  We introduce a new Banach algebra ${\mathcal A}({\mathbb C}_+)$ of bounded
analytic functions on ${\mathbb C}_+=\{z\in{\mathbb C}\, :\, {\rm Re}(z)>0\}$
which is an analytic version of the Figa-Talamenca-Herz algebras on ${\mathbb
R}$. Then we prove that the negative generator $A$ of any bounded
$C_0$-semigroup on Hilbert space $H$ admits a bounded (natural) functional
calculus $\rho_A\colon {\mathcal A}({\mathbb C}_+)\to B(H)$. We prove that this
is an improvement of the bounded functional calculus ${\mathcal B}({\mathbb
C}_+)\to B(H)$ recently devised by Batty-Gomilko-Tomilov on a certain Besov
algebra ${\mathcal B}({\mathbb C}_+)$ of analytic functions on ${\mathbb C}_+$,
by showing that ${\mathcal B}({\mathbb C}_+)\subset {\mathcal A}({\mathbb
C}_+)$ and ${\mathcal B}({\mathbb C}_+)\not= {\mathcal A}({\mathbb C}_+)$. In
the Banach space setting, we give similar results for negative generators of
$\gamma$-bounded $C_0$-semigroups. The study of ${\mathcal A}({\mathbb C}_+)$
requires to deal with Fourier multipliers on the Hardy space $H^1({\mathbb
R})\subset L^1({\mathbb R})$ of analytic functions.
