= C-star unitization by left multiplication
{c}
{title2=$\|a+\lambda1\|=\|L_a+\lambda I\|$}
The left multiplication representation of a <C-star algebra> is isometric: test $L_a$ on $a^*/\|a\|$ to get $\|L_a\|=\|a\|$. For a nonunital algebra, its closed image plus the identity operator is the algebraic <unitization>. For a formal $z=a+\lambda1$ and $b\in A$, $\|zb\|^2=\|b^*z^*zb\|\leq\|b\|\|L_{z^*z}b\|$. Thus $\|L_z\|^2\leq\|L_{z^*z}\|\leq\|L_{z^*}\|\|L_z\|$; applying the same estimate to $z^*$ proves the <algebra involution> is isometric and the <C-star identity> holds. This gives a unitization without first assuming an operator representation on a <Hilbert space>.
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