Let
A
A
be a semisimple right complemented Banach algebra,
L
A
{L_A}
the left regular representation of
A
A
, and
M
l
(
A
)
{M_l}\left ( A \right )
the left multiplier algebra of
A
A
. In this paper we are concerned with
L
A
{L_A}
and its relationship to
A
A
and
M
l
(
A
)
{M_l}\left ( A \right )
. We show that
L
A
{L_A}
is an annihilator algebra and that it is a closed ideal of
M
l
(
A
)
{M_l}\left ( A \right )
. Moreover,
L
A
{L_A}
and
M
l
(
A
)
{M_l}\left ( A \right )
have the same socle. We also show that the left multiplier algebra of a minimal closed ideal of
A
A
is topologically algebra isomorphic to
L
(
H
)
L\left ( H \right )
, the algebra of bounded linear operators on a Hilbert space
H
H
. Conditions are given under which
L
A
{L_A}
is right complemented.