Gradings, graded identities, $*$-identities and graded $*$-identities of an algebra of upper triangular matrices

Abstract: Let $K \langle X\rangle$ be the free associative algebra freely generated over the field $K$ by the countable set $X = {x_1, x_2, \ldots}$. If $A$ is an associative $K$-algebra, we say that a polynomial $f(x_1,\ldots, x_n) \in K \langle X\rangle$ is a polynomial identity, or simply an identity in $A$ if $f(a_1,\ldots, a_n) = 0$ for every $a_1, \ldots, a_n \in A$. Consider $\mathcal{A}$ the subalgebra of $UT_3(K)$ given by: [ \mathcal{A} = K(e_{1,1} + e_{3,3}) \oplus Ke_{2,2} \oplus Ke_{2,3} \oplus Ke_{3,2} \oplus Ke_{1,3} , ] where $e_{i,j}$ denote the matrix units. We investigate the gradings on the algebra $\mathcal{A}$, determined by an abelian group, and prove that these gradings are elementary. Furthermore, we compute a basis for the $\mathbb{Z}_2$-graded identities of $\mathcal{A}$, and also for the $\mathbb{Z}_2$-graded identities with graded involution. Moreover, we describe the cocharacters of this algebra.