# Spatial tensor product of algebras and normed spaces

Let $A_1$ and $A_2$ be two $C^*$-algebras considered as closed $*$-subspaces of some $\mathcal{B}(H_1)$ and $\mathcal{B}(H_2)$. More preciesly there exist faithfull $*$-representations $\pi_1:A_1\to\mathcal{B}(H_1)$, $\pi_2:A_2\to\mathcal{B}(H_2)$. By definition, the uncompleted spatial tensor product $A_1\otimes_{sp}A_2$ of $C^*$-algebras $A_1$ and $A_2$ is $$A_1\otimes_{sp}A_2={\rm span}\{\pi_1(a_1)\otimes_2 \pi_2(a_2) : a_1\in\mathcal{B}(H_1),a_2\in\mathcal{B}(H_2)\}\subset\mathcal{B}(H_1\otimes_2 H_2)$$

This definition doesn't depend on the choice of representations $\pi_1$ and $\pi_2$. This proved in Murphy's book.

Now consider arbitrary normed space $X$. Denote by $B_{X^*}$ closed unit ball of $X^*$ at $0$. It is easy to check that operators $$i_1:X\to l_\infty(B_{X^*}):x\mapsto(f\mapsto f(x))\qquad x\in X,\quad f\in B_{X^*}$$ $$i_2:l_\infty(B_{X^*})\to\mathcal{B}(l_2(B_{X^*})):g\mapsto(f\mapsto g\cdot f)\qquad g\in l_\infty(B_{X^*}),\quad f\in l_2(B_{X^*})$$ are isometric. Thus each normed space $X$ we can embed into some space of bounded operators acting on Hilbert space by isometric operator $i :X\to\mathcal{B}(l_2(B_{X^*})):x\mapsto i_2(i_1(x))$.

Now given some isometric inclusions $i_X:X\to\mathcal{B}(H_X)$ and $i_Y:Y\to\mathcal{B}(H_Y)$ we can define spatial tensor product of normed spaces: $$X\otimes_{sp} Y={\rm span}\{i_X(x)\otimes_2 i_Y(y): x\in X, y\in Y\}\subset\mathcal{B}(H_X\otimes_2 H_Y)$$

My question: Will this definition of spatial tensor product depend on particular choice of inclusions. For what kind of spaces this definition doesn't depend on the choice? May be it is necessary to put some restrictions on embeddings to get independence withing this class of embeddings?

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Don't you mean to have $K_1=H_1$ and $K_2=H_2$? Otherwise what does $\pi_1(a)\pi_1(b)$ mean for example? The definition of the tensor product is also incorrect, because the $a_i$'s should be only in the ranges of the representations. (I'm sure that's what you meant.) – Jonas Meyer Jan 14 '12 at 6:04
Where you wrote $g\circ f$, do you mean the pointwise product of $g$ and $f$? You are talking about $\mathbb C$ valued functions, right? – Jonas Meyer Jan 14 '12 at 6:15
What you are asking about are essentially the spacial tensor products of $C^*$ algebras and of operator spaces. A couple of good references are Pisier's Introduction to operator space theory and Paulsen's Completely bounded maps and operator algebras. – Jonas Meyer Jan 14 '12 at 7:39
Similarly, your first question is interesting and good, but it's answered in any standard text on operator algebras, and I think would not be particularly easy to answer here. I'd recommend Murphy's book for a nice, gentle account of the tensor product of C$^*$-algebras. – Matthew Daws Jan 14 '12 at 13:36
"C*-algebras and operator theory" by Gerard J. Murphy. But really any book on operator algebras covers this (Takesaki, or Kadison and Ringrose are also good). – Matthew Daws Jan 14 '12 at 17:08

This question seems to have been edited recently. The answer to the (new) question is: what you have defined is the "Banach Space Injective Tensor Product" of $X$ and $Y$. There is some information here: http://en.wikipedia.org/wiki/Topological_tensor_product
By the first part of your question, we're not actually interested in the Hilbert space representation. So we only care about what the C$^*$-algebraic minimal tensor product of $\ell^\infty(I)$ and $\ell^\infty(J)$ is. We identify these spaces with $C(\beta I)$ and $C(\beta J)$ (Stone-Cech compactifications). You'll find in standard texts that the minimal tensor product of $C(K)$ with $C(L)$ is merely $C(K\times L)$. Thus, if you follow through the calculation, you will find that for $\tau=\sum x_i\otimes y_i \in X\otimes Y$, we have $$\|\tau\| = \sup\Big\{ \Big| \sum_i \mu(x_i) \lambda(y_i) \Big| : \mu\in B_{X^*}, \lambda\in B_{Y^*} \Big\}.$$ This is by definition the injective tensor norm. For the basics, you can read this in Takesaki, for example (or Kadison and Ringrose I think). For more, I like Ryan's book (see the wikipedia article).
Sorry for my my poor English, but I don't understand the sentence: "As that matters is what the $C^*$-algebraic minimal tensor product of $\ell_\infty(I)$ and $\ell_\infty(J)$ is" – Norbert Mar 19 '12 at 17:59