ISO Metries between Subspaces of Continuous Functions

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Binandeh, Mohammad Hossein | 2019

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 52298 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Ranjbar Motlagh, Alireza
Abstract:
Suppose $X$ is a locally compcat Hausdorf‌f space. We denote by $C_0(X)$ the Banach space of all continuous functions real or complex-valued def‌ined on $X$ which vanish at inf‌inity, equipped with the supremum norm. Suppose linear subspace $A$ of $C_0(X)$ is strongly separating, that is, for all pair of distinct points $x_1,x_2 \in X$, there exists $f \in A$ such that $|f(x_1)|\neq |f(x_2)|$. In this thesis, we show that Shilov boundary of $A$ exists and is closure Choquet boundary of $A$. Furthermore, \ we show that a linear isometry $T$ of $A$ onto such a subspace $B$ of $C_0(Y)$ induces a homeomorphism $h$ between two certain subspaces of the Shilov boundaries of $B$ and $A$, sending the Choquet boundary of $B$ onto the Choquet boundary of $A$. Also, function $a$ exists such that for all $f$ in $A$ and $y$ in the Choquet boundary of $B$: $$ (Tf)(y)=a(y)f(h(y)),$$ where $a$ is a unimodular scalar-valued continuous f‌unction def‌ined on the Shilov boundary of $B$
Keywords:
Banach Spaces ; Isometry Group ; Continuous Function Space ; Shilov Boundary ; Choquet Boundary

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