A unitization that keeps its scalar quotient
GPT-6 Sol (OpenAI), Codex writing thread, Ultra effort, September 2026. New original prose: CC0.
Even when a \(C^*\)-algebra already has an identity, it can be useful to adjoin a new one. The resulting forced unitization always has a quotient onto \(\mathbb C\) whose kernel is the original algebra. That quotient detects when a continuous function of an element of \(A\) returns to \(A\), as needed for approximate identities. We use the programme's existing forced-unitization construction and give an alternative proof of its maximum norm for the C*-weight route. The required abstract self-adjoint calculus and positive-cone theorem are proved in AC1–4. The free comparison is Blackadar, corrected author edition, II.1.2.1, printed page 54. We keep the scalar coordinate in the norm so that the same construction also works for already-unital and zero algebras.
The earlier programme Banach-algebra lesson owns the forced-unitization statement. Equations (UZ.3)–(UZ.12) below give its complete local alternative proof; (UZ.2) and (UZ.14)–(UZ.15) prove the quotient-calculus and inherited-cone applications. The construction through UZ05 uses no functional calculus. Only UZ06–07 invoke AC1–4, which are proved for an already given unital C*-algebra. Thus neither unitization nor the calculus is being assumed circularly.
The programme provider is originally by Claude Opus 5.5 (Anthropic), September 2026, with its October 2026 revision and full self-check by GPT-6.1 Sol (OpenAI), Ultra, CC0. The new bridges to earlier programme lessons here are by GPT-6.1 Sol (OpenAI), Ultra, October 2026, CC0; the earlier local prose retains the credit above. The current free-source reconstruction, analytic prerequisite proof and domain checks are by GPT-6 Astra (OpenAI), Ultra, October 2026, CC0.
For every complex \(C^*\)-algebra \(A\), including \(A=0\), we construct a unital \(C^*\)-algebra \(\widetilde A=A\oplus\mathbb C\), an isometric *-homomorphism \(j\), and a unital contractive *-homomorphism \(q\) satisfying
\[ j(a)=(a,0),\qquad q(a,\lambda)=\lambda,\qquad \ker q=j(A). \tag{UZ.1} \]The image \(j(A)\) will be a closed two-sided ideal, \(\widetilde A/j(A)\cong\mathbb C\) isometrically, and \(j(A_+)=j(A)\cap\widetilde A_+\). In particular, for self-adjoint \(a\in A\) and any continuous \(f\) on an interval containing the spectrum of \(j(a)\),
\[ q(f(j(a)))=f(0). \tag{UZ.2} \]Thus \(f(j(a))\in j(A)\) whenever \(f(0)=0\). The zero is in the spectrum of \(j(a)\), since its image under the unital map \(q\) is zero.
The algebraic extension and its left action
The following algebraic construction and norm calculation prove the programme contract in the scalar-quotient notation needed for weights.
Give \(A\oplus\mathbb C\) its vector-space operations and define
\[ \begin{aligned} (a,\lambda)(b,\mu)&=(ab+\lambda b+\mu a,\lambda\mu),\\ (a,\lambda)^\star&=(a^*,\overline\lambda). \end{aligned} \tag{UZ.3} \]These operations have identity \({\bf1}=(0,1)\). Associativity can be checked without assuming a norm: both bracketings of a triple product have first coordinate
\[ abc+\lambda bc+\mu ac+\nu ab +\lambda\mu c+\lambda\nu b+\mu\nu a \]and second coordinate \(\lambda\mu\nu\). The involution reverses products by the same expansion. Formula (UZ.3) also shows that \(j(A)\) is a two-sided ideal and that \(j,q\) preserve products and involutions.
Let \(\mathcal L(A)\) be the bounded linear operators on the Banach space \(A\). The left action and its seminorm are
\[ T_{(a,\lambda)}b=ab+\lambda b,\qquad p(a,\lambda)=\|T_{(a,\lambda)}\|_{\mathcal L(A)}. \tag{UZ.4} \]The estimate \(\|T_{(a,\lambda)}b\|\leq(\|a\|+|\lambda|)\|b\|\) proves boundedness. Expanding products gives \(T_{uv}=T_uT_v\), so \(p\) is submultiplicative. No Hilbert-space adjoint is involved in \(T_{u^\star}\); it is simply the left action of another algebra element.
For \(L_a=T_{j(a)}\),
\[ \|L_a\|=\|a\|. \tag{UZ.5} \]Submultiplicativity gives the upper bound. If \(a\ne0\), test \(L_a\) on \(a^*/\|a\|\); the \(C^*\)-identity and isometry of involution give \(\|aa^*\|/\|a\|=\|a\|\). The latter isometry follows from \(\|a\|^2\leq\|a^*\|\|a\|\) and the same inequality with \(a^*\), treating zero separately. The zero algebra satisfies (UZ.5) as well.
Checking the C*-seminorm
The next calculation proves the C*-seminorm identity directly from the C*-identity of the original algebra. The left action is on a Banach space; its involution is inherited algebraically and is not an operator adjoint on that space.
Write \(ub=T_ub\) for \(u\in A\oplus\mathbb C\) and \(b\in A\). The algebraic equality \((ub)^*(ub)=b^*(u^\star u)b\) occurs entirely in \(A\). Its \(C^*\)-identity gives
\[ \begin{aligned} \|T_ub\|^2 &=\|b^*T_{u^\star u}b\|\\ &\leq\|b\|\|T_{u^\star u}b\| \leq p(u^\star u)\|b\|^2. \end{aligned} \tag{UZ.6} \]Taking a supremum over unit-ball \(b\), then using \(T_{u^\star u}=T_{u^\star}T_u\), yields
\[ p(u)^2\leq p(u^\star u)\leq p(u^\star)p(u). \tag{UZ.7} \]Apply this also to \(u^\star\). If one of \(p(u),p(u^\star)\) is zero, both are zero; otherwise divide to obtain \(p(u)\leq p(u^\star)\) and its reverse. The two bounds in (UZ.7) then coincide:
\[ p(u^\star)=p(u),\qquad p(u^\star u)=p(u)^2. \tag{UZ.8} \]This argument uses neither a representation theorem nor an approximate identity.
Why the scalar coordinate matters
The scalar coordinate will remove the possible kernel of the left action, including when the original algebra already has an identity.
Define
\[ N(a,\lambda)=\max\bigl(\|L_a+\lambda I_A\|,|\lambda|\bigr) =\max\bigl(p(a,\lambda),|q(a,\lambda)|\bigr). \tag{UZ.9} \]It is a seminorm by the triangle inequality. If it vanishes, \(\lambda=0\) and (UZ.5) forces \(a=0\), so it is a norm. Both terms are submultiplicative; (UZ.8) gives \(N(u^\star)=N(u)\) and
\[ \begin{aligned} N(u^\star u) &=\max\bigl(p(u^\star u),|q(u^\star u)|\bigr)\\ &=\max\bigl(p(u)^2,|q(u)|^2\bigr)=N(u)^2. \end{aligned} \tag{UZ.10} \]Also \(N({\bf1})=\max(\|I_A\|,1)=1\), including \(A=0\).
If \(A\) already has a unit, the formal element \((-1_A,1)\) is nonzero and acts as zero on \(A\). Its left-action seminorm is therefore zero, while \(N(-1_A,1)=1\). For \(A=\mathbb C\), the map \((a,\lambda)\mapsto(a+\lambda,\lambda)\) identifies the forced unitization with \(\mathbb C\oplus\mathbb C\) and gives \(N(a,\lambda)=\max(|a+\lambda|,|\lambda|)\). The extra scalar coordinate is visible even in this smallest example.
Completeness, ideal and quotient norm
From the definition of \(T\),
\[ N(a,\lambda)\leq\|a\|+|\lambda|. \tag{UZ.11} \]Conversely, \(|\lambda|\leq N(a,\lambda)\), and (UZ.5) gives \(\|a\|=\|L_a\|\leq p(a,\lambda)+|\lambda|\|I_A\|\leq2N(a,\lambda)\). Hence
\[ \|a\|+|\lambda|\leq3N(a,\lambda). \tag{UZ.12} \]The sum norm on \(A\oplus\mathbb C\) is complete because both coordinates are complete. The two inequalities show that an \(N\)-Cauchy sequence converges in the sum norm and hence in \(N\). Therefore \((A\oplus\mathbb C,N)\) is a unital \(C^*\)-algebra.
Equation (UZ.5) makes \(j\) isometric. Its image is closed by completeness and is the ideal already identified algebraically. The inequality \(|q(u)|\leq N(u)\), with equality at \({\bf1}\), makes \(q\) contractive of norm one. Every element with scalar coordinate \(\lambda\) has norm at least \(|\lambda|\); the representative \((0,\lambda)\) has norm exactly \(|\lambda|\). Thus the quotient norm of its coset is \(|\lambda|\), proving the asserted isometric quotient.
Continuous functions and the quotient
We now apply the self-adjoint continuous calculus proved in AC1–4 to the constructed unital \(C^*\)-algebra. If \(h=h^\star\), then \(q(h)\) is real and lies in \(\sigma_{\widetilde A}(h)\): otherwise an inverse for \(h-q(h){\bf1}\) would map under \(q\) to an inverse for zero.
Take a compact real interval \(J\) containing the spectrum and a continuous \(f:J\to\mathbb C\). Polynomial approximation on \(J\) can be seen directly from Bernstein polynomials. After rescaling to \([0,1]\), for real continuous \(g\) put
\[ B_ng(t)=\sum_{k=0}^n g(k/n){n\choose k}t^k(1-t)^{n-k}. \]The binomial weights sum to one, have weighted mean \(k/n=t\), and variance \(t(1-t)/n\). Their mass outside \(|k/n-t|<\delta\) is at most \(1/(4n\delta^2)\). Uniform continuity then gives, whenever \(|g(s)-g(t)|<\eta\) for \(|s-t|<\delta\),
\[ \sup_t|B_ng(t)-g(t)| \leq\eta+\frac{\|g\|_\infty}{2n\delta^2}. \tag{UZ.13} \]Send \(n\) to infinity and then \(\eta\) to zero; approximate the real and imaginary parts separately. A singleton interval uses only a constant polynomial.
Choose resulting polynomials \(P_n\to f\) uniformly on \(J\). Since \(q\) is unital and multiplicative, \(q(P_n(h))=P_n(q(h))\). Continuity of \(q\) and isometry of functional calculus pass to the limit, giving
\[ q(f(h))=f(q(h)). \tag{UZ.14} \]For \(h=j(a)\) with \(a=a^*\), this is (UZ.2). If \(f(0)=0\), the result lies in \(\ker q=j(A)\). For positive \(j(a)\), the functions \(t/(t+\varepsilon)\), \(\sqrt{t/(t+\varepsilon)}\), and \(\sqrt t\) therefore return elements of \(j(A)\). The inverse \((j(a)+\varepsilon{\bf1})^{-1}\) need not itself lie there.
The inherited positive cone
Use \(A_+=\{b^*b:b\in A\}\). If \(a=b^*b\), then \(j(a)=j(b)^\star j(b)\geq0\) in \(\widetilde A\). Conversely, suppose \(j(a)\geq0\) there. Its positive square root \(c\) exists by the unital functional calculus. Formula (UZ.14) gives \(q(c)=\sqrt{q(j(a))}=0\), so \(c=j(b)\) for some \(b\in A\). Now \(j(a)=c^\star c=j(b^*b)\), and injectivity of \(j\) yields \(a=b^*b\). Therefore
\[ j(A_+)=j(A)\cap\widetilde A_+. \tag{UZ.15} \]Order comparisons among self-adjoint elements of \(A\) agree with those in \(\widetilde A\), and their norms agree by (UZ.5). The approximate-identity calculation can therefore use the forced unitization without changing either the original order or the original norm.
References
- Bruce Blackadar, Operator Algebras, corrected author edition of 8 February 2017, II.1.2.1 (unitization), II.2.3.1–2 and II.3.1.1–5 (calculus and order), printed pages 54 and 63–66. The local proofs above and AC1–4 supply the arguments; the reference does not replace them.