*-polynomial identities of 4x4 upper triangular matrices with the reflection involution

Abstract

Let $U{T}_4(F)$ be $4\times 4$ upper triangular matrix algebra over a field $F$ of characteristic zero and let $\mathcal{A}$ be the subalgebra of $U{T}_4(F)$ linearly generated by $\{{\mathbf{e}}_{ij}:1\le i\le j\le 4\}\setminus {\mathbf{e}}_{23}$ where $\{{\mathbf{e}}_{ij}:1\le i\le j\le 4\}$ is the standard basis of $U{T}_4(F)$. We describe the set of all $\ast$-polynomial identities for $\mathcal{A}$ with the involution defined by the reflection of second diagonal.