Let $X$, $Y$ be complete metrizable spaces, $\beta X$, $\beta Y$ be their Stone-Čech compactifications. It is known that $C_b(X) \simeq C(\beta X)$. Is it possible to say something about the relation between $C_b(X \times Y)$ and $C(\beta X \times \beta Y)$?
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Henno Brandsma
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Danila Zaev
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For compacts $K$ and $L$ we always have $$ C(K\times L)\cong_1C(K)\mathop{\otimes}^\varepsilon C(L) $$ where $\mathop{\otimes}^\varepsilon$ is a completed injective tensor product of Banach spaces. For the proof of $(1)$ see Ryan's book Introduction to Tensor Products of Banach Spaces (p. 50) – Norbert Jul 01 '13 at 07:57
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@Norbert I liked this as an answer. – Michael Greinecker Jul 01 '13 at 09:05
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@MichaelGreinecker, as you wish – Norbert Jul 01 '13 at 13:20
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For all Tychonoff spaces $X$ and $Y$ we know that $C_b(X \times Y) = C(\beta (X \times Y))$ (isometry as Banach spaces, as well as a ring isomorphism), and $\beta(X \times Y) \cong \beta X \times \beta Y$ iff $X \times Y$ is pseudocompact (see the original paper). The latter happens, in your case, iff $X \times Y$ is compact (pseudocompactness and compactness coincide for metrizable spaces; we only need first countability and normality to see that) iff $X$,$Y$ are compact and everything trivializes, as $\beta X = X$ etc.
Henno Brandsma
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For compacts $K$ and $L$ we always have $$ C(K\times L)\cong_1C(K)\mathop{\otimes}^\varepsilon C(L) $$ where $\mathop{\otimes}^\varepsilon$ is a completed injective tensor product of Banach spaces. For the proof of $(1)$ see Ryan's book Introduction to Tensor Products of Banach Spaces (p. 50)
Norbert
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