Upper triangular $2\times2$-matrices over a Baer *-ring Let $A$ be a Baer $*$-ring. Let us denote $B$ by the space of all upper triangular matrices 
$\left(\begin{array}{cc}
a_1& a_2 \\
0  & a_4 
\end{array}\right)$ where $a_i$'s are in $A$. Is $B$ a Baer *-ring too?
As  for the involution on $B$, I mean any involution that makes the mapping $$a\to \left(\begin{array}{cc}
a& 0 \\
0  & 0 
\end{array}\right)$$ forms an embedding. 

There exist two involutions  on ring $B$ as follows:
$\begin{pmatrix}
a&b\\
0&c
\end{pmatrix}^{\ast_1}= \begin{pmatrix}
c^{\ast}&b^{\ast}\\
0&a^{\ast}
\end{pmatrix},$ 
and
$\begin{pmatrix}
a&b\\
0&c
\end{pmatrix}^{\ast_2}= \begin{pmatrix}
c^{\ast}&-b^{\ast}\\
0&a^{\ast}
\end{pmatrix}$. 
An involution is proper if $aa^{\ast}=0$, then $a=0$. If the $\ast$-ring $R$ is Baer $\ast$, then  $\ast$ is a involution proper.
The involutions $\ast_1$ and $\ast_2$ are not proper, because
$\begin{pmatrix}
1&0\\
0&0
\end{pmatrix}\begin{pmatrix}
1&0\\
0&0
\end{pmatrix}^{\ast_1}=\begin{pmatrix}
1&0\\
0&0
\end{pmatrix}\begin{pmatrix}
1&0\\
0&0
\end{pmatrix}^{\ast_2}=\begin{pmatrix}
1&0\\
0&0
\end{pmatrix}\begin{pmatrix}
0&0\\
0&1
\end{pmatrix}=0$ and $\begin{pmatrix}
1&0\\
0&0
\end{pmatrix}\not=0$. 
Hence $B=T_2(A)$ is not a Baer $\ast$-ring.
