Szlenk index of $C(K)\widehat{\otimes}_\pi C(L)$

TL;DR

This paper calculates the Szlenk index of the projective tensor product of spaces of continuous functions on scattered compact spaces, showing it equals the maximum of the individual indices, with implications for space isomorphism.

Contribution

It provides a precise computation of the Szlenk index for tensor products of $C(K)$ spaces, revealing a simple maximum relation and advancing understanding of their isomorphic properties.

Findings

Szlenk index of $C(K)\widehat{\otimes}_\pi C(L)$ equals max of individual indices

Results on non-isomorphism of tensor products with other $C(M)$ spaces

Clarification of the structure of tensor products of $C(K)$ spaces

Abstract

We compute the Szlenk index of an arbitrary projective tensor product $C(K)\widehat{\otimes}_\pi C(L)$ of spaces $C(K), C(L)$ of continuous functions on scattered, compact, Hausdorff spaces. In particular, we show that it is simply equal to the maximum of the Szlenk indices of the spaces $C(K), C(L)$. We deduce several results regarding non-isomorphism of $C(K)\widehat{\otimes}_\pi C(L)$ and $C(M)$ or $C(M)\widehat{\otimes}_\pi C(N)$ for particular choices of $K,L,M,N$.