Abstract. Given a Banach space $X$, we define the number $\lambda_0(X) = \inf d(X_2, \ell ^1(2))$, where the infimum is taken over all two-dimensional subspaces $X_2$ of $X$. Here, $d(M,N)$ means the Banach--Mazur distance between Banach spaces $M,N$ defined by $d(M,N) = \inf\{\|T\|\|T^{-1}\|: T\colon M\to N$ is an isomorphism$\}$. We establish some facts about $\lambda_0$ and then consider applications to Banach--Stone type theorems for isomorphisms on continuous, vector-valued function spaces. If $Q,K$ are locally compact Hausdorff spaces, and $X,Y$ are Banach spaces for which both $\lambda_0(X^*)$ and $\lambda_0(Y^*)$ are greater than one, it has been shown that if $T$ is an isomorphism from $C_0(Q,E)$ onto $C_0(K,Y)$ with $\|T\|\|T^{-1}\|$ sufficiently small, then $Q$ and $K$ are homeomorphic, a generalization of the Banach--Stone Theorem for isometries. We examine such results for subspaces of these spaces. A closed subspace $M$ of $C_0(Q,X)$ is said to be a $C_0(Q)$-module if it is closed under multiplication by functions in $C_0(Q)$. If $M$ and $N$ are $C_0(Q), C_0(K)$-modules, respectively, then with assumptions similar to those mentioned above, we are able to obtain results in which the homeomorphism is between the strong boundaries of $N$ and $M$. In this case, the strong boundaries are the subsets of $K$ and $Q$, respectively, upon which the functions in $N$ and $M$ have nonzero values. We also obtain a new theorem concerning isometries.

DOI: 10.14232/actasm-014-255-x

AMS Subject Classification (1991): 46B03, 46E40

Keyword(s): isomorphism, isometry, strong boundary, homeomorphism

Received January 7, 2014, and in revised form April 1, 2014. (Registered under 5/2014.)