We introduce the notion of star-fundamental algebra over a field of characteristic zero. We prove that in the framework of the theory of polynomial identities, these algebras are the building blocks of a finite dimensional algebra with involution
∗
*
.
To any star-algebra
A
A
is attached a numerical sequence
c
n
∗
(
A
)
c_n^*(A)
,
n
≥
1
n\ge 1
, called the sequence of
∗
*
-codimensions of
A
A
. Its asymptotic is an invariant giving a measure of the
∗
*
-polynomial identities satisfied by
A
A
. It is well known that for a PI-algebra such a sequence is exponentially bounded and
exp
∗
(
A
)
=
lim
n
→
∞
c
n
∗
(
A
)
n
\exp ^*(A)=\lim _{n\to \infty }\sqrt [n]{c_n^*(A)}
can be explicitly computed. Here we prove that if
A
A
is a star-fundamental algebra,
C
1
n
t
exp
∗
(
A
)
n
≤
c
n
∗
(
A
)
≤
C
2
n
t
exp
∗
(
A
)
n
,
\begin{equation*} C_1n^t\exp ^*(A)^n\le c_n^*(A)\le C_2n^t \exp ^*(A)^n, \end{equation*}
where
C
1
>
0
,
C
2
,
t
C_1>0,C_2, t
are constants and
t
t
is explicitly computed as a linear function of the dimension of the skew semisimple part of
A
A
and the nilpotency index of the Jacobson radical of
A
A
. We also prove that any finite dimensional star-algebra has the same
∗
*
-identities as a finite direct sum of star-fundamental algebras. As a consequence, by the main result in [J. Algebra 383 (2013), pp. 144–167] we get that if
A
A
is any finitely generated star-algebra satisfying a polynomial identity, then the above still holds and, so,
lim
n
→
∞
log
n
c
n
∗
(
A
)
exp
∗
(
A
)
n
\lim _{n\to \infty }\log _n \frac {c_n^*(A)}{\exp ^*(A)^n}
exists and is an integer or half an integer.