Every bounded functional becomes normal in one representation

The exact theorem and prerequisites follow. OA-MOD-UB-02–09 build the canonical structure, normal extension, central kernel and functorial maps. The current prerequisite binding, normal-inverse details and supported-representation example are by GPT-6 Astra (OpenAI), Ultra, October 2026, CC0. Earlier course authorship is retained.

Let \(A\) be any complex C*-algebra, including a nonunital or zero algebra. Write \(j_A:A\to A^{**}\) for the Banach evaluation embedding, \(j_A(a)(f)=f(a)\), \(f\in A^*\). There is a canonical C*-algebra structure on this Banach bidual making it a von Neumann algebra with predual \(A^*\). The embedding is an isometric *-homomorphism with ultraweakly dense image. For every \(\omega\in A^*_+\), its canonical normal extension is

It is positive and has the same norm. Both multiplication with one variable fixed and the involution are continuous for \(\sigma(A^{**},A^*)\), the involution being conjugate linear.

Every nondegenerate *-representation \(\pi:A\to B(K)\) extends uniquely to a normal onto *-homomorphism

There is a central projection \(z\in A^{**}\) with

and this isomorphism and its inverse are normal. We also prove the precise supported version for degenerate representations and functoriality for arbitrary C*-homomorphisms.

The exact mathematical inputs are as follows.

• The unitization construction and the \(f(0)=0\) functional-calculus consequence in Continuous functions and the quotient supply an isometric ideal \(A\subset\widetilde A\). Unital continuous functional calculus is the written abstract calculus and positive-order theorem, AC1–4. No representation theorem is used to construct this unitization.
• Positive complexification and its norm through Passage to a nonunital algebra supplies positive functionals which detect the norm of every positive element and a decomposition of each \(f\in A^*\) as a complex linear combination of four bounded positive functionals. Its inputs are unitization, Hahn--Banach, locally convex separation, and Banach--Alaoglu. Jordan uniqueness is not required here.
• An approximate identity indexed by finite sets through Cyclicity, normalization and nondegeneracy supply the nonunital cyclic GNS representation for a bounded positive functional, its coefficient vector and exact vector norm, and nondegeneracy. The finite GNS construction is The algebra of finite elements through The GNS quotient and its exact domain restricted to a bounded functional. BK01 proves Hilbert completion, bounded extensions, Riesz representation and adjoints, including the completion used for the direct sum below.
• Why self-adjoint weak density gives strong density through Contractive approximation from a nonunital algebra, with the nonunital density proof in The generated algebra and its commutant, supply strong* density of the contractive ball of any nondegenerate *-subalgebra in its bicommutant. These exact selected proofs do not depend on modular theory or on a universal-bidual theorem.
• Its dual is exactly B(H) through Positive functionals, closed cones and norm closure supply the concrete predual, \(M=(M_*)^*\) isometrically, bounded-form duality, and normal vector functionals. Corners have exactly their inherited predual supplies the inherited predual of a concrete corner, used for the normal inverse in OA-MOD-UB-08. OA-MOD-BK-03--OA-MOD-BK-04 supply fixed multiplication, bounded strong-to-ultraweak convergence and increasing projection suprema; OA-MOD-BK-06 gives support projections, and OA-MOD-BK-08 gives the unitary test for centrality.
• AB05 proves Banach–Alaoglu for the specified real or complex dual unit ball; AB04 proves the compact-image and Hausdorff-closedness facts used in UB08. AB01 proves the Banach-adjoint identities, norm equality, weak* continuity of second adjoints and the annihilator range of an inclusion. NP1 proves real and complex Hahn–Banach with norm preservation. These statements have full written proofs; none is an unproved external contract.

The spanning statement used here concerns the whole Banach dual, as proved in OA-MOD-ST-02–07; the concrete-predual result OA-MOD-CP-03–07 alone does not provide it.

Mathematical antecedents: Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979: Chapter III, §2, Proposition 2.1, Lemma 2.2, Definition 2.3 and Theorem 2.4, printed pp. 120–123, for positive decomposition, universal extensions and the bidual realization; Chapter II, §4, Theorem 4.8, printed pp. 82–84, for Kaplansky density. The exact edition has been checked. HAP04–05 gives the complete alternative density proof using self-adjoint resolvents and a two-by-two matrix argument. Below we prove multiplicativity of the extension, central-kernel reduction, normality of the inverse, and the supported version for degenerate representations explicitly.

Elementary facts about homomorphisms

An algebraic *-homomorphism \(\theta:A\to C\) is contractive. Indeed it extends to a unital algebraic *-homomorphism between forced unitizations by \((a,\lambda)\mapsto(\theta(a),\lambda)\). Square-root factorization in the domain shows that this map preserves positivity. Therefore

in the target unitization, proving \(\|\theta(a)\|\leq\|a\|\). This argument does not presume continuity of the homomorphism.

An injective *-homomorphism is isometric. For \(a\geq0\), suppose \(\|\theta(a)\|<\|a\|\). Choose a continuous real function \(g\) on \([0,\|a\|]\), zero on \([0,\|\theta(a)\|]\) and at zero, with \(g(\|a\|)\ne0\). Unital functional calculus and the unitization \(g(0)=0\) rule give \(g(a)\in A\) and \(g(a)\ne0\). Polynomial approximation with the constant term removed, and the contractivity just proved, give \(\theta(g(a))=g(\theta(a))=0\), a contradiction. Thus norms agree on positive elements. The C*-identity gives agreement on every \(a\):

The zero algebra causes no exception. The range of any *-homomorphism will be used below only as a *-subalgebra, until its closedness is proved.

The universal representation is indexed by all states

Suppose first \(A\ne0\), and let \(S(A)\) be the set of bounded positive functionals of norm one. ST's norm detection makes this set nonempty. For every \(\omega\in S(A)\), BG supplies \((H_\omega,\pi_\omega,\xi_\omega)\) such that

Inner products are linear in their first variable. Put

The Hilbert direct sum is the completion of finitely supported vector families for the sum of squared norms. The uniform estimate \(\|\pi_\omega(a)\|\leq\|a\|\) defines a bounded operator on it. Products and adjoints hold first on finite families and then by density, so \(\pi_u\) is a *-representation.

It is nondegenerate: a vector supported in one summand is approximated there by vectors of the form \(\pi_\omega(a)\xi_\omega\), which are \(\pi_u(a)\) applied to the vector \(\xi_\omega\) in that summand. Finite sums of these vectors are dense in \(H_u\).

For \(a\geq0\), ST and (UB.5) give

The C*-identity gives \(\|\pi_u(a)\|=\|a\|\) for all \(a\). Thus \(\pi_u\) is faithful. There is no countable selection of states in this construction. If \(A=0\), use the empty direct sum \(H_u=0\), \(\pi_u=0\), and \(M=0\); all the following maps are the unique zero maps.

OA-MOD-HAP-05 now gives, for every \(x\in M\) with \(\|x\|\leq1\), a net \(a_i\in A\) with

HAP produces contractions in \(\pi_u(A)\), and its faithful isometry gives their unique contractive lifts. OA-MOD-BK-03 supplies the ultraweak limit from the explicitly bounded net. No bound is inferred from convergence of an arbitrary net.

The restriction of the concrete predual is an onto isometry

By OA-MOD-CP-06, \(M_*\) is a Banach space with \((M_*)^*=M\) isometrically. Define the bounded complex-linear restriction map

It is contractive. For \(x\) in the unit ball of \(M\), take (UB.8). Ultraweak continuity gives

Taking the supremum over that ball proves

In particular \(R\) is injective.

It is onto, for the following reason. If \(\omega\in A^*_+\) is nonzero, write \(c=\|\omega\|\) and \(\omega_0=\omega/c\in S(A)\). On \(M\) the vector functional belonging to \(\sqrt c\,\xi_{\omega_0}\), supported in its one summand of \(H_u\), is positive and normal, and restricts to \(\omega\). Zero has the zero extension. ST writes an arbitrary \(f\in A^*\) as a linear combination of four such positive functionals; the same combination of their normal extensions is an element of \(M_*\) mapped to \(f\). Uniqueness follows from injectivity.

Consequently

is an onto isometry, and it sends positive functionals to positive normal functionals. Its restriction is not an arbitrary Hahn--Banach extension: uniqueness on the whole \(M\) has just been proved.

Transfer of the algebra structure to the Banach bidual

Take the Banach adjoint of (UB.9), followed by CP's concrete duality:

Because \(R\) is an onto isometry, \(U\) is an onto isometry as well: the adjoint of \(R^{-1}\) is its inverse. It is a homeomorphism for the respective weak-star topologies, directly from evaluation. For every \(a\in A\) and \(\rho\in M_*\),

Since the predual separates points, \(Uj_A(a)=\pi_u(a)\). Define

The Banach norm already on \(A^{**}\) is unchanged. Transporting the concrete C*-identities proves all algebra and C*-norm axioms. The map \(j_A\) is an isometric *-homomorphism. Transporting the ultraweak topology proves that this is a von Neumann algebra with the specified predual \(A^*\). The dense image assertion follows from (UB.8) and the homeomorphism \(U\).

For \(\omega\in A^*_+\), put \(\rho=R^{-1}\omega\in M_*^+\). Equation (UB.12) reads

This proves positivity, normality and the norm equality in (UB.1). Its uniqueness among ultraweakly continuous extensions follows from the density of \(j_A(A)\).

This algebra structure is canonical. Any two separately weak-star continuous bilinear products extending the product on \(j_A(A)\) agree first when their second input is in \(j_A(A)\), by approximation of the first input, and then on both arbitrary inputs by approximation of the second. The analogous one-limit argument applies to a weak-star continuous conjugate-linear involution. Thus (UB.13) is independent of the chosen realizations of the cyclic GNS spaces. This is a uniqueness argument for the specified algebra structure, not a theorem about uniqueness of all possible Banach preduals.

Extension of a representation by its coefficient functionals

Let \(\pi:A\to B(K)\) be any *-representation, initially possibly degenerate. It is contractive by OA-MOD-UB-02. The concrete predual of \(B(K)\) from CP defines a bounded map

Its Banach adjoint, under the two established dualities, defines

Thus \(T\) is bounded and ultraweakly continuous, and \(Tj_A=\pi\). Equivalently, all its vector coefficients are specified by

These bounded coefficient forms are another direct description through the Hilbert bounded-form theorem.

We verify multiplicativity, rather than infer it from the adjoint construction. For \(a\in A\) and \(X\in A^{**}\), choose \(j_A(b_i)\to X\) weak-star. Fixed multiplication in each von Neumann algebra is ultraweakly continuous, so

The analogous argument proves \(T(j_A(a)X)=\pi(a)T(X)\). For arbitrary \(Y\), approximate it by \(j_A(a_i)\) in (UB.16). Again fixed multiplication is continuous, giving

Continuity of the involution and a one-variable approximation give \(T(X^*)=T(X)^*\). Hence \(T\) is a normal *-homomorphism. No joint ultraweak continuity of multiplication has been used.

If \(\pi\) is nondegenerate, the projection \(T(1)\) satisfies \(T(1)\pi(a)=\pi(a)\) and is therefore the identity on the dense span of \(\pi(A)K\). Thus \(T(1)=I_K\). The range is contained in \(\pi(A)''\), because every \(T(X)\) is an ultraweak limit of elements \(\pi(a_i)\), and that bicommutant is ultraweakly closed. Its surjectivity is proved in OA-MOD-UB-08.

Uniqueness holds before surjectivity: two ultraweakly continuous extensions agree on the dense subspace \(j_A(A)\).

An ultraweakly closed two-sided ideal is central

Let \(N\) be a concrete von Neumann algebra and \(I\) an ultraweakly closed two-sided *-ideal. For \(b\in I_+\), the elements

are bounded by one and increase strongly to the support \(s(b)\) as \(\varepsilon\) decreases to zero, by OA-MOD-BK-06. Hence \(s(b)\in I\). For two positive elements, the null space of their sum is the intersection of their null spaces: evaluate its nonnegative quadratic form. Therefore \(s(b+c)=s(b)\vee s(c)\). The supports of the positive elements of \(I\) form an upward-directed family of projections. Their supremum \(p\) belongs to \(I\), by bounded monotone convergence and ultraweak closedness. Include the support of zero, so this assertion also covers \(I=0\).

For every unitary \(v\in N\), conjugation carries \(I_+\) onto itself and \(s(vbv^*)=vs(b)v^*\), so \(vpv^*=p\). By OA-MOD-BK-08, \(p\) commutes with all of \(N\): it is central. For \(a\in I\), both \(a^*a\) and \(aa^*\) lie in \(I_+\), and their supports are at most \(p\). Since \(\|a(1-p)\xi\|^2=\langle a^*a(1-p)\xi,(1-p)\xi\rangle=0\), one has \(a=ap\); similarly \(a=pa\). Conversely \(p\in I\) and the ideal property give \(Np\subseteq I\). Thus

The projection is unique, because it is the identity of that ideal.

Apply this result to \(\ker T\) from OA-MOD-UB-06, after the concrete identification \(A^{**}\cong M\). Set \(z=1-p\). Then

OA-MOD-UB-02 makes this restriction isometric.

Surjectivity uses a compact ball and the exact density theorem

Assume again that \(\pi\) is nondegenerate and put \(N=\pi(A)''\). The central corner \(zA^{**}\) is weak-star closed, since multiplication by \(z\) is weak-star continuous. Its closed unit ball is compact: it is the intersection of the compact unit ball of \(A^{**}=(A^*)^*\) with the closed conditions \(X=zX\). Here Banach--Alaoglu is applied to that particular dual ball.

Since \(T\) is continuous and isometric on the corner, the set

is ultraweakly compact, hence closed in the Hausdorff space \(N\). It contains every contraction \(b\in\pi(A)\). Indeed choose \(a\) with \(\pi(a)=b\). Then \(T(zj_A(a))=b\), and the isometry in (UB.18) gives \(\|zj_A(a)\|=\|b\|\leq1\). We did not require a contractive lift in \(A\).

OA-MOD-HAP-05 applied to the nondegenerate *-subalgebra \(\pi(A)\) says that every contraction in \(N\) is a strong* limit of contractions in \(\pi(A)\). This bounded net also converges ultraweakly, so its limit lies in \(D\). Hence \(D\) is exactly the unit ball of \(N\). Scaling proves \(T(A^{**})=N\) and proves (UB.2)--(UB.3).

For completeness, the inverse is normal by a direct predual argument. Write \(C=zA^{**}\), with its inherited concrete corner predual from OA-MOD-CP-11, and let \(V=T|_C:C\to N\). Ultraweak continuity gives the bounded map

Because \(V\) maps the unit ball onto the unit ball isometrically, \(\|V_*\rho\|=\|\rho\|\). Its range is therefore norm closed, since \(N_*\) is complete. The annihilator of that range in \((C_*)^*=C\) is zero: if \(\rho(Vc)=0\) for every \(\rho\in N_*\), predual separation gives \(Vc=0\), and injectivity gives \(c=0\). If a norm-closed linear subspace of \(C_*\) were proper, Hahn--Banach, applied to the quotient or to a point outside that subspace, would supply a nonzero bounded linear functional annihilating it. Here is the annihilator step explicitly. If \(W\subset C_*\) is a proper norm-closed subspace, choose \(u\notin W\) and set \(d=\operatorname{dist}(u,W)>0\). The functional

is well-defined and satisfies \(|f(w+\lambda u)|\leq\|w+\lambda u\|/d\): for \(\lambda\ne0\), factor out \(\lambda\) and use the distance to \(W\); the zero case is immediate. NP1 extends this nonzero functional to \(C_*\), contradicting the zero-annihilator assertion. Thus \(V_*\) is onto. Its inverse is an isometry, and the Banach adjoint of \(V_*^{-1}\), under concrete duality, is \(V^{-1}\). As an adjoint, that inverse is weak-star continuous. This proves global normality without an order-normality criterion, weight theory, or an unjustified norm bound on arbitrary convergent nets.

For a degenerate \(\pi\), let \(q\) project onto \(K_0=\overline{\operatorname{span}\pi(A)K}\). This is a reducing subspace, since \(\pi(A)\) is self-adjoint; \(\pi\) vanishes on \(K_0^\perp\). Indeed, for \(\eta\in K_0^\perp\) and \(\xi\in K\),

because \(\pi(a^*)\xi\in K_0\). Hence \(\pi(A)K=\pi(A)K_0\), whose linear span is dense in \(K_0\) by its definition. This proves nondegeneracy of the restricted representation, including \(K_0=0\). Apply the preceding theorem to its nondegenerate restriction on \(K_0\). Formula (UB.15) then shows that the full \(T\) is this extension on \(K_0\), direct-summed with zero on \(K_0^\perp\), and \(T(1)=q\). Its range is the represented von Neumann algebra on \(K_0\), extended by zero. It is not in general the full ambient \(\pi(A)''\). For example, the zero representation on a nonzero Hilbert space extends to the zero map, whereas the bicommutant of its zero image contains the ambient identity.

A nonzero supported example. Let \(A=\mathbb C\), \(K=\mathbb C^2\), and

Every functional on \(\mathbb C\) is multiplication by a scalar, so the evaluation embedding identifies \(A^{**}\) with \(\mathbb C\). Its normal extension is exactly \(T(\lambda)=\lambda q\), with range \(\mathbb Cq\). A two-by-two matrix commutes with \(q\) precisely when its off-diagonal entries vanish. Therefore \(\pi(A)'\) is the diagonal algebra. An operator commuting with every diagonal matrix is again diagonal, giving

This strictly contains \(\mathbb Cq\), since it contains \(1-q\). On \(K_0=\mathbb C\oplus0\), the represented algebra is \(\mathbb C\) and the extension is onto, as the supported statement requires.

Second adjoints preserve homomorphisms and composition

Let \(\alpha:A\to C\) be any C*-homomorphism, without a unitality or nondegeneracy assumption. Its Banach second adjoint

is weak-star continuous and satisfies \(\alpha^{**}j_A=j_C\alpha\). The same two successive one-variable limits used in OA-MOD-UB-06 prove multiplicativity; one limit proves preservation of the involution. Thus it is a normal *-homomorphism for the structures just constructed. Banach adjoints give

These statements include the zero map and zero algebras.

If \(j:B\hookrightarrow A\) is an isometric C*-subalgebra inclusion, Hahn--Banach makes \(j^*:A^*\to B^*\) onto with norm-preserving lifts. Consequently \(j^{**}\) is an isometry: the supremum over the unit ball of \(A^*\) in its dual norm equals the supremum over the unit ball of \(B^*\). Its range is

To see surjectivity onto this annihilator, an \(X\in(B^\perp)^\perp\) defines \(Y(f)=X(\widetilde f)\) on \(B^*\), where \(\widetilde f\) is any extension to \(A\). Two such extensions differ in \(B^\perp\); norm-preserving lifts give \(|Y(f)|\leq\|X\|\|f\|\). Thus \(Y\in B^{**}\) and \(j^{**}Y=X\). The annihilator is weak-star closed. This is the precise functorial bidual input used by the hereditary-corner proof HF-05 and ND-02.