Algebraic unitization of a star algebra

Let $A$ be a complex C*-algebra (C star algebra), not assumed unital. The algebraic unitization of $A$ is the complex vector space

$A^{+}:=A\oplus \mathbb{C}=\{(a,\lambda ):a\in A,\lambda \in \mathbb{C}\},$

equipped with the multiplication, involution and unit

$(a,\lambda )(b,\mu ):=(ab+\lambda b+\mu a,\lambda \mu ),(a,\lambda {)}^{\ast }:=(a^{\ast },\bar{\lambda }),{\mathbf{1}}_{A^{+}}:=(0,1)\mathrm{.}$

The pair is written $a+\lambda 1$ for $(a,\lambda )$, so that the displayed product is the expansion of $(a+\lambda 1)(b+\mu 1)$ with the convention $\lambda b=b\lambda$. The map ${\chi }_{\mathrm{\infty }}(a,\lambda ):=\lambda$ is the quotient character of $A^{+}$; it is complex-linear, multiplicative and nonzero, and it vanishes exactly on $A\oplus \{0\}$.

Three algebraic facts are recorded and verified here because they are used without further comment:

• the product is associative and complex-bilinear, and $(0,1)$ is a two-sided identity: expanding $((a,\lambda )(b,\mu ))(c,\nu )$ and $(a,\lambda )((b,\mu )(c,\nu ))$ gives in both cases $(abc + \lambda bc + \mu ac + \nu ab + \lambda\mu c
  • \lambda\nu b + \mu\nu a,\ \lambda\mu\nu)$;
• the involution is involutive and anti-multiplicative: $(a,\lambda {)}^{\ast \ast }=(a,\lambda )$ and $((a,\lambda )(b,\mu ){)}^{\ast }=(b^{\ast },\bar{\mu })(a^{\ast },\bar{\lambda })$, both direct computations from the C*-algebra axioms;
• $A\cong A\oplus \{0\}$ is a two-sided ideal of $A^{+}$ with $(a,\lambda )(b,0)=(ab+\lambda b,0)$ and $(b,0)(a,\lambda )=(ba+\lambda b,0)$ in $A\oplus \{0\}$.

No norm is defined here. For nonzero genuinely nonunital $A$, Minimal C star unitization constructs a C*-norm on $A^{+}$; its operator construction uses nonunitality to be injective. That theorem is not invoked here for unital $A$. If $A$ is the zero algebra then $A^{+}\cong \mathbb{C}$ is the complex numbers with their usual structure, and the quotient character is the identity map.

• The direct sum is algebraic. The definition does not require $A$ to be nonunital; if $A$ happens to be unital then $A^{+}$ is still the direct sum with its product, but the unit of $A^{+}$ differs from the unit of the ideal $A$, and for that reason the C*-norm theorem below is stated only for genuinely nonunital $A$.
• The quotient character is the point at infinity. For commutative, genuinely nonunital $A$ (with the zero algebra treated separately), the later representation theorem identifies ${\chi }_{\mathrm{\infty }}$ with evaluation at the added point of the one-point compactification (Character space of the unitization is one-point compactification).