Small isomorphisms of C(X,E) spaces
A linear map T between Banach spaces A and B is called an \(\epsilon\)- isometry if \((1-\epsilon)\| f\| \leq \| Tf\| \leq (1+\epsilon)\| f\|\) for all \(f\in A\). If \(\epsilon >0\) is small, is there an isometry S of A into (onto) B with \(\| T-S\|\) small? In this paper A and B are subspaces of various \(C_ 0(X,E):\) the continuous E-valued functions which vanish at \(\infty\) on X, E a Banach space, X a locally compact Hausdorff space. The circle of ideas represents several directions in which the classical Banach-Stone theorem has been generalized. That theorem says that for \(X_ 1\), \(X_ 2\) compact and E either \({\mathbb{R}}\) or \({\mathbb{C}}\), every linear isometry T of \(C(X_ 1,E)\) onto \(C(X_ 2,E)\) has the form \(Tf=u\cdot f\circ \tau\), where \(\tau\) is a homoeomorphism of \(X_ 2\) onto \(X_ 1\) and u is a unimodular function in \(C(X_ 2,E).\) We can relax the hypothesis on T to (1) its being a linear bijection satisfying \(\| T\| \| T^{- 1}\| <1+\epsilon\) [with \(\epsilon\) not too large, since \(\exists\) non- homeomorphic compact metric \(X_ 1\), \(X_ 2\) and surjective linear T: C(X\({}_ 1)\to C(X_ 2)\) with \(\| T\| \| T^{-1}\| =2]\) or to (2) its being only an ``into'' linear isometry. We can keep T a surjective linear isometry but (3) defined between subspaces (or subalgebras) A, B of \(C(X_ j,E)\). [Here the following example of the author's is instructive: for \(A=B=H^{\infty}=:H^{\infty}({\mathbb{D}})\) and any \(\delta >0\), \(\exists\) a surjective linear map T: \(H^{\infty}\to H^{\infty}\) with \(\| T\| \leq 1+\delta,\) \(\| T^{-1}\| \leq 1+\delta\) and \(\| S- T\| \geq 2-\delta\) for every linear isometry S: \(H^{\infty}\to H^{\infty}.]\) Finally, in the original Banach-Stone set-up, as well as in (1), (2) and (3), \(E={\mathbb{R}}\) or \({\mathbb{C}}\) can be replaced by other Banach spaces. Since the original formulation fails in the case of certain C(X) in the role of E, for positive results when E is a general Banach space hypotheses about the geometry of the unit ball in E are needed which will insure that E is far from having any C(X)-structure. These involve the Banach-Mazur distance \(d_{BM}(E,F):=\inf \{\| T\| \| T^{- 1}\|:\) T a linear isomorphism of E onto \(F\}\) between Banach spaces, and the numbers \(\lambda_ 0(E):=\inf \{d_{BM}(\ell^{\infty}_ 2,E_ 0):\) \(E_ 0\) 2-dimensional subspace of \(E\}\) and \(\lambda (E):=\inf \{\max \{\| e_ 1+\lambda e_ 2\|:\) \(| \lambda | =1\}:\) \(e_ 1\), \(e_ 2\) unit vectors in \(E\}\). Notice that \(\lambda (E)\leq \lambda_ 0(E)\) and that \(\lambda_ 0(E)=\lambda (E)=\sqrt{2}\) if E is a Hilbert space of dimension two or more. The author gets a common generalization of (1), (2) and (3) when \(\lambda_ 0(E)>0\). Mostly the locally compact spaces X are metrizable and the subspaces C(X,E) considered are completely injective tensor products A\({\check \otimes}E\), where A is an extremely regular (closed) subspace of \(C_ 0(X)\), meaning that every point x of X is almost an A-peak point: for every neighborhood U of x and every \(\epsilon >0\) there is an \(f\in A\) with \(| f| <\epsilon\) outside U and \(1=f(x)=\| f\|.\) Here are two sample results: If \(\lambda (E_ j)>1\) and T is a linear isometry from \(A_ 1{\check \otimes}E_ 1\) onto \(A_ 2{\check \otimes}E_ 2\), then there exists a homeomorphism \(\phi\) : \(X_ 2\to X_ 1\) and a norm continuous map \(x_ 2\mapsto T_{x_ 2}\) into the set of isometries from \(X_ 1\) onto \(X_ 2\) such that \[ T(f)(x_ 2)=T_{x_ 2}(f(\phi (x_ 2)))\quad \forall f\in A_ 1{\check \otimes}E_ 1,\quad x_ 2\in X_ 2. \] If E is a Hilbert space and T a linear isomorphism from \(C_ 0(X_ 1,E)\) onto \(C_ 0(X_ 2,E)\) such that \(\| T\| <1+\epsilon <5/4\) and \(\| T^{-1}\| \leq 1\), then \(\exists\) a (canonical) linear isometry S from \(C_ 0(X_ 1,E)\) onto \(C_ 0(X_ 2,E)\) such that \(\| S- T\| <4\epsilon /(1-4\epsilon).\) Although the results and proofs are quite technical, the paper is very well organized: section 1 describes the many generalizations of the Banach-Stone theorem with references to the literature; section 2 defines notation and motivates hypotheses; section 3 states the four main theorems; a long section 4 proves these and a shorter final section, containing two theorems, discuss the situation where the \(X_ j\) are not metrizable. [Definitive results in the problem (2) mentioned above (but with the linearity hypothesis suppressed) have recently been achieved by \textit{G. M. Lövblom}, Isr. J. Math. 56, 143-159 (1986; Zbl 0637.46027).]
scientific article; zbMATH DE number 4144733