and a metric structure (distance between two numbers is defined). It is a natural mathematical problem to find out the interconnections among these structures when they are all present in the same space. In the first half of the 20th century, three famous theorems that addressed various parts of the problem in spaces of continuous functions appeared. The Banach-Stone Theorem showed that the metric structure determines a C(K) space uniquely. The Gelfand-Komolgorov Theorem and Kaplansky’s Theorem proved that the algebraic structure, respectively, the order structure plus the linear structure, also determine a C(K) space uniquely. In joint work with Lei Li of Nankai University, the author devised a unified approach to the three classical theorems above, and extended the results to many classes of function spaces including spaces of uniformly continuous, Lipschitz and differentiable functions.

A team led by Prof LEUNG Denny from the Department of Mathematics in NUS, together with Prof Wee-Kee Tang from NTU has considered the problem of determining a space from the order structure alone (without the linear structure). Among the significant results obtained is that for certain types of Banach lattices, the order alone determines both the linear and the metric structures of the space. General structure theorems for comparing sequence and function spaces via their order structures are also proved. In particular, a comprehensive study of nonlinear order isomorphisms between many types of function spaces was undertaken. The results obtained substantially generalize all previous results in the area and they greatly enhance the understanding of the role that order plays in general function spaces.

Besides sequence and (continuous) function spaces, many classical spaces are in the form of spaces of measurable functions. It is therefore of natural interest to investigate linear and nonlinear order isomorphisms in spaces of measurable functions. In a related direction, linear disjointness preserving operators have been well studied. The methods may well be applicable to the study of nonlinear disjointness preserving operators.

Table shows the comparison for order isomorphisms between variofunction spaces. [Image credit: Denny Leung]

References

1. Cabello Sanchez F, Cabello Sachez J. “Some preserver problems on algebras of smooth functions”. Ark. Mat. 48 (2010) 289.

2. Cabello Sanchez F, Cabello Sachez J. “Lattices of uniformly continuous functions”. Topology Appl. 160 (2013) 50.

3. Leung DH, Li L. “Order isomorphisms on function spaces.” Studia Math. 219 (2013) 123.

4. Leung DH, Tang WK. “Nonlinear order isomorphisms on function spaces” submitted.