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Li, Lei; Liao, Ching-Jou; Shi, Luoyi; Wang, Liguang; Wong, Ngai-Ching
Journal of mathematical analysis and applications, 12/2023, Letnik: 528, Številka: 1Journal Article
Let F(X),F(Y) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces X,Y, respectively. Let Φ:F(X)→F(Y) be a surjective map satisfying thatf∨g>0⟺Φ(f)∨Φ(g)>0,∀f,g∈F(X). In many cases, we show that there is a homeomorphism τ:Y→X such thatΦ(f)(y)≠0⟺f(τ(y))≠0,∀f∈F(X),∀y∈Y. Assume X,Y are locally compact Hausdorff (resp. separable and metrizable) and Φ:C0(X)+→C0(Y)+ (resp. Φ:Cb(X)+→Cb(Y)+) is a surjective map. We show that Φ preserves the norms of infima, i.e.,‖Φ(f)∧Φ(g)‖=‖f∧g‖,∀f,g∈C0(X)+ (resp.Cb(X)+), if and only if there is a homeomorphism τ:Y→X such thatΦ(f)(y)=f(τ(y)),∀f∈C0(X)+ (resp.Cb(X)+), ∀y∈Y.
