Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question.
For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a semi-norm on $A$ with $\parallel f \parallel=\parallel \omega_{f}\parallel_{\infty}$ where $\omega_{f}(x)$ is the oscilation of $f$ at $x$. It defines a norm on $A/C(X)$. The completion of this normed space is denoted by $DC(X)$.
How can $DC(X\times Y)$ be written in term of $DC(X)$ and $DC(Y)$?(An appropriate completion of the algebraic tensor product $DC(X)\otimes DC(Y)$? (Or some thing else?)
Note: We can not expect to have an algebra structure on $DC(X)$, since $C(X)$ is not an ideal. But in the following particular case we may expect that we have a Banach algebra structure: When $X$ is a compact topological group with its Haar measure, $A$ is the algebra of bounded measurable functions with the convolution product. again we consider the oscillation norm. For abelian $X$, it would be interesting to study the Gelfand spectrum of the resulting commutative Banach algebra. In particular, what is the resulting algebra and its Gelfand spectrum in the particular case of $X=S^{1}$?
