Algebras with involution with linear codimension growth

We study the *-varieties of associative algebras with involution over a field of characteristic zero which are generated by a finite-dimensional algebra. In this setting we give a list of algebras classifying all such *-varieties whose sequence of *-codimensions is linearly bounded. Moreover, we exhibit a finite list of algebras to be excluded from the *-varieties with such property. As a consequence, we find all possible linearly bounded *-codimension sequences.