Cfrgtkky

If $(A_i)$ is a sequence of nuclear $C^*$ algebras,Is $oplus_{c_0}A_i$ ($c_0$ direct sum)and $prod A_i$($ell ^infty $ direct sum) also nuclear?

Fix $ain bigoplus_nA_n$ and $varepsilon>0$. For each $n$, there exist ucp maps $varphi_n:A_nto M_{k(n)}(mathbb C)$ and $psi_varepsilon:M_{k(n)}(mathbb C)to A_n$ such that $|psi_ncircvarphi_n(a_n)-a_n|<varepsilon$.
There is also $m$ such that $|a_n|<varepsilon$ for all $ngeq m$. Write $a_0$ for the truncation of $a$ to its first $m$ entries; then $|a-a_0|<varepsilon$. Then the maps
$$
varphi:bigoplus_nA_nto bigoplus_{n=1}^mM_{k(n)}(mathbb C), psi:bigoplus_{n=1}^mM_{k(n)}(mathbb C)to bigoplus_nA_n
$$
given by $$varphi(b)=bigoplus_{n=1}^mvarphi_n(b_n), psi(bigoplus_{n=1}^m c_n)=bigoplus_{n=1}^m psi_n(c_n)$$
satisfy
begin{align}
|psicircvarphi(a)-a|
&leq|psicircvarphi(a)-psicircvarphi(a_0)|+|psicircvarphi(a_0)-a_0|+|a_0-a|\ \
&leq 2|a_0-a|+|psicircvarphi(a_0)-a_0|<3varepsilon.
end{align}
By making this work over the finite sets $Fsubset bigoplus_nA_n$ we obtain net ${varphi_F}$ and ${psi_F}$ such that $|psi_Fcircvarphi_F(a)-a|to0$ for all $ain bigoplus_nA_n$.

The diret product, on the other hand is not nuclear; it's not even exact. For instance $M=prod_n M_n(mathbb C)$ is not nuclear. It is well-known that nuclear algebras are exact, and that exactness passes to subalgebras. The full C$^*$-algebra of $mathbb F_2$ is known to be non-exact, and to be residually finite; this means that there exists a faithful representation $pi:C^*(mathbb F_2)to M=prod_nM_n(mathbb C)$. So $M$ cannot be exact, and in particular it is not nuclear.