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 $*$-polynomialidentities of $A$ and $c_n(A, *),$ $n=1, 2, \ldots$, thecorresponding 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 withinvolution of countable rank.\parWhen $A$ is a finite dimensional algebra, Giambruno and Zaicev [J.Algebra 222 (1999), no. 2, 471–484; MR1734235 (2000i:16046)] provedthat the limit $$\exp(A,*)=\lim_{n\to \infty}\sqrt[n]{c_n(A, *)}$$ exists and is an integercalled the $*$-exponent of $A.$\parAmong finite dimensional algebras with the same $*$-exponent aprominent role is played by the so called $*$-minimal algebras. Thisnotion was introduced by Di Vincenzo and La Scala in [J. Algebra 317(2007), no. 2, 642–657; MR2362935 (2008j:16095)]. Recall that afinite dimensional algebra $(A, *)$ is $*$-minimal if for anyfinite dimensional algebra $B$ with involution such that$T_*(A)\subset T_*(B)$ we have that $\exp(A,*)>\exp(B,*).$\parIn this paper the authors review recent results of the first authoret al regarding $*$-minimal algebras and prove further propertiestowards a complete classification of $*$-minimal algebras.