Let
A
A
be a Banach algebra,
X
X
a closed subspace of
A
∗
A^*
,
Y
Y
a dual Banach space with predual
Y
∗
Y_*
, and
π
\pi
a continuous representation of
A
A
on
Y
Y
. We call
π
\pi
subordinate to
X
X
if each coordinate function
π
y
,
λ
∈
X
\pi _{y,\lambda }\in X
, for all
y
∈
Y
,
λ
∈
Y
∗
y\in Y, \lambda \in Y_*
. If
X
X
is topologically left (right) introverted and
Y
Y
is reflexive, we show the existence of a natural bijection between continuous representations of
A
A
on
Y
Y
subordinate to
X
X
, and normal representations of
X
∗
X^*
on
Y
Y
. We show that if
A
A
has a bounded approximate identity, then every weakly almost periodic functional on
A
A
is a coordinate function of a continuous representation of
A
A
subordinate to
W
A
P
(
A
)
WAP(A)
. We show that a function
f
f
on a locally compact group
G
G
is left uniformly continuous if and only if it is the coordinate function of the conjugate representation of
L
1
(
G
)
L^1(G)
, associated to some unitary representation of
G
G
. We generalize the latter result to an arbitrary Banach algebra with bounded right approximate identity. We prove the functionals in
L
U
C
(
A
)
LUC(A)
are all coordinate functions of some norm continuous representation of
A
A
on a dual Banach space.