Finitely generated algebras with involution and multiplicities bounded by a constant

Let A be an algebra with involution ⁎ over a field F   of characteristic zero, and let χn⁎(A), n=1,2,⋯n=1,2,⋯ , be the sequence of ⁎-cocharacters of A  . For every n≥1n≥1, let ln⁎(A) denote the nth ⁎-colength of A   which is the sum of the multiplicities in χn⁎(A). In this article, we classify in two different ways the finitely generated ⁎-algebras satisfying an ordinary polynomial identity whose multiplicities of the ⁎-cocharacters χn⁎(A) are bounded by a constant. As a consequence this also yields a characterization of the ⁎-varieties whose ⁎-colength ln⁎(A), n=1,2,…n=1,2,… , is bounded by a constant.