Results on bounded Hochschild cohomology and their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Vanishing of higher bounded Hochschild cohomology — Theorem 1.2. For every von Neumann algebra M and k >= 2, every bounded cocycle in C_b^k(M,M) is the coboundary of a bounded cochain. The proof is completed in the fifth lesson and uses two classical theorems, K1 and K2.
• Ultrapower extensions — Lemma 2.2. A bounded map of a tracial von Neumann algebra that is continuous on operator-norm balls for the 2-norm has a uniform modulus and extends to every tracial ultrapower with the same norm bound and the same continuity.
• Normal maps are continuous on balls — Lemma 2.3. A bounded normal map of a von Neumann algebra with faithful normal tracial state is continuous on operator-norm balls for the 2-norm; Exercise 4.5 proves the converse.
• Rigidity of first-letter identities — Theorem 0.1. If a bounded map, continuous on balls for the 2-norm, multiplies every word with pairwise distinct indices in a free Haar family on the left by a or by b according to the sign of its first letter, then a = b.
• Powers of one unitary — Lemma 4.2. A bounded linear map with S(V^k) = aV^k and S(V^-k) = bV^-k for all k >= 1, for a unitary V of a finite von Neumann algebra with faithful normal tracial state, has a = b.
• Distinguished letters — Lemma 1.4. For Haar unitaries free from a coefficient algebra and from each other, a word in path expressions, each containing a letter that occurs nowhere else in the list, has trace zero.
• Finitely many trace conditions — Proposition 2.5. In an algebra of type II1 with separable predual, finitely many such trace conditions can be met approximately by unitaries: in a factor through Popa's free independence theorem, in general fibre by fibre in the central decomposition.
• An asymptotically free random walk — Theorem 3.1. A symmetric probability measure on the unitary group with countably many atoms, including a 2-norm dense sequence, whose independent long products have asymptotically the joint moments of free Haar unitaries.
• The Liouville theorem — Theorem 0.1. A bounded linear map that is harmonic for the walk and continuous on balls for the 2-norm is left multiplication by its value at 1.
• Primitives in the separable tracial case — Proposition 1.2. A separately normal cocycle of degree k >= 2 on an algebra of type II1 with separable predual and faithful normal tracial state has a primitive of norm at most its own.
• Type II1 from matrix units — Lemma 2.2. A von Neumann algebra with separable predual and faithful normal tracial state that contains unital matrix units of every size is of type II1.
• Primitives in the tracial case — Proposition 2.3. The same bound for every algebra of type II1 with a faithful normal tracial state, through conditional expectations onto countably generated subalgebras and pointwise ultraweak compactness.
• A central homotopy — Lemma 3.1. For a central projection z, an explicit operator J_z on cochains with values in zM satisfies dJ_z + J_z d = id - cut_z and has norm at most n - 1 in degree n.
• Tracial pieces — Lemma 4.1. Every von Neumann algebra of type II1 is the sum of central summands of type II1 that have faithful normal tracial states; with the central homotopy and the classical K1 and K2 this gives Theorem 1.2 of the first lesson for every von Neumann algebra.

Reading order

Five lessons, to be read in order. The first states the problem and proves the facts about the tracial 2-norm. The second proves the rigidity theorem, the third constructs the random walk, the fourth proves the Liouville theorem from them, and the fifth constructs the primitives. Two classical theorems are used without proof in the programme and are stated in the first lesson: the normal reduction of Johnson, Kadison and Ringrose, and the vanishing of the cohomology for algebras without a type II1 central summand.