We establish a general framework for representability of a metric group on a (well-behaved) class of Banach spaces. More precisely, let
G
\mathcal {G}
be a topological group, and
A
\mathcal {A}
a unital symmetric
C
∗
C^*
-subalgebra of
U
C
(
G
)
\mathrm {UC}(\mathcal {G})
, the algebra of bounded uniformly continuous functions on
G
\mathcal {G}
. Generalizing the notion of a stable metric, we study
A
\mathcal {A}
-metrics
δ
\delta
, i.e., the function
δ
(
e
,
⋅
)
\delta (e, \cdot )
belongs to
A
\mathcal {A}
; the case
A
=
W
A
P
(
G
)
\mathcal {A}=W\hskip -0.7mm A\hskip -0.2mm P(\mathcal {G})
, the algebra of weakly almost periodic functions on
G
\mathcal {G}
, recovers stability. If the topology of
G
G
is induced by a left invariant metric
d
d
, we prove that
A
\mathcal {A}
determines the topology of
G
\mathcal {G}
if and only if
d
d
is uniformly equivalent to a left invariant
A
\mathcal {A}
-metric. As an application, we show that the additive group of
C
[
0
,
1
]
C[0,1]
is not reflexively representable; this is a new proof of Megrelishvili [Topological transformation groups: selected topics, Elsevier, 2007, Question 6.7] (the problem was already solved by Ferri and Galindo [Studia Math. 193 (2009), pp. 99–108] with different methods and later the results were generalized by Yaacov, Berenstein, and Ferri [Math. Z. 267 (2011), pp.129–138]). Let now
G
\mathcal {G}
be a metric group, and assume
A
⊆
L
U
C
(
G
)
\mathcal {A}\subseteq \mathrm {LUC}(\mathcal {G})
, the algebra of bounded left uniformly continuous functions on
G
\mathcal {G}
, is a unital
C
∗
C^*
-algebra which is the uniform closure of coefficients of representations of
G
\mathcal {G}
on members of
F
\mathscr {F}
, where
F
\mathscr {F}
is a class of Banach spaces closed under
ℓ
2
\ell _2
-direct sums. We prove that
A
\mathcal {A}
determines the topology of
G
\mathcal {G}
if and only if
G
\mathcal {G}
embeds into the isometry group of a member of
F
\mathscr {F}
, equipped with the weak operator topology. As applications, we obtain characterizations of unitary and reflexive representability.