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.
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.
