Let $K$ and $L$ be compact spaces. Consider the Banach algebra $V(K,L)=C(K)\mathbin{\hat\otimes} C(L)$ , which is the completion of the $C(K)\otimes C(L)$ with respect to the projective tensor norm. It is known that $V(K,L)$ is a subalgebra of $C(K\times L)$ , not isometrically. I am trying to establish this result . Heres what I have figured out. Due to the universal property of tensor product , there is an algebra homomorphism $\theta : C(K)\otimes C(L)\to C(K\times L)$ such that $$\theta(\sum_{i=1}^nf_i\otimes g_i)(x,y)=\sum_{i=1}^nf_i(x)g_i(y)$$

I want to show that this map is injective. Once this is done then we can extend it to its completion.

  • $\begingroup$ But be careful: The extension of an injective uniformly continuous map to the completion needn't be injective! Consider a Banach space $(X,\|\cdot\|)$ and a strictly finer norm $\||\cdot\||$ on it (e.g., $\|x\|+|f(x)|$ for a discontinuous linear functional). Then the identity from $(X,\||\cdot\||)$ to $(X,\|\cdot\|)$ is continuous and injective but its extension to the completion is not injective (because otherwise its inverse would be continuous by the closed graph theorem). $\endgroup$ – Jochen Jan 15 at 13:47
  • $\begingroup$ @jochen Thanks for reminding , I was actually thinking the same . $\endgroup$ – Lav Kumar Jan 16 at 13:55

Suppose $h\in C(K)\otimes C(L)$ and $\theta(h)=0$. Write $h=\sum_{i=1}^nf_i\otimes g_i$, where the $g_i$ are linearly independent. But then this implies that for all $x\in K$, $$0=\theta(h)(x,\cdot)=\sum_{i=1}^nf_i(x)g_i(\cdot),$$ whence $f_i=0$ for all $i$, and thus $h=0$.

  • $\begingroup$ Thank you so much. $\endgroup$ – Lav Kumar Jan 14 at 16:59
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    $\begingroup$ You're welcome. Glad to help! $\endgroup$ – Aweygan Jan 14 at 17:00
  • $\begingroup$ Is the choice of representation of h with g's linearly independent comes from smallest representation of h as sums of elementary tensors ? $\endgroup$ – Lav Kumar Jan 14 at 17:02
  • $\begingroup$ I wouldn't think so, but possibly. You might be able to do it with the $f_i$ linearly independent and obtain a representation with less elementary tensors. $\endgroup$ – Aweygan Jan 14 at 17:09

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