Some formulae for the projective operator space tensor product

1. If $E$ and $F$ are normed spaces, we have [BP91, Prop. 4.1]
   

   $$MAX(E) \stackrel{\scriptscriptstyle \wedge}{\otimes}MAX(F) \stackrel{\mathrm{cb}}{=} MAX(E \otimes_\gamma F)$$

   
   
   completely isometrically.
2. Taking the projective operator space tensor product of various combinations of the column Hilbert spaces ${\mathcal{C}}$, the row Hilbert spaces ${\mathcal{R}}$ and an arbitrary operator space $X$ one obtains the following completely isometric identifications: [ER91], [Ble92b, Prop. 2.3]
   1. $X \stackrel{\scriptscriptstyle \wedge}{\otimes}{\mathcal{C}}_\H\stackrel{\mathrm{cb}}{=}X \otimes_{h}{\mathcal{C}}_\H$
   2. ${\mathcal{R}}_\H\stackrel{\scriptscriptstyle \wedge}{\otimes}X \stackrel{\mathrm{cb}}{=}{\mathcal{R}}_\H\otimes_{h}X$
   3. ${\mathcal{C}}_\H\stackrel{\scriptscriptstyle \wedge}{\otimes}{\mathcal{C}}_\ma... ...}}_\mathcal{K}\stackrel{\mathrm{cb}}{=} {\mathcal{C}}_{\H\otimes_2 \mathcal{K}}$
   4. ${\mathcal{R}}_\H\stackrel{\scriptscriptstyle \wedge}{\otimes}{\mathcal{R}}_\ma... ...}}_\mathcal{K}\stackrel{\mathrm{cb}}{=} {\mathcal{R}}_{\H\otimes_2 \mathcal{K}}$
   5. ${\mathcal{R}}_{\overline{\H}} \stackrel{\scriptscriptstyle \wedge}{\otimes}{\m... ...\otimes_{h}{\mathcal{C}}_\mathcal{K}\stackrel{\mathrm{cb}}{=} T(\H,\mathcal{K})$
   6. $X \stackrel{\scriptscriptstyle \wedge}{\otimes}T(\H,\mathcal{K}) \stackrel{\ma... ...{\mathcal{R}}_{\overline{\H}} \otimes_{h}X \otimes_{h}{\mathcal{C}}_\mathcal{K}$
   7. $\mathit{CB}(X,B(\mathcal{K},\H)) \stackrel{\mathrm{cb}}{=} ({\mathcal{R}}_{\ov... ...thcal{R}}_{\overline{\H}} \otimes_{h}X \otimes_{h}{\mathcal{C}}_\mathcal{K})^*$,
   where the space of trace class operators $T(\H,\mathcal{K})$ is endowed with its natural operator space structure: $T({\H,\mathcal{K}}) :\stackrel{\mathrm{cb}}{=}K({\mathcal{K}, \H})^*$.
3. Let $M$ and $N$ be von Neumann algebras and denote by $M \overline{\otimes}N$ the von Neumann tensor product³⁵. For the preduals one has
   $$M_* \stackrel{\scriptscriptstyle \wedge}{\otimes}N_* \stackrel{\mathrm{cb}}{=}(M \overline{\otimes}N)_*$$
   completely isometrically [ER90a].

   Let $G$ and $\H$ be locally compact topological groups and denote by $\mathit{VN}(G)$, $\mathit{VN}(H)$ the corresponding group von Neumann algebras.³⁶ It is well-known that $\mathit{VN}(G)\, \overline{\otimes}\, \mathit{VN}(H) = \mathit \mathit{VN}(G \times H)$. Since the Fourier algebra³⁷ $A(G)$ can be identified with the predual of the group von Neumann algebra $\mathit{VN}(G)$ [Eym64], this implies that

   $$A(G) \stackrel{\scriptscriptstyle \wedge}{\otimes}A(H) \stackrel{\mathrm{cb}}{=}A(G \times H)$$
   holds completely isometrically³⁸ [ER90a].

Footnotes

... pro\-duct³⁵

If $M \subset B(\H )$, $N \subset B(\mathcal{K})$ are von Neumann algebras, $M \overline{\otimes}N$ is defined to be the closure in the weak operator topology of the algebraic tensor product $M \otimes N \subset B(\H \otimes_2 \mathcal{K})$.

... algebras.³⁶

The group von Neumann algebra $\mathit{VN}(G)$ of a locally compact group $G$ is defined to be the von Neumann algebra generated by the left regular representation of $G$ in $B(L_2(G))$.

... algebra³⁷

For a locally compact group $G$, the set $\{ f * {\check{g}} \vert f, g \in L_2(G) \} \subset {\mathrm{C}}_0(G)$ turns out to be a linear space and even an algebra (with pointwise multiplication). Its completion with respect to the norm $\Vert u\Vert=\inf \{ \Vert f\Vert _2 \Vert g\Vert _2 ~\vert~ u = f * {\check{g}} \}$ is a Banach algebra and is called the Fourier algebra of $G$.

... isometrically³⁸

If $G$ is a locally compact abelian group, then $A(G)$ is identified - via the Fourier transform - with $L_1({\widehat{G}})$, where ${\widehat{G}}$ denotes the dual group of $G$. Thus, for locally compact groups $G$ and $\H$, the identification $A(G) \stackrel{\scriptscriptstyle \wedge}{\otimes}A(H) \stackrel{\mathrm{cb}}{=}A(G \times H)$ can be thought of as a non commutative analogue of the well-known classical identification $L_1(G) \otimes_\gamma L_1(H)=L_1(G \times H)$.