= Unitization of a C-star algebra
{title2=$\widetilde A=A\oplus\mathbb C1$}
= C-star unitization
{c}
{synonym}
For a nonunital <C-star algebra>, adjoining an identity gives multiplication $(a,\lambda)(b,\mu)=(ab+\lambda b+\mu a,\lambda\mu)$ and involution $(a,\lambda)^*=(a^*,\overline\lambda)$. Its standard C-star norm extends that of $A$, with the new identity of norm one. It can be constructed by adjoining the identity operator to a faithful nondegenerate operator representation and taking the <operator norm>. The original algebra is a closed ideal. The usual <spectrum of an element> in a nonunital algebra is taken in this unitization, permitting unital spectral arguments without requiring the original algebra to contain an identity.
Back to article page
