  Let A be any associative algebra graded by a finite abelian group G, then if
we denote by GKdim_k(A) and GKdim^G_k (A) the Gelfand-Kirillov dimension of its
relatively free algebra and its relatively free G-graded algebra in k variables
respectively, then GKdim_k(A)\leq GKdim^G_k (A). We show a counterexample of
the previous result for Jordan algebras (hence non-associative). In particular,
there exists a $Z_2$-grading on $UJ_n$, the Jordan algebra of $n\times n$ upper
triangular matrices, n equal to 2 or 3, such that the previous inequality does
not hold.
