Functions in
Op
(
I
D
,
Re
A
+
L
)
{\text {Op}}\left ( {{I_D},\operatorname {Re} A + L} \right )
are harmonic on
D
D
for a closed subalgebra
A
A
of
C
0
(
Y
)
{C_0}\left ( Y \right )
, an ideal
I
I
of
A
A
and a linear subspace
L
L
of finite dimension in
C
0
,
R
(
Y
)
{C_0}{,_R}\left ( Y \right )
unless the uniform closure of
I
I
is selfadjoint.