Let $A$ and $B$ be Banach algebras. Then the map $\phi:(A\widehat\otimes A) \oplus\_\infty (B\widehat\otimes B) \to (A\oplus\_\infty B)\widehat\otimes(A\oplus\_\infty B)$ is a contractive embedding.


Can you give me a proof for the above proposition? Or give me a reference for its proof.



Projective tensor product
I guess that by $E\oplus\_\infty F$ you mean the direct sum with the maximum as norm. I also guess that you consider the natural mapping. Let us look at the dual mapping:
\begin{align}
&((A\oplus\_\infty B)\widehat\otimes(A\oplus\_\infty B))' 
= L(A\oplus\_\infty B, (A\oplus\_\infty B)')
= L(A\oplus\_\infty B, A'\oplus\_1 B')\to
\\&
\to ((A\widehat\otimes A)\oplus\_\infty (B\widehat\otimes B))' 
= (A\widehat\otimes A)'\oplus\_1 (B\widehat\otimes B)'
=L(A,A')\oplus\_1 L(B,B')
\end{align}
In block matrix representation the mapping is now projecting out the main diagonal. The operator norms are easily handled. This projection is now obviously a contraction. This proof works for general Banach spaces.


