Vanishing of higher bounded Hochschild cohomology

Proof. Apply Theorem 0.1(a) of [Pop14] with every coordinate algebra and every coordinate subalgebra equal to P. The required non-intertwining condition holds: a nonzero corner of the diffuse factor P cannot embed in the scalar relative commutant, or in a finite matrix amplification of it. Moreover, P^\omega is a factor, so the relative commutant of the full ultraproduct subalgebra is \C1. The conclusion for the separable centered subspace B\ominus\C1 is therefore ordinary scalar-trace freeness. This is the case in which every coordinate subalgebra equals its ambient factor, discussed after that theorem and in [Pop14, Sections 1.5–1.6].

A diffuse abelian tracial algebra contains a Haar unitary. Choose one, adjoin it to B, and repeat. Finite adjunction preserves separable predual: a countable generating set gives a separable tracial Hilbert space by polynomial approximation. Thus all d choices take place in the same ultrapower. At each step the new algebra is free from the algebra generated at the preceding steps, giving the asserted family.□

Proof. The labels h_1,\ldots,h_r are distinct. Put all nondistinguished unitaries into the coefficient algebra:

Then B_0 and C are free. This grouping property follows directly by expanding centered polynomials in a free family into alternating centered products; bounded approximation extends it to the generated von Neumann algebras. Write

If w=x_{i_1}^{\sigma_1}\cdots x_{i_t}^{\sigma_t} is nonempty and reduced, its evaluated product has the form

The displayed distinguished-letter string is itself reduced. Indeed, an adjacent inverse pair would require i_j=i_{j+1} and \sigma_j=-\sigma_{j+1}, contrary to the reduction of w.

Expand every c_j, including the two outside coefficients, as its scalar part plus its centered part. In each resulting term, removing scalar coefficients joins distinguished letters into nonempty contiguous subwords of that reduced string. Every such block is a centered element of C. The remaining factors alternate between centered elements of B_0 and centered elements of C, and there is at least one C factor. Freeness makes the trace zero. If there are no centered coefficient factors, the term is just a scalar multiple of a nonempty reduced Haar word and again has trace zero. This proves the assertion, with no condition on the first and last indices of w.□

Proof. First suppose that M=P is a factor. The constant coefficient algebra W^*(F)\subset P^\omega has separable predual. Apply Lemma 2.2 and then Lemma 2.3. The same free d-tuple makes every prescribed trace exactly zero in P^\omega, even though different tests can designate different labels.

Every unitary of P^\omega has unitary representatives. To see this, take a bounded self-adjoint logarithm there, lift it to a bounded sequence of self-adjoint elements of P, and exponentiate coordinatewise. Choose such representatives for the d unitaries. Each test trace converges to zero along \omega. Since the family of tests is finite, the coordinate sets on which their absolute values are less than \varepsilon have a common member. Its d coordinate unitaries solve all the requirements in P.

For general M, the factor case will give successful tuples in almost every fiber. The remaining task is to choose them measurably. Use the central direct integral [Bla06, Sections III.1.6 and III.5.1]

where (\Omega,\nu) is a probability space and, almost everywhere, M_s is a \mathrm{II}_1 factor with its normalized trace \tau_s. Standard separable direct-integral theory provides countably many bounded measurable sections a_j whose values generate M_s for almost every s. After removing one null set, all these properties and the unitary identities for the finitely many coefficients hold simultaneously.

There is a countable family (v_j) of measurable unitary sections which is dense in each fiber unitary group for \norm{\cdot}_{2,s}. Here is an explicit construction. Enumerate the sections

where q ranges over self-adjoint *-polynomials in finitely many variables with coefficients in \mathbb Q+i\mathbb Q. This includes symmetrizations of arbitrary rational *-polynomials. In a fixed fiber, any unitary is \exp(ih) for a bounded self-adjoint h. Kaplansky density and norm approximation by rational polynomials approximate h in 2-norm by the indicated self-adjoint polynomials. For bounded self-adjoint a,b, the integral identity

implies \norm{e^{ia}-e^{ib}}_2\leq\norm{a-b}_2. Thus these sections have the required fiberwise density. Each section is globally bounded, and exponentiation makes its norm one.

Enumerate all ordered d-tuples drawn from (v_j) as \boldsymbol v^{(1)},\boldsymbol v^{(2)},\ldots. Let E_k be the set of fibers where candidate k satisfies every test with absolute trace less than \varepsilon/2. These sets are measurable, since the traces of products of measurable sections are measurable. They cover almost every fiber. Indeed, the factor case gives a successful tuple in each such fiber, and the requirements are open in the product 2-norm topology. For this last assertion, telescoping products of unitaries gives

and inversion is also a 2-norm isometry on unitaries. The countable dense tuples therefore meet the open set of successful choices.

Take the first successful tuple: on the disjoint measurable sets

use \boldsymbol v^{(k)}, and use the identity tuple on the null complement. The resulting sections u_1,\ldots,u_d are measurable and unitary, so belong to M. Each test has fiber trace of absolute value less than \varepsilon/2 almost everywhere. Integrating gives an absolute global trace at most \varepsilon/2<\varepsilon, as required.□

Proof of Theorem 2.1. Choose a 2-norm dense sequence (v_j)_{j\geq1} in \U(M) and put

For \ell\geq1, fix the numerical parameters

Here n_\ell and d_\ell are positive integers, and p_0>0 because p_\ell\leq4^{-\ell}. At time n_\ell we will sample \ell paths, choosing level j with probability p_j at each increment. These parameters make level \ell appear in every path with high probability while later levels remain unlikely. The much larger label count d_\ell makes repeated level-\ell labels unlikely across the K_\ell increments. We construct finite symmetric measures

successively, and finally set

At stage \ell, choose a finite symmetric set F_\ell of base atoms with \mu_0(F_\ell)>1-K_\ell^{-2}. The coefficient alphabet consists of F_\ell and all u_{j,h}^{\pm1} from stages j<\ell; it is finite. Introduce d_\ell unknown unitary labels. Consider every ordered list of \ell paths of length n_\ell in this alphabet and the signed unknown labels such that:

1. 1.

   each path contains at least one unknown label;

2. 2.

   no unknown label occurs twice anywhere in the entire list, irrespective of sign.

For each such list and every nonempty reduced word of length at most \ell on \ell generators, require its evaluated trace to have absolute value less than 1/\ell. There are finitely many lists and finitely many words: all alphabet sizes, path lengths, and word lengths have been fixed before making this choice. Designating any one unknown occurrence in each path puts every test under Lemma 2.4. Apply that lemma to choose u_{\ell,1},\ldots,u_{\ell,d_\ell} satisfying all these requirements simultaneously. This completes the induction.

We estimate probabilities on a labeled sampling space. An increment first chooses a level j with probability p_j. At level zero it chooses an atom with law \mu_0; at level j\geq1 it chooses one of d_j labels uniformly and an independent uniform sign. Its value is then the corresponding unitary. This description gives law \mu even when distinct labels happen to represent the same element of M.

Sample \ell independent paths of length n_\ell, involving K_\ell increments in all. Let \mathcal B_\ell be the union of the following four events: a level greater than \ell appears; a level-zero increment lies outside F_\ell; a level-\ell label is repeated; or some path has no level-\ell increment. Union bounds give

For the collision term, any specified pair of increments has probability p_\ell^2/d_\ell\leq1/d_\ell of using the same level-\ell label; the signs do not matter. For the last term, a specified path misses that level with probability (1-p_\ell)^{n_\ell}.

Every term on the right of (2.2) tends to zero. Indeed, p_{\ell+1}=p_\ell^2 and p_\ell\leq1/4, so

Consequently, the four terms admit the respective bounds

all converging to zero.

Outside \mathcal B_\ell, the sampled paths form exactly one of the lists tested at stage \ell. All their earlier-level factors belong to the coefficient alphabet, each path has a level-\ell label, and all occurrences of those labels are distinct. Thus, simultaneously for every w\in\mathbb F_\ell with 1\leq|w|\leq\ell, the trace of the evaluated word has absolute value less than 1/\ell.

Now fix r, a nonempty finite \mathcal W, and \eta>0. For \ell\geq\max\{r,\max_{w\in\mathcal W}|w|\}, append independent paths to the given r paths to obtain \ell paths. On the preceding good event every word in \mathcal W has absolute trace below 1/\ell. Once 1/\ell<\eta, the probability in the theorem is at most \Pr(\mathcal B_\ell) and therefore tends to zero.

The measure \mu is symmetric and countably supported by construction. Its base component has positive weight, so each v_j is a positive-mass atom. Since bounded 2-norm convergence implies strong convergence in the tracial representation, the von Neumann algebra generated by these atoms contains every unitary and hence is M. Finally, the identity group word always has trace one, and every *-monomial in unitaries reduces to a group word. The asserted convergence therefore describes all joint *-moments.□

Proof. If (3.2) fails for some r,\varepsilon, choose x_j with \norm{x_j}\le r, \norm{x_j}_2<1/j, and \norm{R(x_j)}_2\ge\varepsilon. This contradicts (3.1); the converse is immediate. Applying the uniform assertion to differences gives uniform 2-norm continuity on each fixed operator norm ball.

Use the tracial ultrapower and its ideal \mathcal I_\omega from (2.1), now for (N,\rho). Its tracial norm is

By (3.2), the coordinate map induced by R preserves \mathcal I_\omega. Applying this observation to the difference of two bounded representatives proves that (3.3) is well-defined. The quotient norm gives

To transfer the modulus, fix r<\infty and \varepsilon>0, and choose \delta in (3.2) for the radius r+1 and the output tolerance \varepsilon/2. If \norm{X}\le r, the quotient norm allows a representative with \sup_j\norm{x_j}\le r+1. When \norm{X}_2<\delta/2, the set

belongs to \omega. Consequently

Thus \delta/2 is a valid input tolerance for R^\omega on the radius-r ball. The proof used no separability of N and applies again with N^\omega in its place, proving the last assertion.□

Proof. Write C=\norm{T} and H=L^2(M,\tau). Expectations in (3.4) are norm-convergent sums, since \mu is countably supported and \norm{T(xU)U^*}\le C\norm{x}. For g\in\U(M) put

Let U_1,U_2,\ldots be independent with law \mu, and set W_0=1, W_n=U_1\cdots U_n, and \mathcal F_n=\sigma(U_1,\ldots,U_n). The H-valued random variables Z_n=D(W_n) have countable range and satisfy \norm{Z_n}_2\le\norm{Z_n}\le C. Moreover,

Thus (Z_n) is a martingale in L^2(\Omega;H). Its increments are orthogonal, and

It follows that Z_n converges in L^2(\Omega;H) to a random variable Z_\infty, and continuity of conditional expectation gives Z_n=\E[Z_\infty\mid\mathcal F_n]. In particular, with

orthogonality gives the uniform Cauchy estimate

The operator norm ball K_C=\{x\in M:\norm{x}\le C\} is closed in H. Indeed, from a 2-norm-convergent sequence in K_C, ultraweak compactness provides an ultraweakly convergent subnet with limit in K_C. Testing against \tau(y^*\,\cdot), for y\in M, identifies this limit with the H-limit, since M is dense in H. A subsequence of Z_n converges almost surely in H, so Z_\infty\in K_C almost surely.

If Z_\infty=c almost surely, conditional expectation gives Z_0=Z_1=c almost surely. Hence c=T(1) and D(u)=c at every atom u of \mu. These atoms contain a 2-norm-dense subset of \U(M) by Theorem 2.1. For any u\in\U(M), choose atoms u_j with \norm{u_j-u}_2\to0. Bounded-ball continuity gives T(u_j)\to T(u) in 2-norm, while \norm{cu_j-cu}_2\le\norm{c}\norm{u_j-u}_2\to0. Since T(u_j)=cu_j, we obtain T(u)=cu. Every element of a unital complex C^*-algebra is a finite linear combination of unitaries, proving (3.5) in this case.

Suppose, for a contradiction, that Z_\infty is not deterministic. Its law is a Borel probability measure on the separable Hilbert space H, so its support contains two distinct points a,b. Because K_C is closed and has full measure, a,b\in K_C\subset M.

Forward and reverse endpoints. We claim that, for every \delta>0, there are numbers \eta_\delta>0 and N_\delta such that, whenever n\ge N_\delta and G has law \mu^{*n},

Realize G=U_1\cdots U_n, let h=\lfloor n/2\rfloor, and put

The two variables use disjoint sets of increments and are independent. Symmetry of \mu implies that both have the law of Z_h. Apply (3.7) to the forward string and separately to the reversed inverse string, which is also a string of independent increments with law \mu. It gives

For fixed \delta, the support property gives

Mean-square convergence implies, for all sufficiently large h,

Independence of A_n,B_n, the triangle inequality, and Markov’s inequality therefore bound the probability in (3.8) from below by

For large n this is at least p_ap_b/8, proving the claim. The independence used here is that of the disjoint prefixes A_n,B_n; the variables G and G^{-1} remain dependent.

Concatenation of independent blocks. Let G_1,\ldots,G_m be independent with law \mu^{*n}, and consider

where i_1,\ldots,i_t are pairwise distinct. Each inverse block can be represented by reversing and inverting its own increments. By symmetry, concatenating these distinct signed blocks is a length-tn walk with independent increments of law \mu. In particular,

Consequently

The restriction is on unsigned indices: using both G_i and G_i^{-1} would reuse a block and would not justify this estimate.

Finite requirements and diagonal selection. Fix an integer m\ge1 first, and use tolerance 1/m in (3.8). For m independent length-n products, let E_{m,n} be the event

Independence between the paths gives numbers c_m>0 and N_m such that

For example, one may take c_m=(\eta_{1/m})^m.

Let \mathcal Q_m be the finite set of nonidentity reduced words of length at most m on the symbols x_1,\ldots,x_m. Let \mathcal V_m be the finite set of all nonempty signed words on these symbols with pairwise distinct unsigned indices. Along the times n_\ell in Theorem 2.1, the probability that any of the requirements

fails tends to zero. For the first line, this is a finite union of the moment convergence statements in Theorem 2.1. For the second line, the union bound and (3.10) give the upper bound |\mathcal V_m|m^2e_{n_\ell}.

Choose \ell_m>\ell_{m-1}, after fixing m, so large that n_{\ell_m}\ge N_m and this combined failure probability is less than c_m/2. Then the intersection of E_{m,n_{\ell_m}} with all the requirements (3.12)–(3.13) has probability at least c_m/2>0. Choose a deterministic tuple

from that intersection. The constants c_m may tend to zero as fast as they please: the choice of \ell_m is made separately for each fixed m, after the positive lower bound has been obtained.

Write a_1=a and a_{-1}=b. The endpoint conditions and (3.13) imply, for every v=x_{i_1}^{\sigma_1}\cdots x_{i_t}^{\sigma_t}\in\mathcal V_m,

The equality uses T(v)=D(v)v and the 2-norm isometry of right multiplication by the unitary v.

The exact family in the ultrapower. Fix a free ultrafilter \omega_1 on the tuple index m. Set u_i^{(m)}=1 when m<i, and define

Lemma 3.2 makes R a bounded linear map on N, with \norm{R}\le C, and supplies its bounded-ball 2-continuity. Each g_i is unitary. Every fixed nonidentity reduced word q lies in \mathcal Q_m for all sufficiently large m, so (3.12) gives

These are precisely the joint moments of a countable free Haar family: powers of one generator are centered, and expansion of alternating centered Laurent polynomials in different generators reduces to nonidentity words. The corresponding freeness of their generated von Neumann algebras follows by bounded 2-norm approximation.

Identify a,b\in M with their constant images in N. For every nonempty word w=g_{i_1}^{\sigma_1}\cdots g_{i_t}^{\sigma_t} with pairwise distinct unsigned indices, (3.14) holds for all sufficiently large tuple indices. Hence it becomes the exact identity

These are all the hypotheses of Theorem 4.1, applied to (N,\tau^{\omega_1}) and R. That theorem yields a=b in N. The constant embedding preserves the faithful trace, so this also means a=b in M, contrary to their choice. Thus Z_\infty is deterministic, and the earlier argument proves (3.5).□

Proof. Write L=\sum_{|w|=q}c_w\lambda(w). On the standard basis of \ell^2(\mathbb F_r), separate each product wx according to its exact number t\in\{0,\ldots,q\} of cancellations at the joining boundary. This gives L=\sum_{t=0}^q L_t on finitely supported vectors. Fix t. Such a product has the form

where all three displayed concatenations uv, v^{-1}h, and uh are reduced. Conversely, these conditions describe exactly the products with t cancellations, allowing any of u,v,h to be empty when its required length permits this. In particular, h may be empty when the input is exhausted.

For this fixed t, an input basis word of length at least t has a unique prefix of length t and hence a unique suffix h. An output basis word of length at least q-t likewise determines h uniquely by removing its prefix of length q-t. Thus the input spaces indexed by different h are orthogonal, as are the corresponding output spaces. Relative to these two decompositions, L_t is a direct sum of matrices B_h, with columns indexed by v and rows by u. An entry is c_{uv} when the three reduction conditions hold and is zero otherwise. The fixed cutting length makes (u,v)\mapsto uv injective within each matrix. Therefore

Taking the direct-sum norm bounds L_t by the same quantity. Summing over the q+1 possible values of t proves (4.3).

For the norm transfer, recall that if A\geq0 belongs to a von Neumann algebra with faithful tracial state \rho, then

The upper bound is immediate. If 0<c<\norm{A}, the spectral projection e=1_{(c,\infty)}(A) is nonzero and has positive trace, so \rho(A^n)\geq c^n\rho(e). Taking roots and then letting c\uparrow\norm{A} proves the formula. For a word polynomial p, the free Haar moments give, for every n\geq1,

Both traces are faithful; here \tau_{\mathbb F_r} is the canonical trace on the free group von Neumann algebra. Applying (4.4) gives \norm{p(g)}=\norm{p(\lambda)}. The asserted 2-norm formula follows from orthonormality of reduced words.□

Proof. Fix this product, and let L=\sum_{j=1}^k(2d_j+1) be its degree. Every term in the expansion of one block B_d(s_r) has the odd alternating sign pattern

Deleting adjacent inverse letters deletes two consecutive opposite signs. The remaining pattern is still alternating, begins and ends positively, and is nonempty: the excess of positive over negative signs remains one. Hence the free reduction of each block is a nonempty word with positive endpoint signs. After reducing the blocks internally, their boundaries join two positive letters and admit no further cancellation. Every reduced word in P(s_r) is therefore nonempty and begins and ends positively. Taking adjoints reverses the block order and all signs, so the corresponding assertion for P(s_r)^* has negative endpoint signs.

We next bound the coefficient of every possible reduced word, including words much shorter than L. Write

For a fixed reduced word w of length q, record a free reduction of an original term by pairing its original positions whenever their letters cancel. For definiteness, the reduction may be performed by reading from left to right and canceling against the last surviving letter whenever possible. The recorded pairs form a noncrossing partial matching of the L positions: there are (L-q)/2 pairs, and the unmatched positions spell w in order. All nested and newly adjacent cancellations are included in this record.

There are at most 3^L noncrossing partial matchings of L positions. Indeed, mark each position as an opening endpoint, a closing endpoint, or unmatched; noncrossing forces a closing endpoint to pair with the last unpaired opening endpoint, so this encoding is injective. We may count every such partial matching, including ones not realized by a reduction. For a fixed pattern and fixed w, the labels at the unmatched positions are prescribed. Each canceled pair has equal unsigned labels and opposite signs and hence has at most r label choices. Coincidences between different pairs or with unmatched positions can only reduce the number of choices. Each original term has coefficient r^{-L/2}. Consequently, for every q and w,

An impossible parity or reduction pattern contributes zero. The same bound applies to the adjoint polynomial.

Let E_{r,q} be the sum of those length-q terms whose surviving reduced word has a repeated unsigned index. Original repetitions which disappear in the reduction remain included in the coefficients of the retained words. For q=1 this sum is zero. For q\geq2 the number of possible words in the sum is at most

choose a pair of equal-index positions, choose their common label and the other q-2 labels, and choose all signs. Counting nonreduced strings here only enlarges the bound. By orthonormality and (4.8),

Lemma 4.2 now yields the aggregate operator norm estimate

There is no scalar term, by the block reduction argument.

Put Q_r=P(s_r)-E_r. Each word in this finite sum has pairwise distinct unsigned indices and begins positively. Thus (4.2) and linearity give R(Q_r)=aQ_r. Moreover, \norm{P(s_r)}\leq2^L, so (4.10) also proves that the sequence (Q_r) is bounded. In N we have

Passing to \widehat N proves the first identity in (4.7). For the adjoint, the retained sum is Q_r^*: its reduced words have distinct unsigned indices and begin negatively. Therefore R(Q_r^*)=bQ_r^*, and the identical estimate with b proves the second identity.□

Proof. We first show that s has zero kernel, which makes its polar part unitary in the finite algebra. We then transfer the block identities to powers of that unitary. Set A=s^*s, so 0\leq A\leq4. To determine its kernel, we compute its moments. For d\geq1, the quantity \tau((s_r^*s_r)^d) is r^{-d} times the number of assignments of labels in \{1,\ldots,r\} to the alternating 2d positions

whose word reduces to the identity. An identity reduction pairs all positions noncrossingly, so it uses at most d unsigned labels. Assignments using at most d-1 labels number O_d(r^{d-1}): for each number of labels, choose an ordered list of those labels and then assign a member of that list to each of the 2d positions.

If exactly d labels occur in an identity word, every label occurs twice. Its two occurrences must be paired in any cancellation matching, so the matching is uniquely determined by the labeling. Conversely, in a noncrossing perfect matching, every pair has an even number of positions inside it, since those positions must be matched among themselves. The endpoints therefore have opposite parities, and hence opposite signs in our alternating string. Assigning distinct labels to its d pairs gives an identity word: erase innermost pairs successively. It follows that the number of leading assignments is exactly

where C_d is the number of noncrossing perfect matchings on 2d positions. Choosing the partner of the first position gives C_0=1 and C_d=\sum_{j=0}^{d-1}C_jC_{d-1-j}. The generating function thus satisfies C(z)=1+zC(z)^2; its solution with constant term one gives

Consequently,

The zeroth moment is also C_0=1.

Consider the measure

For every integer d\geq0, substitution x=4t and the beta integral give

In particular, \nu is a probability measure. Both \nu and the spectral distribution of A are supported in [0,4]. Equality of all their moments implies equality on polynomials and then on all continuous functions by uniform polynomial approximation. They are therefore the same measure. Since (4.13) has no atom at zero,

by faithfulness. In the polar decomposition s=VA^{1/2} this gives V^*V=1. Traciality and faithfulness then give

Thus V is a unitary.

It remains to transfer the identities. For a fixed regularization parameter we will use polynomial approximation in operator norm; we will then remove the regularization using bounded-ball 2-continuity. For \varepsilon>0 put

Fix k\geq1. A polynomial p(t)=\sum_{d=0}^D c_dt^d has the literal ordered expansion

Every summand is a product covered by Lemma 4.3. That lemma and complex linearity yield

For the second identity one takes the adjoint of the expansion (4.14), conjugating its scalar coefficients and reversing the block order, and applies the separate adjoint identity in Lemma 4.3. Choose polynomials converging uniformly on [0,4] to (t+\varepsilon)^{-1/2}. Functional calculus and boundedness of S pass the last two identities to the norm limit, giving

The x_\varepsilon are contractions, and their distance to V satisfies

by dominated convergence and \nu(\{0\})=0. Multiplication satisfies \norm{xyz}_2\leq\norm{x}\norm{y}_2\norm{z} in a tracial algebra. The noncommutative telescoping identity

therefore shows that \norm{x_\varepsilon^k-V^k}_2\leq k\norm{x_\varepsilon-V}_2\to0; the same holds for their adjoints. All these powers are contractions. Using the bounded-ball 2-norm continuity of S in (4.15), along for example \varepsilon=1/n, proves (4.11).□

Proof. We use analytic polynomials whose imaginary parts remain uniformly bounded on the unit circle while their real parts grow near 1. Rotating these polynomials will bound a-b on each short spectral arc; the trace will then combine the compression estimates without a loss depending on the number of arcs. Write \mathbb T=\{z\in\C:|z|=1\} and

We first verify the uniform sine-sum bound

By periodicity and oddness it suffices to use the angular distance t\in[0,\pi] from 2\pi\Z. At t=0 the sum is zero. For t>0 put K=\lfloor1/t\rfloor. The part with k\leq\min(K,m) has absolute sum at most Kt\leq1, since |\sin(kt)|\leq kt. Geometric summation gives, for every interval of integers p\leq q,

If m\geq K+1, put p=K+1 and A_j=\sum_{k=p}^j\sin(kt). Abel summation gives

Its absolute value is at most \pi/(tp)\leq\pi, including when K=0. This proves (4.16).

For \lambda\in\mathbb T, write

Continuous functional calculus and (4.16) give \norm{Y_{m,\lambda}}\leq B. The identities for positive and negative powers imply

Since the argument of S has norm at most 2B, we obtain

uniformly in m and \lambda.

Fix m and any half-open arc I\subset\mathbb T of angular length at most 1/m, and let e=1_I(V). Choose \lambda to rotate the midpoint of I to 1. On the rotated arc the argument \theta satisfies |\theta|\leq1/m, so

Thus, for X=X_{m,\lambda}, Borel functional calculus gives an element F\in W^*(V) equal to 1/X on I and to zero off I, with

Writing c=a-b and multiplying (4.17) on the right by F gives

In the corner e\mathcal A e, this says ec^*ce\leq\delta_m^2 e.

Partition \mathbb T into finitely many disjoint half-open arcs of angular length at most 1/m, and denote their spectral projections by e_1,\ldots,e_{J_m}. They are orthogonal and sum to 1. Traciality and (4.18) imply

because h_m\geq\log(m+1)\to\infty. Each arc estimate is weighted by the trace of its own projection; the number of arcs introduces no factor. Faithfulness gives c=0, as required. The right multiplication by F and the tracial compression identity apply to arbitrary a,b; the argument imposes no commutation condition and uses no particular spectral distribution of V.□

Proof of Theorem 4.1. Construct s\in\widehat N and S=R^{\omega_2} as in (4.6). Lemma 4.3 propagates the first-letter identities to every ordered block product. Lemma 4.4 then provides a unitary V satisfying the positive and negative power identities. Lemma 4.5 gives equality of the constant images of a and b in \widehat N. The constant embedding is trace preserving and faithful, so a=b in N.□

Proof. We argue by contradiction. A failure of continuity will give elements with mutually orthogonal two-sided supports whose images stay uniformly away from zero in 2-norm. Random signs will then contradict the boundedness of B.

If the first assertion fails, there are R>0, \eta>0, and a sequence (x_j) such that

Choose \alpha>0 with \norm{B}\alpha<\eta/4. In the polar decomposition x_j=v_j|x_j|, set

Here r_j and \ell_j are the right and left support projections of y_j. In particular,

The trace and the spectral truncation give

Pass to a subsequence for which \sum_j\tau(e_j)<\infty, and put p_j=\bigvee_{h\ge j}e_h. Then p_j decreases to zero: indeed, \tau(p_j)\le\sum_{h\ge j}\tau(e_h)\to0, and \tau is faithful and normal.

For fixed j and h>j, the compressions

converge ultraweakly to y_j as h\to\infty. Applying B preserves this convergence. The 2-norm is ultraweakly lower semicontinuous, as is seen from

Starting with j_1=1, we may therefore choose successively j_{s+1}>j_s so that

The t_s have pairwise orthogonal two-sided supports and \norm{t_s}\le R. Hence \norm{\sum_{s=1}^N\epsilon_st_s}\le R for every choice of signs \epsilon_s\in\{-1,1\}. Taking independent uniform signs, Hilbert space orthogonality of these scalar random variables yields

which is impossible for large N. This proves the asserted modulus. Applying it to differences, which lie in the ball of radius 2R, gives continuity on the ball of radius R.□

Proof. We first construct an averaged cochain that is a right module map in its last input. Evaluation at 1 will give a (k-1)-cochain, and averaging the original equation df=0 will identify its differential.

Choose the symmetric countably supported measure \mu for which Theorem 3.3 applies, and write \mu=\sum_{s\ge1}p_s\delta_{u_s}. For a bounded k-cochain \Phi, define

The sum converges in operator norm, and P is a contraction for the cochain norm. If W_j=U_1\cdots U_j, where the U_i are independent with law \mu, and W_0=1, induction gives

Set

We use compactness for pointwise ultraweak convergence. For each fixed tuple \mathbf x=(x_1,\ldots,x_k), all values F_n(\mathbf x) lie in the ultraweakly compact ball of radius \norm{f}\prod_i\norm{x_i}. The product of these balls, indexed by all tuples in M^k, is compact. Multilinearity is closed for this topology, since addition and scalar multiplication are ultraweakly continuous. Thus some subnet of (F_n) converges at every tuple to a bounded multilinear map F with \norm{F}\le\norm{f}.

On any common cochain norm ball, P is continuous for this topology. To see this, suppose \Phi_\lambda\to\Phi pointwise ultraweakly and \norm{\Phi_\lambda},\norm{\Phi}\le C. For a fixed tuple (\mathbf a,x) and a normal functional \rho\in M_*, truncate (5.2) after L terms. The finite sum passes to the limit, and the remaining difference has absolute value at most

Letting L\to\infty proves the claim. Since

the subnet limit satisfies PF=F.

Fix \mathbf a, and put B(x)=f(\mathbf a,x). This is a bounded normal linear map. For any R<\infty and \varepsilon>0, Lemma 5.1 provides \delta>0 such that

Indeed, right multiplication by a unitary preserves both the operator norm and the 2-norm. The same bound holds after averaging over any W_j and after taking the Cesàro average. Ultraweak lower semicontinuity of the 2-norm then gives \norm{F(\mathbf a,x)}_2\le\varepsilon under these input bounds. Using \varepsilon/2 in place of \varepsilon proves the required continuity modulus for T(x)=F(\mathbf a,x). The equality PF=F is exactly the harmonicity identity for this map. Consequently Theorem 3.3 gives

In particular h is a bounded (k-1)-cochain and \norm{h}\le\norm{f}.

It remains to obtain its coboundary identity. Let \nu_n=n^{-1}\sum_{j=0}^{n-1}\mu^{*j} and h_n(\mathbf a)=F_n(\mathbf a,1). Evaluate the original equation df=0 at (a_1,\ldots,a_k,v), multiply the output on the right by v^*, and average with respect to v\sim\nu_n. This gives the exact equality

The final term is unchanged because vv^*=1. Along the chosen subnet, each displayed evaluation converges ultraweakly; multiplication by a fixed a_i preserves that convergence. Passing to the limit and using (5.3) therefore yields

Thus g=(-1)^kh satisfies (5.1). This calculation averages df at the displayed tuples; it requires neither a commutation relation between P and d nor separate normality of the limiting cochain F.□

Proof. Order the finite subsets E\subset M by inclusion. For each E, choose a countably generated unital von Neumann subalgebra N_E containing E, the finite set f(E^k), and unital matrix systems of every size m\ge2. Such systems exist in a type \mathrm{II}_1 algebra: the unit can be divided into m equivalent orthogonal projections [Bla06, Examples III.2.5.4(iii)], and partial isometries between them supply the matrix units. Adjoining one system for each m adds only countably many generators.

The restriction of the trace to N_E is faithful and normal. The tracial Hilbert space L^2(N_E) is separable: *-polynomials with coefficients in \mathbb Q+i\mathbb Q in a countable set of generators are dense in the generated C^*-algebra in norm, and bounded strong approximation makes that algebra 2-norm dense in N_E. The faithful tracial representation thus gives N_E separable predual. Moreover N_E has no type I central part. A nonzero homogeneous type \mathrm{I}_n central summand would inherit a unital copy of M_m(\C) for every m, whereas its n-dimensional fibers cannot admit a unital copy when m>n. Every finite type I algebra decomposes into its homogeneous type \mathrm{I}_n summands. Since N_E is finite, it is therefore of type \mathrm{II}_1.

Let \mathcal E_E:M\to N_E be the normal trace-preserving conditional expectation. Its existence and bimodularity are classical; see [Dix53, Théorème 8] and [Ume54, Theorem 1]. Define

This map is separately normal and has norm at most \norm{f}. For inputs in N_E, bimodularity of \mathcal E_E gives df_E=\mathcal E_E(df)=0. By Proposition 5.2, choose g_E\in C_b^{k-1}(N_E,N_E) such that dg_E=f_E and \norm{g_E}\le\norm{f}. Extend it to a cochain on M by

This extension has the same norm bound. If (a_1,\ldots,a_k)\in E^k, all its inputs and adjacent products belong to N_E. Expanding the differential consequently gives

where the final equality uses f(E^k)\subset N_E.

The common norm bound and the same product compactness used above give a pointwise ultraweakly convergent subnet of (G_E), with limit g\in C_b^{k-1}(M,M) and \norm{g}\le\norm{f}. For a fixed k-tuple (a_1,\ldots,a_k), the indices of this subnet eventually contain E_0=\{a_1,\ldots,a_k\}. Equation (5.5) therefore holds eventually along the subnet. Passing to the limit in its finitely many coboundary terms gives dg=f at this tuple. This holds at every tuple, proving the proposition. The choices of N_E and \mathcal E_E can be made independently for different E; eventual exactness is supplied by the directed finite subsets.□

Proof. The norm estimate follows directly from (5.6). To verify the identity, expand each input as a_i=za_i+qa_i. By multilinearity it suffices to use tuples whose entries each belong to zM or to qM. On such a tuple with first qM entry at position r, formula (5.6) consists of the single term inserting q immediately before that entry, with sign (-1)^r; on an all-zM tuple it is zero.

Consider such an n-tuple with first qM entry a_r. In (J_zd)\psi, merging the inserted q with a_r has total sign (-1)^r(-1)^r=1 and contributes \psi(a_1,\ldots,a_n). A merger strictly before the insertion has the same arguments as its counterpart in (dJ_z)\psi, but the latter inserts at position r-1, so their signs are opposite. A merger strictly after the pair (q,a_r) also has the same arguments as its counterpart in (dJ_z)\psi, and its merger index is larger by one, again reversing the sign. The merger of a_{r-1} with the inserted q, when r>1, vanishes because zMqM=0; the corresponding merger across a_{r-1},a_r in (dJ_z)\psi vanishes for the same reason.

The left outer terms cancel when r>1, with insertion indices r and r-1. When r=1, both left outer terms vanish because qM annihilates the output module. For the right outer terms, if r<n the same insertion index and the two consecutive differential degrees give opposite signs. If r=n, the last input is in qM, so both right outer terms vanish. Thus the only surviving term is \psi(a_1,\ldots,a_n), as required because the cut of this tuple is zero. On an all-zM tuple, both sides of (5.7) vanish.

For completeness, in degree one the formula reads

since both outer module actions in (d\psi)(q,qa) vanish. In degree zero both sides are zero by the definitions. This proves the identity in all degrees.□

Proof of Theorem 1.1. The zero algebra is immediate, so assume M\ne0. Fix k\ge2 and an arbitrary bounded cocycle f_{\mathrm{orig}}\in C_b^k(M,M). By the normal reduction in Proposition 1.2, there are a separately normal cocycle f and a bounded (k-1)-cochain b such that

Let z_{\mathrm{II}_1} be the central projection of the type \mathrm{II}_1 summand, and let z_0=1-z_{\mathrm{II}_1}. We first partition z_{\mathrm{II}_1} into central projections whose algebras admit faithful normal tracial states.

In the abelian algebra Z(z_{\mathrm{II}_1}M), choose a maximal orthogonal family (z_i)_{i\in I} of nonzero supports of normal states. Its join is z_{\mathrm{II}_1}: a nonzero complementary projection would support a further normal state, contradicting maximality. A state is faithful on its support, so each z_iZ(M) has a faithful normal state \varphi_i. Composing \varphi_i with the faithful normal center-valued trace of z_iM gives a faithful normal tracial state \tau_i on z_iM; see [Bla06, Theorem III.2.5.7 and Corollary III.2.5.8]. The family I may be uncountable. Omit z_0 if it is zero, and otherwise keep the entire complementary summand as one piece.

For each piece define the restricted cocycle

Multiplication by a central projection is a bimodule map, so f_i is a separately normal cocycle and \norm{f_i}\le\norm{f}. For i\in I, Proposition 5.3 supplies a primitive b_i\in C_b^{k-1}(z_iM,z_iM) with

If z_0\ne0, the vanishing result for algebras without a type \mathrm{II}_1 central summand in Proposition 1.2 supplies a bounded primitive b_0 of f_0. Its norm is some finite number.

Pull each primitive back to M through the homomorphism x\mapsto z_ix:

The outer module actions also pull back, since the outputs lie in z_iM. Thus

Define cochains with values in z_iM on all of M by

Equation (5.9) gives

and, when present,

There is consequently a single finite constant C bounding the norms of all these cochains; one can take the maximum of the two displayed bounds, omitting the second when z_0=0.

For an arbitrary central partition of unity, the map

is an isometric *-isomorphism onto the bounded direct product. For surjectivity, the finite partial sums of a bounded family (x_i)_i, x_i\in z_iM, form a net directed by inclusion of the finite index sets. This net converges strongly, since the central supports are orthogonal; its limit belongs to M. For each tuple (x_1,\ldots,x_{k-1}), the family (H_i(x_1,\ldots,x_{k-1}))_i belongs to this product and has norm at most C\prod_j\norm{x_j}. Let G(x_1,\ldots,x_{k-1})\in M be the corresponding element. Coordinatewise multilinearity shows that G is a bounded multilinear cochain with \norm{G}\le C. Central projection commutes with every term of its coboundary, so

Since the projections have join 1, it follows that dG=f. Finally,

We have obtained an actual bounded (k-1)-cochain whose coboundary is the given cocycle. Hence H_b^k(M,M)=0 for every complex von Neumann algebra M and every k\ge2, with the actual image of d in the definition of cohomology and with no complete-boundedness hypothesis.□