Toeplitz operators with $C+H^\infty$ symbols on $l^p$

We show that the algebra of all multipliers on $l^p$ $(1<p<\infty)$ contains a closed subalgebra, $C_p+H_p^\infty$, which coincides with the familiar algebra $C+H^\infty$ in the case $p=2$. We also prove that a Toeplitz operator with $C_p+H_p^\infty$ symbol is Fredholm on $l^p$ if and only if its symbol is invertible in $C_p+H_p^\infty$.