Some results on ∗-minimal algebras with involution

Let $(A, *)$ be an associative algebra with involution over a field $F$ of characteristic zero, $T_*(A)$ the ideal of $*$-polynomial identities of $A$ and $c_n(A, *),$ $n=1, 2, \ldots$, the corresponding sequence of $*$-codimensions. Recall that $c_n(A, *)$ is the dimension of the space of multilinear polynomials in $n$ variables in the corresponding relatively free algebra with involution of countable rank. \par When $A$ is a finite dimensional algebra, Giambruno and Zaicev [J. Algebra 222 (1999), no. 2, 471–484; MR1734235 (2000i:16046)] proved that the limit $$\exp(A, *)=\lim_{n\to \infty}\sqrt[n]{c_n(A, *)}$$ exists and is an integer called the $*$-exponent of $A.$ \par Among finite dimensional algebras with the same $*$-exponent a prominent role is played by the so called $*$-minimal algebras. This notion was introduced by Di Vincenzo and La Scala in [J. Algebra 317 (2007), no. 2, 642–657; MR2362935 (2008j:16095)]. Recall that a finite dimensional algebra $(A, *)$ is $*$-minimal if for any finite dimensional algebra $B$ with involution such that $T_*(A)\subset T_*(B)$ we have that $\exp(A,*)>\exp(B,*).$ \par In this paper the authors review recent results of the first author et al regarding $*$-minimal algebras and prove further properties towards a complete classification of $*$-minimal algebras.