Algebras with involution with linear codimension growth

We study the ∗-varieties of associative algebras with involution over a field of characteristic zero whichare 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 ofalgebras to be excluded from the ∗-varieties with such property. As a consequence, we find all possiblelinearly bounded ∗-codimension sequences.