Algebras with Unit - x Has Quasi-Inverse y IFF 1 - x Has Inverse 1 - y

# Algebras with Unit - x Has Quasi-Inverse y IFF 1 - x Has Inverse 1 - y

 Proposition 1: Let $\mathfrak{A}$ be an algebra with unit and let $x, y \in \mathfrak{A}$. Then $x$ is quasi-invertible with quasi-inverse $y$ if and only if $1 - x$ is invertible with inverse $1 - y$.

Note that the above proposition uses both the terms "quasi-inverse" AND "inverse".

• Proof: $\Rightarrow$ Suppose that $x$ is quasi-invertible with quasi-inverse $y$. Then $x \circ y = y \circ x = 0$. We see that:
(1)
\begin{align} \quad \quad (1 - x)(1 - y) = 1 - y - x + xy = 1 - (x + y - xy) = 1 - (x \circ y) = 1 - 0 = 1 \end{align}
• And similarly we have that:
(2)
\begin{align} \quad \quad (1 - y)(1 - x) = 1 - x - y + yx = 1 - (y + x - yx) = 1 - (y \circ x) = 1 - 0 = 1 \end{align}
• Therefore $1 - x$ is invertible and the inverse of $1 - x$ is $1 - y$.
• $\Leftarrow$ Suppose that $1 - x$ is invertible with inverse $1 - y$. Then $(1 - x)(1 - y) = 1$ and $(1 - y)(1 - x) = 1$, or equivalently:
(3)
\begin{align} \quad 1 - x - y + xy = 1\quad \Leftrightarrow \quad x \circ y = 0 \end{align}
• And:
(4)
\begin{align} \quad 1 -y - x + yx = 1 \quad \Leftrightarrow \quad y \circ x = 0 \end{align}
• So $x$ is quasi-invertible with quasi-inverse $y$. $\blacksquare$