Abstract. We consider the space $E=E(\Omega,\|. \|)$ as the commutative C*-algebra ${\cal C}_0(\Omega )$ equipped with a norm $\|. \|$ having the monotonicity property $\|f \|\ge\|g \|$ if $| f | \ge | g | $. We show there exists a finest partition $\Pi $ of the underlying space $\Omega $ along with a function $m\colon\Omega \to{\msbm R}_+$ with the following properties: $\sup_{S\in\Pi } \# S < \infty $, $0<\inf m\le\sup m < \infty $ and each $E$-Hermitian operator $A$ can be written in the matrix form $Af(\omega ) = \sum_{\eta\in S} a^{(S)}_{\omega\eta } f(\eta )$, $\omega\in S \in\Pi _E$ with some system $[ a^{(S)} : S \in\Pi ]$ of matrices $a^{(S)} = [ a^{(S)}_{\omega\eta } ]_{\omega,\eta\in S}$ indexed with the elements of $\Omega $ and we have $\{f| _{S} : \|f \|\le1\} = \{\varphi\in {\cal C}(S): \sum_{\omega\in S} | \varphi(\omega ) | ^2 \le1\} $ for any partition member $S\in\Pi $. Hence, generalizing the Banach--Stone theorem, we obtain matrix descriptions for surjective isometries $E(\Omega,\|. \|) \to E(\widetilde{\Omega },\|. \|^\sim )$. We apply this result to show that unlike in the classical case of spectral norms, the linear isometric equivalence of the spaces $E(\Omega,\|. \|)$ and $E(\widetilde{\Omega },\|. \|^\sim )$ does not imply the existence of a positive surjective linear isometry in general, disproving a conjecture on Sunada type theorems for generalized Reinhardt domains.

AMS Subject Classification (1991): 46E15, 46B42, 28C05

Keyword(s): Banach lattice, Reinhardt domain, C*-algebra

Received November 25, 2006. (Registered under 5962/2009.)