Countable modular sums and the boundary of finite-weight tests

An analytic identity can determine a weight on every element where a reference weight is finite, yet miss an infinite value. This distinction matters when constructing operator-valued weights: their scalar composites must agree on the entire positive cone. We prove the safe part of the countable modular test, then build a counterexample from two closed forms on the same Hilbert space. All Hilbert inner products are linear in the first variable.

What the analytic strong sum actually proves

Let \(\varphi\) be a faithful normal semifinite weight on a von Neumann algebra \(M\). Write \(\mathfrak n_\varphi\), \(H_\varphi\), \(\Lambda_\varphi\), \(J_\varphi\), and \(\sigma^\varphi\) for its GNS and modular data. Given a sequence \((a_j)_{j\geq1}\subset M\), define the normal weight for \(X\in M_+\) by

The sum is the supremum of its finite partial sums; normality follows from normality of each summand and interchange of two directed positive suprema.

We name the precise analytic input rather than hiding it. The fixed-operator theorem of Takesaki II, VIII.3.18(i)–(ii), says that the global inequality \(\varphi(aXa^*)\leq\varphi(X)\) for every \(X\in M_+\) is equivalent to a bounded negative-half-strip modular orbit of \(a\), with endpoint \(b=\sigma^\varphi_{-i/2}(a)\) satisfying \(\|b\|\leq1\). Under these conditions it also gives the typed right-action formula

The full proof of this analytic theorem is an explicit open course prerequisite, OA-MOD-ACI-DEP-RIGHT-ACTION. The source's proof of IX.4.20 cites VIII.3.17 at this step; VIII.3.18 is the direct fixed-operator statement.

Suppose first that \(\Psi_A=\varphi\) on all of \(M_+\). Each summand in (ACI.1) is then at most \(\varphi\), so (ACI.2) gives bounded endpoints \(b_j\). Put \(B_k=\sum_{j=1}^k b_j^*b_j\) and \(\eta_x=J_\varphi\Lambda_\varphi(x)\). For \(x\in\mathfrak n_\varphi\), the right-action formula yields

Density of the GNS vectors and the antiunitarity of \(J_\varphi\) give \(0\leq B_k\leq1\). The assumed equality of weights turns (ACI.3) into equality as \(k\to\infty\). Thus the increasing contractions \(B_k\) converge strongly to \(1\): their quadratic forms converge to the identity on a dense set, and \((1-B_k)^2\leq1-B_k\). This is the valid forward implication.

Conversely, suppose each \(a_j\) has such a contractive analytic endpoint and \(B_k\uparrow1\) strongly. Applying (ACI.2) again gives

Here \(x\in\mathfrak n_\varphi\); the first equality follows by taking the strong limit of \(B_k\) in (ACI.3). Every positive \(X\) with \(\varphi(X)<\infty\) is \(x^*x\) for \(x=X^{1/2}\in\mathfrak n_\varphi\). Hence the strong sum proves equality on the finite \(\varphi\)-cone. It gives no value at a positive \(X\) of infinite \(\varphi\)-weight.

The printed IX.4.20 permits a nonfaithful semifinite normal weight and formulates its analytic test in the support corner. The faithful argument above applies there once the exact support-reduction contract is supplied. That reduction and the complete proof of VIII.3.18 remain open course dependencies; the counterexample below already has support \(1\), so neither issue can repair the printed all-positive reverse implication.

A one-dimensional extension of a closed form

Let \(H=\ell^2(\mathbb N)\) with standard basis \((e_n)\). Write \(f_n=\langle f,e_n\rangle\) and \(\lambda_n=n^2+3\). Define the diagonal operator by

For \(f\in H\), put \(q_D(f)=\sum_n\lambda_n|f_n|^2\), allowing the value \(\infty\). The form domain \(Q_D=\{f:q_D(f)<\infty\}\), with inner product induced by \(q_D\), is a Hilbert space and \(q_D(f)\geq4\|f\|^2\). Choose the unit vector

It lies in \(H\) but not in \(Q_D\), because the summands in \(q_D(v)=c^2\sum_n(n^2+3)n^{-2}\) do not tend to zero.

Form the algebraic direct sum \(Q_K=Q_D\oplus\mathbb Cv\). For \(f\in Q_D\), define

It extends \(q_D\) on its whole form domain, but \(q_K(v)=4<\infty\). The decomposition is unique. Its direct-sum form norm is complete, and its inclusion into \(H\) is continuous:

Thus \(q_K\) is a densely defined closed positive form bounded below by the Hilbert norm.

We construct its operator explicitly from a compact embedding. Let \(j:(Q_K,q_K)\to H\) be the inclusion. The unit ball of \(Q_D\) is relatively compact in \(H\): its tail past coordinate \(N\) has squared norm at most \((N+1)^{-2}\), while its first \(N\) coordinates form a bounded finite-dimensional set. Adding the bounded one-dimensional \(\alpha v\) part preserves compactness. Hence \(j\) is compact, injective, and has dense range. The positive compact operator \(T=jj^*\) is injective. Let \((v_j)_{j\geq1}\) be an orthonormal eigenbasis with \(Tv_j=\mu_j^{-1}v_j\), where \(\mu_j\geq1\) and \(\mu_j\to\infty\).

For clarity, the spectral form of \(q_K\) follows directly from this construction. The vectors \(w_j=\sqrt{\mu_j}\,j^*v_j\) form an orthonormal basis of \(Q_K\): orthogonality follows from \(T=jj^*\), and a vector orthogonal to all \(w_j\) has zero image under \(j\). Since \(\langle jf,v_j\rangle_H =\mu_j^{-1/2}\langle f,w_j\rangle_{Q_K}\), Parseval gives

Here \(h\in Q_K\). Conversely, any \(h\in H\) for which the sum is finite is \(jf\) for the element of \(Q_K\) with coefficients \(\sqrt{\mu_j}\langle h,v_j\rangle\) in the \(w_j\) basis. Thus (ACI.9) is the full form domain, and the diagonal operator \(Kv_j=\mu_jv_j\) is positive self-adjoint with compact inverse. The two forms agree at every point of \(Q_D\), yet the vector \(v\) belongs only to the larger domain.

Analytic rank-one operators with the wrong infinite value

The extended trace weight associated to \(D\) is

Here \(X\in B(H)_+\). It is faithful and normal: it is the supremum of finite sums of positive normal functionals and dominates the usual trace. Let \(P_N\) project onto the first \(N\) basis vectors. For every positive \(X\), the compressions \(P_NXP_N\) have finite weight and converge ultraweakly to \(X\), proving semifiniteness. Write \(E_{mn}=|e_m\rangle\langle e_n|\) and \(\lambda_n=n^2+3\). The vectors \(f_{mn}=\lambda_n^{-1/2}\Lambda_\varphi(E_{mn})\) are an orthonormal GNS basis. On their finite span the Tomita map is \(Sf_{mn}=\sqrt{\lambda_m/\lambda_n}\,f_{nm}\), hence \(\Delta f_{mn}=(\lambda_m/\lambda_n)f_{mn}\). Coordinate truncation is a graph core for this diagonal map. The modular theorem therefore gives \(\sigma_t^\varphi(E_{mn}) =(\lambda_m/\lambda_n)^{it}E_{mn}\), and normality extends the formula to \(\sigma_t^\varphi(X)=D^{it}XD^{-it}\) on \(B(H)\).

Put \(u=e_1/2\), so \(D^{1/2}u=e_1\), and use the \((v_j,\mu_j)\) from (ACI.9). With \(|r\rangle\langle s|\xi=\langle\xi,s\rangle r\), define

Every \(a_j\) and \(b_j\) is bounded. For \(-1/2\leq\operatorname{Im}z\leq0\), the orbit

is bounded on the closed strip for each fixed \(j\), sigma-weakly continuous there, and holomorphic inside. Indeed \(u\) is a \(D\)-eigenvector, while \(D^{i\bar z}\) has nonpositive real exponent on this strip and is uniformly bounded. Scalar spectral integrals give the claimed continuity and holomorphy. For real \(t\), (ACI.12) is \(D^{it}a_jD^{-it}\), and \(F_j(-i/2)=b_j\). The source requires boundedness for each orbit, not a common bound over all \(j\).

The endpoint-square partial sums are positive. From (ACI.11), \(b_j^*b_j=\mu_j|D^{-1/2}v_j\rangle\langle D^{-1/2}v_j|\). For every \(\xi\in H\), put \(\eta=D^{-1/2}\xi\). The spectral form (ACI.9) then gives

Here \(D^{-1/2}H=Q_D\), exactly the smaller form domain where the two forms agree. Each partial sum is at most \(1\); the increasing sums therefore converge strongly to \(1\). They cannot converge in norm, since each has finite rank.

For \(X\in B(H)_+\), the rank-one formula and \(\varphi(|u\rangle\langle u|)=1\) give

This is the normal extended trace weight determined by \(K\). It agrees with \(\varphi\) whenever \(\varphi(X)<\infty\): then \(X\) is trace class, and a positive rank-one spectral decomposition expresses both values as sums of \(q_D(f)\) and \(q_K(f)\) for vectors \(f\in Q_D\). At the rank-one projection \(P_v=|v\rangle\langle v|\), however,

Thus the full analytic strip condition and the strong endpoint identity hold for a faithful normal semifinite weight on a separable type I factor, while the printed all-positive reverse conclusion fails. Even \(a_ja_j^*=\mu_j|u\rangle\langle u|\) lies in the \(\varphi\)-centralizer.

The mechanism is a proper closed-form extension: the endpoint identity tests the common smaller domain \(Q_D\), not the extra vector \(v\in Q_K\setminus Q_D\). No nonzero positive finite- \(\varphi\) operator lies below \(P_v\), so monotone finite cutdowns cannot fill this gap. Indeed any \(0\leq Y\leq P_v\) is \(tP_v\) for some \(0\leq t\leq1\), and its \(\varphi\)-weight is infinite if \(t>0\).

Source and boundary. The target is Takesaki, Theory of Operator Algebras II, IX.4 Lemma 4.20, printed p. 225/PDF p. 245, with VIII.3 Lemma 3.18(i)–(ii), printed pp. 125–126/PDF pp. 145–146, as its direct analytic input. The source's forward implication and finite-cone reverse calculation survive; its reverse claim on all positive elements does not. This counterexample does not refute the separate existence theorem IX.4.18. That theorem needs an independent all-positive construction, which is not given in these lessons. The analytic right-action theorem behind (ACI.2) and the modular theorem used for the type I model are not proved in this lesson.