Separating maps on spaces of continuous functions

Biseparating maps between operator algebras

“…It is interesting to notice that for compact sets X, Y a separating invertible map T : C (X) → C (Y ) is automatically biseparating and consequently of the form (1.1) [10]. However, whether this is also true for noncompact sets X, Y, is an open problem (see [4] for a partial solution). A reader interested in separating and biseparating maps in more general setting may want to check a recent monograph [1].…”

Automatic continuity of biseparating maps

“…One of such areas includes composition operators considered in ergodic theory and harmonic analysis, as any composition operator is separating. Separating and biseparating maps have been studied intensively by many authors; the reader may want to check [3,[5][6][7][8][9]12] for a large number of interesting results, examples, and applications of such maps. A recent monograph by Y. Abramovich and A. K. Kitover [2] may serve as the best first step into that area; lattice isomorphisms, Banach algebra isomorphisms, and weighted composition operators are the basic examples of separating maps, but the reader will also find there separating maps that do not belong to any of these categories.…”

Biseparating maps on generalized Lipschitz spaces

“…This explains the interest and amount of work devoted to the characterization of separating or biseparating operators. See, e.g., [1,2,3,4,5,7,8,16,19,27].…”