SOME GENERALIZATIONS OF M-FINITE BANACH SPACES

We will show that let X and Y be M -finite Banach spaces with canonical M-decompositions $X{\cong}{{\prod}^{{\gamma}_{\infty}}_{i=1}}{X^{n_i}}_{i}\;and\;Y{\cong}{{\prod}^{{\bar{\gamma}}_{\infty}}_{j=1}}{\tilde{Y}^{m_j}}_{j}$, respectively and M and N nonzero locally compact Hausdorff spaces. Then I : $C_{0}$(M,X) ${\longrightarrow}\;C_{0}$(N,Y) is an isometrical isomorphism if and only if r = $\bar{r}$ and there are permutation and homeomorphisms and continuous maps such that I = ${I^{-1}}_{N.Y}\;{\circ}I_{w}^{-1}{\circ}({{\prod}^{\gamma}}_{i=1}I_{t_i,u_i}){\circ}I_{M,X}$.