Star[a_,b_] := ((a+b-2 a b)/(1-a b))

d={Star[Star[a,b],c], Star[a, Star[b,c]]}

d // Simplify

{1,-1}.d // Simplify

Star[1,a]

Simplify[D[Star[f[x], f[y]], y]]

eqn=f'[x]==Simplify[
D[Star[f[x], f[y]], y] /. {f[y] ->0, f'[y] -> c}
]

DSolve[eqn, f[x], x]

g[x_,c1_,c2_]:=(x c1+c2-1)/(c1 x+c2)

Simplify[
g[x+y] == Star[g[x], g[y]]
]

h[x_] = x/(x+1)

Simplify[
h[x+y]
]