The Banach–Stone theorem revisited

Banach–Stone Theorems for maps preserving common zeros

“…The following corollary of Theorem 10 contains the result conjectured in [10]. The original conjecture was solved by Chen, Chen and Wong [8] and independently by Ercan andÖnal [11].…”

“…Professor N. C. Wong has informed us that he and his co-authors have independently solved the conjecture of Ercan andÖnal [8]. The conjecture was also solved by its proposers Ercan andÖnal [11].…”

The uniform separation property and Banach–Stone theorems for lattice-valued Lipschitz functions

“…We can cite, among others, the generalizations obtained by J. X. Chen, Z. L. Chen and N.-C. Wong [5], Z. Ercan and S.Önal [6,7] and X. Miao, J. Cao and H. Xiong [12].…”

“…According to [5,Theorem 3] or [7,Theorem 5], for compact Hausdorff spaces X and Y and Banach lattices E and F , if C(X, E) and C(Y, F ) denote the Banach lattices of continuous E-valued and F -valued functions defined on X and Y , respectively, endowed with the pointwise order and the supremum norm, then every vector lattice isomorphism T : C(X, E) → C(Y, F ) preserving the nowhere vanishing functions in both directions can be written as a weighted composition operator in the form: (…”

Common zeros preserving maps on vector-valued function spaces and Banach modules

“…For some results concerning non-vanishing preserving maps (on lattices) one can see [4,6,5,8,9,15,18,19,20]. We also refer to [14] for some relevant concepts in the case of scalar-valued continuous functions.…”