If $A$ is a commutative Banach algebra, can we identify $l^\infty(A)$ as the dual of $ l_1\hat\otimes A$ (the projective tensor product of $l_1$ with $A$? Also, what do we know about the quotient space $l^\infty(A)/c_0(A)$?
Thank you!
Asked
Active
Viewed 150 times
1 Answers
2
No. Take $A=c_0$. Then $c_0$ is complemented in $\ell_\infty(c_0)$ (just project down on the very first coordinate). Note that $c_0$ is never complemented in a dual Banach space as this is equivalent to being complemented in $c_0^{**}\cong \ell_\infty$, which is apparently not the case.
See t.b.'s answer for Complementability of von Neumann algebras
Regarding the quotient space $\ell_\infty(A)/c_0(A)$, it is mysterious enough even for $A=\mathbb{C}$. What would you like to know?
Tomasz Kania
- 16,996