Recovering a representation from its positive cone

Self-checked by the writing AI.

A standard form can be specified by axioms, without naming the weight that produced it. To compare two such forms, first find the part seen by one positive vector. Its cyclic graph identifies the local modular conjugation. Small corners can then be compared canonically, and their comparisons agree on overlaps. This procedure also explains how projections are recorded by the closed faces of the cone.

The mathematical antecedents are Takesaki, Theory of Operator Algebras II, IX.1, especially the supported-corner and conjugation results, principal-face description, orthogonality criterion and equivalence theorem in 1.8–1.14. The proof below uses the previously constructed weight forms and gives the domain argument and the local-to-global construction explicitly. Hilbert spaces and algebras may have arbitrary cardinality.

Axioms and the geometric facts available before comparison

Let \(M\subset B(H)\) be a von Neumann algebra. A standard form is a quadruple \((M,H,J,P)\), where \(J\) is an antiunitary involution, \(P\) is a self-dual cone, and

Inner products are linear in the first variable. Self-duality means that \(\eta\in P\) exactly when every \(\langle\xi,\eta\rangle\), \(\xi\in P\), is real and nonnegative. Thus \(P\) is norm closed, convex and pointed. This is the complex Hilbert-space dual cone convention of SF-01.

We use the proofs of SF-06–07 at their axiomatic level. They require self-duality, pointwise \(J\)-fixedness, the commutant identity and \(xJxJ\)-invariance, but not the central identity or the existence of representatives for every normal functional. Consequently, in every form under discussion, the following statements hold before any equivalence theorem:

where \(s_\xi=s_M(\omega_\xi)\) and \(\omega_\xi(x)=\langle x\xi,\xi\rangle\). Moreover,

In particular a unitary intertwining the algebra and mapping one cone onto another is unique if it exists. Indeed the ratio of two such unitaries commutes with the algebra and preserves every cone-vector functional; the first inequality makes it fix every cone vector, and (SE.2) makes it the identity.

For clarity about prerequisites, the analytic inputs are the finite GNS construction WG-004–006, the cyclic involution and antilinear adjoint conventions TC-03–05/08, right-bounded multipliers HA-05/08 and the actual right-involution graph core RD-04. SF-04 supplies the cone generators in a weight form; SF-09 supplies comparison between two weight forms. NW-03 supplies the directed family of sigma-finite projections. No spatial derivative, relative modular identification, cocycle theorem or assumption of a faithful normal state on \(M\) is used.

Compression and the commutant

We first prove a bounded-operator fact independently of cone geometry. If \(A\) is a von Neumann algebra on \(L\) and \(p\in A\) is a projection, then

Here the right side is the set of restrictions; elements of \(A'\) commute with \(p\), so these restrictions make sense.

Proof. Let \(T\in B(pL)\) commute with \(pAp\). On finite sums prescribe

Let \(X:(pL)^n\to L\) be the row operator with entries \(a_i|_{pL}\). The entries of \(B=X^*X\) are \(pa_i^*a_jp\). The diagonal operator \(D_T\) commutes with this positive matrix and its square root. Bounded continuous functional calculus therefore gives

Thus (SE.5) is well-defined and bounded. Extend it to \(L_0=\overline{A(pL)}\), then by zero on \(L_0^\perp\). The space \(L_0\) reduces \(A\), and the defining formula commutes with each element of \(A\) on its dense finite-sum domain. The extension lies in \(A'\) and restricts to \(T\). The other inclusion in (SE.4) follows by multiplication. \(\square\)

If \(B\) is a unital *-algebra on \(L\) and \(q\in B'\) is a projection, one also has

To see the nontrivial inclusion, extend a commuting operator on \(qL\) by zero on \((1-q)L\); the reducing decomposition shows that it commutes with \(B\).

For a standard form and a projection \(e\in M\), put

The two factors of \(Q_e\) commute. Apply (SE.4) on \(eH\), then (SE.7) with \(f|_{eH}\), to obtain

The same two arguments with \(M,M'\) exchanged give the reverse commutant identity. Thus \(N_e=N_e''\) on \(K_e\), in particular it is a von Neumann algebra. Since \(JQ_eJ=Q_e\), the restricted conjugation \(J_e\) satisfies \(J_eN_eJ_e=N_e'\). No assertion about the image of a general normal representation is needed for this argument.

Every projection has a faithful cone corner

For a projection \(e\in M\), define

Then \(P_e=Q_eP\), this cone is self-dual in \(K_e\), and the restriction of \(eMe\) to \(K_e\) is faithful. Together with \(J_e\), these objects form a standard form of \(eMe\), even when this corner has no faithful normal state.

Detection of a support. Cone invariance gives \(Q_eP\subset P\). For \(\eta\in P\) and real \(t\), expand the nonnegative cone pairing

The equality of the two linear coefficients follows from \(J\eta=\eta\); the quadratic coefficient is that of the orthogonal projection \(Q_e\). If \(Q_e\eta=0\), positivity for every real \(t\) forces \(e\eta=0\). In particular

because cone vectors span \(H\).

Every vector in \(P\cap K_e\) is fixed by \(Q_e\), proving \(P_e=Q_eP\). For \(v\in K_e\), its pairings against \(Q_eP\) are its pairings against \(P\). Hence if they are all real nonnegative, \(v\in P\cap K_e\). This proves self-duality in the smaller Hilbert space and thus spanning of \(K_e\).

Suppose \(x\in eMe\) acts by zero on \(K_e\). Then \(xQ_e=0\). Its right support \(r=s(x^*x)\le e\) has \(rQ_e=0\); this follows either from the kernel of \(x\) or its spectral projections. Thus \(rJeJ=0\), and \(JrJ\le JeJ\) gives \(Q_r=0\). Equation (SE.12) makes \(r=0\), hence \(x=0\). This proves faithfulness.

Pointwise \(J_e\)-fixedness follows by restriction. If \(x\in eMe\), then \(xJxJ\) preserves both \(P\) and \(K_e\), giving the cone-invariance axiom. The commutant identity was proved in SE-02. To verify the central axiom without assuming a description of \(Z(eMe)\), we record why it follows from the other axioms. For a central unitary \(u\) in a represented algebra with those axioms, \(uJuJ\) is a unitary commuting with that algebra and mapping its cone onto itself; its inverse is \(u^*Ju^*J\). The geometric uniqueness in SE-01 makes \(uJuJ=I\), so \(JuJ=u^*\). A central self-adjoint contraction \(a\) is the real part of the central unitary \(a+i(1-a^2)^{1/2}\). Real linearity therefore gives \(JaJ=a\), and decomposition into real and imaginary self-adjoint parts gives \(JzJ=z^*\) for every central \(z\). Apply this to \(N_e\), using its identity on \(K_e\), to complete all the corner axioms. \(\square\)

The two cyclic domains are graph cores

Let \(N\subset B(K)\) have a cyclic separating vector \(\Omega\). Define the antilinear maps

Then

In particular both \(N\Omega\) and \(N'\Omega\) are graph cores for the corresponding closed involutions. Hilbert-space density alone would not imply this assertion.

Proof. TC-04 proves closability and \(F_0\subset S_0^*=F\). The space \(\mathcal A=N\Omega\), with product \((a\Omega)(b\Omega)=ab\Omega\) and involution \(a\Omega\mapsto a^*\Omega\), is a left Hilbert algebra: left multiplication is bounded, its adjoint relation follows from the Hilbert inner product, its involution is closable, and its algebra identity vector \(\Omega\) makes \(\mathcal A^2=\mathcal A\). Its left von Neumann algebra is \(N\).

HA-05 assigns to every right-bounded vector \(\eta\) a multiplier \(R_\eta\in N'\). Substituting the algebra unit in its defining formula gives \(\eta=R_\eta\Omega\). Conversely, for \(b\in N'\), the vector \(b\Omega\) is right bounded: its multiplier on \(a\Omega\) is \(a b\Omega=b a\Omega\), hence is the bounded operator \(b\). Thus the right-bounded space equals \(N'\Omega\). TC-04 puts every such vector in \(D(F)\), with \(F(b\Omega)=b^*\Omega\). The full right algebra \(\mathcal A_r=\mathcal B_r\cap D(F)\) of HA-08 is consequently exactly \(N'\Omega\). RD-04 says that this algebra is a graph core for \(F\). This proves (SE.14). The core assertion for \(S\) is its definition. \(\square\)

Recognizing a modular conjugation by its exact graph

For \(N,\Omega\) as in SE-04, let \(C\) be an antiunitary involution such that

Then \(C\) is the modular conjugation \(J_\Omega\). Conversely \(J_\Omega\) has these properties.

Proof, including self-adjointness. The first two hypotheses send \(N\Omega\) onto \(N'\Omega\), and direct calculation on the latter gives \(CS_0C=F_0\). Antiunitary transport preserves graph closure. Equation (SE.14) therefore gives the equality of closed antilinear operators

The operator \(L=CS\) is closed, densely defined and complex-linear. The antilinear adjoint convention is \(\langle Su,v\rangle=\langle Fv,u\rangle\). For its ordinary linear adjoint one consequently has

Indeed \(\langle CSu,v\rangle=\langle Cv,Su\rangle\); conjugate the antilinear adjoint test to obtain exactly (SE.17), including its necessity. Equation (SE.16) makes this domain \(D(S)\) and its action \(CS\). Hence \(L=L^*\).

For \(a\in N\), its quadratic form on the graph core is

The two bounded factors \(a^*\) and \(Ca^*C\) commute, so the last inequality is (SE.15) with \(a^*\). The graph norms of \(L\) and \(S\) agree; graph approximation makes this inequality hold throughout \(D(L)\). Thus \(L\) is positive self-adjoint, not just positive symmetric. The factorization \(S=CL\) has a positive self-adjoint second factor. Both \(S\) and its range are dense with zero kernel, by TC-05, so polar uniqueness TC-08 gives \(C=J_\Omega\), \(L=\Delta_\Omega^{1/2}\).

Conversely \(S\Omega=F\Omega=\Omega\) implies \(\Omega\in D(S^*S)\) and \(\Delta_\Omega\Omega=\Omega\), hence \(J_\Omega\Omega=\Omega\). The commutant identity is MF-05, applied to the cyclic GNS Hilbert algebra. Positivity of \(\Delta_\Omega^{1/2}\) in (SE.18), with \(a^*\) in place of \(a\), gives the final condition in (SE.15). \(\square\)

The equality in (SE.16) is essential. Positivity on \(N\Omega\) by itself would only produce a positive symmetric operator; it would not identify the self-adjoint square root in a polar decomposition.

The corner seen by a positive vector

Fix \(\xi\in P\), put \(e=s_\xi\), and let \(\omega_e\) be \(\omega_\xi\) restricted to \(eMe\). It is faithful, normal and finite. The GNS map defines a unique unitary

and it satisfies

Among unitaries to \(K_e\), the algebra-intertwining and cone-mapping properties alone characterize \(U_\xi\).

Proof. If \(e=0\), the spaces and map are zero. Otherwise (SE.2) gives

The equality uses \(e\xi=\xi\). The norm identity \(\omega_e(a^*a)=\|a\xi\|^2\) and polarization give the isometry and its onto extension. Multiplication proves intertwining. If \(a\in eMe\) kills \(\xi\), it kills \(M'\xi\) and hence \(eH\), so \(a=0\). Thus \(\xi\) is separating for the represented corner; it is cyclic by (SE.21). Faithfulness of \(\omega_e\) follows by applying this argument to positive square roots.

The restriction \(J_e\) fixes \(\xi\) and exchanges the corner and its commutant by SE-02. The last condition in (SE.15) holds because \(aJaJ\xi\in P\), \(\xi\in P\), and the bounded left and right factors commute. SE-05 now identifies \(J_e\) with the transported modular conjugation, with all closed domains accounted for.

Because \(\omega_e\) is finite, its finite-star algebra is all of \(eMe\). SF-04 describes its cone as the closure of \(\pi_{\omega_e}(a)J_{\omega_e}\Lambda_{\omega_e}(a)\), \(a\in eMe\). Their images are \(aJaJ\xi\in P_e\). The cone \(U_\xi P_{\omega_e}\) is self-dual in \(K_e\). Every vector of \(P_e\) pairs nonnegatively with it and therefore belongs to it, proving the reverse inclusion. This cone equality follows from the graph identification; it was not a premise of that identification. Uniqueness follows from SE-01. \(\square\)

Closing one order ideal gives its entire supported face

For \(\xi\in P\), define its principal cone ideal by

Then

The closure is in the Hilbert norm. The right side is the smallest closed cone face containing \(\xi\). Here a cone face is a subcone \(F\subset P\) such that \(\eta+\zeta\in F\), \(\eta,\zeta\in P\), implies \(\eta,\zeta\in F\).

Proof. Let \(e=s_\xi\), \(q=1-e\). If \(\eta\in[0,c\xi]_P\), the two cone vectors \(Q_q\eta\) and \(Q_q(c\xi-\eta)\) sum to zero. Pointedness makes \(Q_q\eta=0\), and (SE.11) gives \(q\eta=0\). Since \(J\eta=\eta\), both \(e\) and \(JeJ\) fix it; hence \(\eta\in P_e\). This proves one inclusion after closure.

For the other inclusion use SE-06 to work in the finite faithful GNS corner, with \(\xi\) as its GNS identity vector. Its norm need not be one. The proof in SE-05 gives \(\Delta\xi=\xi\). SF-03 identifies the left square cone here with \(\overline{(eMe)_+\xi}\): every positive bounded \(a\) is a square of its positive square root, and every algebra square is of this kind. SF-04 and its graph-continuity proof consequently give

The vectors \(a\xi\) lie in \(D(\Delta^{1/2})\), because they lie in the initial involution domain; thus the quarter powers are defined. Each generator satisfies

It therefore belongs to \(I_\xi\). This proves density.

For any projection \(e\), if \(\eta+\zeta\in P_e\) with \(\eta,\zeta\in P\), apply \(Q_{1-e}\), pointedness and (SE.11) to conclude that both summands belong to \(P_e\). Thus \(P_e\) is a closed face. Every cone face containing \(\xi\) contains every \([0,c\xi]_P\), and every closed such face contains their closure. This proves minimality. \(\square\)

Projections and all closed cone faces

The map \(e\mapsto P_e\) is an order-preserving bijection between projections of \(M\) and norm-closed cone faces of \(P\), and its inverse is

For projections \(e,f\),

In particular, for positive vectors \(\xi,\eta\), all three conditions are equivalent: \(\xi\perp\eta\), \(s_\xi s_\eta=0\), and \(P_{s_\xi}\perp P_{s_\eta}\).

Proof. Supports of vectors in \(P_e\) have supremum \(e\). They are at most \(e\). If their supremum \(r\) were smaller than \(e\), put \(q=e-r\). By (SE.12), \(K_q\ne0\); self-duality and spanning of \(P_q\) give a nonzero vector in \(P_q\subset P_e\). Its nonzero support would be both below \(q\) and below \(r\), a contradiction.

For a finite family \(\xi_1,\ldots,\xi_n\in P\),

The right side fixes the sum, giving one inequality. If \(s\) is the support of the sum, apply \(Q_{1-s}\) to the summands. They are cone vectors with sum zero, so each is zero. Detection (SE.11) says that \(1-s\) kills every \(\xi_j\), giving the other inequality.

Let \(F\) be a closed cone face and \(e=\bigvee_{\xi\in F}s_\xi\). It is contained in \(P_e\). For each finite subset \(A\subset F\), let \(\xi_A=\sum_{\xi\in A}\xi\) and \(e_A=s_{\xi_A}\), allowing the empty sum. SE-07 shows \(P_{e_A}\subset F\). The projections \(e_A\) increase strongly to \(e\). Their conjugates do also, and the products \(Q_{e_A}\) increase strongly to \(Q_e\). To verify the product convergence, subtract the two products and use that all factors are contractions and each factor converges strongly. For \(\eta\in P_e\), the vectors \(Q_{e_A}\eta\in P_{e_A}\subset F\) converge to \(\eta\). Closedness gives \(P_e\subset F\).

If \(e\le f\), then \(Q_e\le Q_f\), giving \(P_e\subset P_f\). Conversely the support-supremum characterization just proved implies \(e\le f\) from that inclusion. This establishes the order bijection.

If \(ef=0\), vectors in the two cones have orthogonal left supports, proving orthogonality. Conversely, if the two cones are orthogonal, (SE.3) makes every support from the first orthogonal to every support from the second. Their suprema are \(e,f\), so \(ef=0\). The same argument with the support of one arbitrary vector gives the second formula in (SE.27). Apply these facts to \(e=s_\xi,f=s_\eta\) and use (SE.3) for the last assertion. \(\square\)

Finding a full-support vector on each small corner

A projection \(e\) is called sigma-finite when \(eMe\) has a faithful normal state; include \(0\). Every such \(e\) is the support of a vector in \(P\). This conclusion concerns an arbitrary axiomatic form, so a normal-functional cone representation theorem cannot be assumed in its proof.

Proof. For \(e\ne0\), choose a faithful normal state \(\rho\) on \(eMe\). Supports of vectors in \(P_e\) have join \(e\), by SE-08. For finite subsets \(A\subset P_e\), the projections \(r_A=\bigvee_{\eta\in A}s_\eta\) increase to \(e\); normality gives \(\rho(r_A)\uparrow1\). Choose finite \(A_n\) with \(\rho(r_{A_n})>1-2^{-n}\). Their countable join is \(e\): its complement has \(\rho\)-value at most \(2^{-n}\) for every \(n\), and faithfulness removes that complement.

Enumerate the union of these finite sets as a sequence \((\eta_n)\), with repetitions or zeros if it is finite. The norm-convergent positive series

belongs to \(P_e\). Let \(s=s_\xi\). For each \(n\), separate the corresponding positive multiple of \(\eta_n\) from the series; the remainder lies in \(P\) by closedness. Applying \(Q_{1-s}\) makes two cone vectors sum to zero. Pointedness and (SE.11) imply \((1-s)\eta_n=0\). Thus every selected support lies below \(s\), so \(s=e\). This also proves that \(\xi\) is cyclic and separating for \(eMe\) on \(K_e\), by SE-06. The zero corner is immediate. \(\square\)

NW-03 proves that the sigma-finite projections form an upward-directed set \(\mathcal E\) with supremum \(1\). Finite joins need not commute: the support of a positive linear combination of their faithful corner states is their join. Every nonzero projection contains the nonzero support of an ordinary normal positive functional, which proves exhaustion. Consequently

There is also an exact cone union \(P=\bigcup_{e\in\mathcal E}P_e\), since a positive vector's finite normal functional is faithful on its support corner. By four-vector spanning and finite directed joins, the union of the \(K_e\) is in fact all of \(H\). The density statement alone will suffice below. The index set \(\mathcal E\) need not have a countable cofinal subset.

Patching the unique comparisons

Let \((M_1,H_1,J_1,P_1)\) and \((M_2,H_2,J_2,P_2)\) be arbitrary standard forms, and let \(\theta:M_1\to M_2\) be a unital *-isomorphism. There is exactly one unitary \(V:H_1\to H_2\) such that

It is enough for uniqueness to require the first and third properties.

Proof. A *-isomorphism preserves the positive order in both directions. It therefore preserves bounded increasing positive suprema and is normal. Identify the two algebras abstractly through \(\theta\), writing them as representations \(\pi_i\) of one \(M\). In each representation form the corner spaces \(K_{i,e}\), conjugations and cones, for \(e\in\mathcal E\).

SE-09 gives \(\xi_i\in P_{i,e}\) with support exactly \(e\). SE-06 identifies this corner with the natural-cone GNS form of a faithful finite functional on \(eMe\). SF-09 compares these two weight forms. Composing the three actual unitaries gives a unitary

intertwining \(eMe\) and mapping \(P_{1,e}\) onto \(P_{2,e}\). It is unique by SE-01 and therefore independent of the chosen vectors. Since the cones span their Hilbert spaces and their conjugations fix them pointwise, \(V_eJ_{1,e}=J_{2,e}V_e\).

If \(f\le e\) are in \(\mathcal E\), intertwining of \(\pi_i(f)\) and of the conjugations gives

Thus \(V_e\) maps \(K_{1,f}\) onto \(K_{2,f}\). Its restriction maps the intersection cones onto one another and intertwines \(fMf\). Uniqueness identifies this restriction with \(V_f\). These statements check both range and cone equality, not merely inclusion of the corner spaces.

Directedness and (SE.33) define a single linear isometry on \(\bigcup_eK_{1,e}\). Its range contains \(\bigcup_eK_{2,e}\), so completeness and (SE.30) extend it to a surjective unitary \(V\). It intertwines the conjugations there by continuity. If \(\xi\in P_1\), the vectors \(Q_{1,e}\xi\) are in the local cones and converge to \(\xi\); their images are in \(P_2\). Closedness gives \(VP_1\subset P_2\), and the same argument for the inverse proves equality.

To prove the identity for the whole algebra, fix \(\eta\in K_{1,e}\) and \(x\in M\). For \(f\in\mathcal E\) above \(e\), the local intertwining gives

Both normal representations send \(f\uparrow1\) to strongly increasing projections with supremum \(I\). Hence \(\pi_i(fxf)\to\pi_i(x)\) strongly, with norms bounded by \(\|x\|\). Pass to the limit in (SE.34), then use density of the union of the corners. This proves the algebra identity in (SE.31). Uniqueness is SE-01, completing the proof. \(\square\)

Compositions of these unitaries implement compositions of isomorphisms and map the cones onto each other, so uniqueness makes the construction compatible with composition and inverses. It also proves that every arbitrary axiomatic standard form is equivalent to the form produced by any n.s.f. weight: WH-13 supplies such a weight, and SF-05 supplies its standard form. This conclusion is obtained after the local patching argument, not used to justify it.

Transporting positive functionals and automorphisms

In every axiomatic standard form, each \(\omega\in M_*^+\) has a unique representative \(\xi_\omega\in P\) with

Existence follows by SE-10 from SF-10's weight-constructed form; uniqueness is already (SE.3). In this transport, the equality for all \(x\) follows from the algebra intertwining, and cone membership follows from the onto cone identity. Thus no functional changes when the representation is changed. The inequalities (SE.3), positive homogeneity \(\xi_{c\omega}=\sqrt c\,\xi_\omega\) for \(c\ge0\), and the norm-continuous and bounded monotone-net conclusions of SF-11 hold unchanged.

For a normal automorphism \(\alpha\), let \(u_\alpha\) be SE-10's unique unitary in the given form. Then

The functional formula follows by evaluating the vector functional of \(u_\alpha\xi_\omega\), and the product formula follows from uniqueness. SF-12's topology proof now applies to every form: convergence of \(\alpha_i\) and \(\alpha_i^{-1}\) pointwise in the predual norm is equivalent to strong convergence of \(u_{\alpha_i}\) to \(u_\alpha\). Indeed (SE.3) applied to (SE.36) proves convergence on cone vectors, hence on \(H\). Conversely, strong convergence of unitaries to a unitary gives strong convergence of their adjoints, and the upper bound in (SE.3) gives predual norm convergence on positive normal functionals in both directions. Decompose a general normal functional into a linear combination of positive ones to finish. This is a specified topology assertion; no completeness assertion for an unspecified uniformity is implicit in it.

The transported cone is intrinsic even when \(M\) is not sigma-finite. A normal positive functional sees one sigma-finite support corner; different functionals need not share a common faithful normal state on the whole algebra.

A face can require a genuine closure

Consider the diagonal algebra \(M=\ell^\infty(\mathbb N)\) on \(H=\ell^2(\mathbb N)\), with coordinatewise conjugation and the cone of nonnegative real sequences. The standard axioms hold directly: the diagonal algebra is its commutant, and \(xJxJ\) multiplies each coordinate by \(|x_n|^2\).

Take

Both belong to \(P\), and \(s_\xi=1\), so \(P_{s_\xi}=P\). Yet \(\eta\notin I_\xi\), since a bound \(\eta\le c\xi\) would require \(n\le c\) for every \(n\). For \(\eta^{(k)}\), the truncation to the first \(k\) coordinates, one has \(0\le\eta^{(k)}\le k\xi\) and \(\eta^{(k)}\to\eta\) in \(\ell^2\). This verifies explicitly why the closure in (SE.23) cannot be omitted.

For any index set \(I\), the same standard form on \(\ell^2(I)\) has \(K_e=\ell^2(E)\) and \(P_e=\ell^2(E)_+\) for the projection \(e=1_E\). A vector has at most countably many nonzero coordinates: for each integer \(k\), only finitely many coordinates can have squared magnitude at least \(1/k\), and the union of those finite sets contains its support. SE-09 therefore makes every sigma-finite projection have countable support. Conversely, on any nonempty countable \(E\), choose strictly positive coordinate masses summing to one; summation against these masses is a faithful normal state on \(\ell^\infty(E)\), with normality following by monotone convergence of nonnegative series. Include the empty corner by convention. Thus the sigma-finite projections are precisely the projections with countable support. Their directed set exhausts the representation for uncountable \(I\), although no single positive vector has support \(1\). This is an example of the patching mechanism, not an assumption in its proof.

Problems with complete solutions

Problem 1: two local choices. In SE-10, choose different full-support cone vectors on one sigma-finite corner. Prove that the resulting local unitary does not change, and identify the same unitary after restricting to a smaller sigma-finite corner.

Solution. Each construction is an onto unitary intertwining the same represented corner algebra and mapping its cone onto the same target cone. Their ratio commutes with the algebra and preserves the cone. Its action preserves every cone-vector functional, so (SE.3) makes it fix the cone pointwise; spanning makes the ratio the identity. For a smaller corner, (SE.33) proves onto restriction of the Hilbert spaces and of their cones. Its restriction is therefore the unique local comparison there. No compatibility assumption about the chosen vectors is necessary.

Problem 2: complement versus set difference. Show that \(P_{1-e}\) is the set of cone vectors orthogonal to all of \(P_e\). Explain why it is generally not the set difference \(P\setminus P_e\).

Solution. Equation (SE.27) proves the orthogonal-complement assertion. In the diagonal two-coordinate form, let \(e\) project onto the first coordinate. The vector \((1,1)\) lies in \(P\setminus P_e\) but is not orthogonal to \((1,0)\in P_e\). The vector \((0,1)\) does belong to the orthogonal cone. Complement in the projection lattice corresponds to cone orthogonality, not ordinary removal of a subset.

Problem 3: a faithful corner representation. Suppose only the self-dual cone, its pointwise fixed conjugation, the commutant identity and cone invariance are given. Prove that a nonzero projection cannot satisfy \(eJeJ=0\), and deduce that compression to \(eJeJH\) does not kill a nonzero element of \(eMe\).

Solution. With \(Q_e=0\), (SE.11) is a nonnegative affine function of every real \(t\). Its coefficient forces \(e\eta=0\) for every \(\eta\in P\), so spanning gives \(e=0\), a contradiction. If \(x\) were killed by compression, its right support \(r\le e\) would satisfy \(rJeJ=0\), hence \(rJrJ=0\). The result just proved gives \(r=0\) and \(x=0\). This proof also explains why it is safe to infer the central conjugation axiom on the corner after the other axioms have been established.

Problem 4: why the adjoint core matters. For the operator \(L=CS\) in SE-05, isolate the equality that proves self-adjointness and explain what would remain unproved if one only knew \(F_0\subset S^*\).

Solution. Self-adjointness uses \(D(S^*)=CD(S)\) and \(S^*C=CS\), both coming from the closed equality \(CSC=S^*\). If only \(F_0\subset S^*\) were known, closing that inclusion would give \(CSC\subset S^*\); the domain in (SE.17) could then be larger than \(D(S)\). Positivity on the initial core would give a positive symmetric operator but would not exclude different self-adjoint extensions. SE-04's right-algebra graph-core theorem is the input that replaces the inclusion by equality.

What these comparisons make available next

The corner theorem, principal-face closure and abstract equivalence are now proved for arbitrary standard forms. Their explicit dependency order is: bounded commutant compression; cone detection; the cyclic adjoint graph core; modular conjugation recognition; supported GNS identification; faces and small-support vectors; compatible local comparisons; global transport. The new steps do not use spatial derivatives or Connes cocycles.

The remaining standard-form topics include the intrinsic realization of general nonfaithful semifinite GNS maps, polar decompositions and order/concavity properties of cone vectors, cone isometries and Jordan structures, continuity of relative operators under varying functionals, and the spectral commutator estimates and modular orbit bounds. These require their own exact proofs; the present equivalence theorem is a prerequisite for some of them. In particular a proof of equivalence does not itself prove the full exercise program or every later callback in IX.1.

A useful research route starts with SE-08's recovery of projections from closed cone faces. It asks which additional structure on the cone recovers multiplication rather than only the projection order. Another starts with SE-10's compatible corner maps and asks how unbounded weight domains can be transported through them. In that question, the corner \(eJeJH\), the one-sided spaces \(eH\) and \(JeJH\), and the graph domain of an unbounded map must be kept distinct.