Continuous Hochschild Cohomology Research Articles

Thompson’s semigroup and the first Hochschild cohomology

Abstract In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson’s group $\mathcal {F}$ . We introduce the notion of unique factorization semigroup which contains Thompson’s semigroup $\mathcal {S}$ and the free semigroup $\mathcal {F}_n$ on n ( $\geq 2$ ) generators. Let $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal {S}$ and $\mathcal {F}_n$ , respectively. We prove that all derivations on $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ are automatically continuous, and every derivation on $\mathfrak {B}(\mathcal {S})$ is induced by a bounded linear operator in $\mathcal {L}(\mathcal {S})$ , the weak-operator closed Banach algebra consisting of all bounded left convolution operators on $l^2(\mathcal {S})$ . Moreover, we prove that the first continuous Hochschild cohomology group of $\mathfrak {B}(\mathcal {S})$ with coefficients in $\mathcal {L}(\mathcal {S})$ vanishes. These conclusions provide positive indications for the left amenability of Thompson’s semigroup.

Derivations and cohomologies of Lipschitz algebras

For a compact metric space (M, d), $${\mathrm {Lip}}M$$ denotes the Banach algebra of all complex-valued Lipschitz functions on (M, d). Motivated by a classical result of de Leeuw, we give a canonical construction of a compact Hausdorff space $${\hat{M}}$$ and a continuous surjection $$\pi :{\hat{M}} \rightarrow M$$ which may viewed as a metric analogue of the unit sphere bundle over a Riemannian manifold. It is shown that, for each $$n \ge 1$$ the continuous Hochschild cohomology $${\mathrm {H}}^{n}({\mathrm {Lip}}M, C({\hat{M}}))$$ has the infinite rank as a $${\mathrm {Lip}}M$$-module, if the metric space (M, d) admits a local geodesic structure, for example, if M is a compact Riemannian manifold or a non-positively curved metric space. Here $$C({\hat{M}})$$ denotes the algebra of all complex-valued continuous functions on $${\hat{M}}$$. On the other hand, if the coefficient $$C({\hat{M}})$$ is replaced with C(M), then it is shown that $${\mathrm {H}}^{1}({\mathrm {Lip}}M,C(M)) = 0$$ for each compact Lipschitz manifold M.

Point derivations and cohomologies of Lipschitz algebras

AbstractFor a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

Approximate amenability and a variant of continuous Hochschild cohomology

We study a variant of continuous Hochschild cohomology of a Banach algebra in connection with a higher-dimensional analogue of the approximate amenability of the algebra. Some results on higher-dimensional amenability have natural analogues in our context. Alternating cocycles, due to Johnson \cite {johnson}, are studied, and a previous result of the author B. E. Johnson on Lipschitz algebras over compact metric spaces is improved.

Hochschild cohomology of reduced incidence algebras

We compute the continuous Hochschild cohomology of four reduced incidence algebras: the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation for the coalgebra Cotor-groups of their pre-dual coalgebras.

Hochschild cohomology of tensor products of topological algebras

AbstractWe describe explicitly the continuous Hochschild and cyclic cohomology groups of certain tensor products of $\widehat{\otimes}$-algebras which are Fréchet spaces or nuclear DF-spaces. To this end we establish the existence of topological isomorphisms in the Künneth formula for the cohomology of complete nuclear DF-complexes and in the Künneth formula for continuous Hochschild cohomology of nuclear $\widehat{\otimes}$-algebras which are Fréchet spaces or DF-spaces for which all boundary maps of the standard homology complexes have closed ranges.

Hochschild Cohomology Theories in White Noise Analysis

We show that the continuous Hochschild cohomology and the differential Hoch- schild cohomology of the Hida test algebra endowed with the normalized Wick product are the same.

BARRY EDWARD JOHNSON 1937–2002

Barry Johnson made major contributions to the theory of Banach algebras, by stimulating research on automatic continuity and cohomology in these algebras. His research on the continuous Hochschild cohomology of Banach and operator algebras led to major developments in these areas, and to a recognition of ‘amenability’ as more than simply a group-theoretic idea, but also one that is widely applicable in modern analysis.

Hochschild cohomology of factors with property Γ

The main result of this paper is that the k th continuous Hochschild cohomology groups H k (M, M) and H k (M ,B (H)) of a von Neumann factor M⊆ B(H )o f type II 1 with property Γ are zero for all positive integers k. The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps. We prove joint continuity in the �·� 2-norm of separately ultraweakly continuous multilinear maps, and combine these results to reduce to the case of completely bounded cohomology which is already solved.

Property gamma factors and the Hochschild cohomology problem.

The main result of this article is that the kth continuous Hochschild cohomology groups H(k)(M,M ) and H(k)(M, B(H)) of a von Neumann factor subset, dbl equals B(H) of type II(1) with property Gamma are 0 for all positive integers k. The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps. We prove joint continuity in the parallel. parallel (2) norm of separately ultraweakly continuous multilinear maps and combine these results to reduce to the case of completely bounded cohomology, which is already solved.

Barry Edward Johnson. 1 August 1937 – 5 May 2002

Barry Johnson made major contributions to the theory of Banach algebras, stimulating research on automatic continuity and cohomology in these algebras. His research on the continuous Hochschild cohomology of Banach and operator algebras led to major developments in these areas, and to ‘amenability’ moving from a group theoretic idea to one that is widely applicable in modern analysis.

Finite von Neumann Algebra Factors with Property Γ

Techniques introduced by G. Pisier in his proof that finite von Neumann factors with property Γ have length at most 5 are modified to prove that the length is 3. It is proved that if such a factor is a complemented subspace of a larger C*-algebra then there exists a projection of norm one from the larger algebra onto the smaller and a new proof of the fact that the continuous Hochschild cohomology group H2c(M, M) vanishes is also included.

Cartan subalgebras of finite von Neumann algebras

Sorin Popa [Po 1] [Po 2] has proved several results on the existence and properties of hyperfinite subfactors R of a type II1 factor M with the relative commutant R0 \M of R in M equal to C1. These theorems have been used in various cohomology calculations [CES, CS, PS, CPSS], as averaging over an amenable subgroup that generates the hyperfinite subfactor is a major step in showing that the continuous and completely bounded Hochschild cohomology groups are equal. It has seemed reasonable that Popa's results could be extended from factors to general type II1 von Neumann algebras by direct integral theory [KR, Chapter 4]. However, we do not know of such an attempt. Direct integral theory can be used directly to prove cohomology is zero and deduce results like [CPSS, Theorems 5.4 and 5.5] however these theorems on the continuous Hochschild cohomology for a von Neumann algebra with Cartan subalgebras are deduced from the theorems in this paper. This paper provides direct proofs of Popa's main two results in [Po 1] by modifying his proofs using an interpolation type result for projections in a maximal abelian selfadjoint sub-algebra (masa) of the type II1 algebra. This introduction contains a more detailed description of how our results extend Popa's, the basic definitions, and a brief reference to their use in the calculation of Hochschild cohomology groups in von Neumann algebras. Though averaging plays an important role in calculating the Hochschild cohomology of von Neumann algebras for all types I ; II1; II1; III of von Neumann algebras (see [Ri]), the results proved here are only used in the type II1 situation. The reason is that the type I 's are already trivially hyperfinite, and the type II1 and III von Neumann algebras may be handled by their stability under tensoring with B H†. This tensor factor B H† of M in the II1 and III cases gives a suitable hyperfinite algebra over which to average. Popa [Po 1] restricts his attention to type II II1 and II1† factors. MATH. SCAND. 85 (1999), 105^120

Hochschild cohomology groups of certain algebras of analytic functions with coefficients in one-dimensional bimodules

We compute the algebraic and continuous Hochschild cohomology groups of certain Fréchet algebras of analytic functions on a domain U in $ℂ^n$ with coefficients in one-dimensional bimodules. Among the algebras considered, we focus on A=A(U). For this algeb

On Continuous Hochschild comology and Cohomology Groups

Using the method of continuous projective resolutions established by Alain Connes, we calculate the continuous Hochschild homology and cohomology groups of the Frechet algebra of smooth functions on a manifold M. These (co)homology groups are allowed to take values in the Frechet space E of smooth sections of a vector bundle E over M and in the strong dual E*. Moreover, we show that in the cohomological case the differential and continuous Hochschild cohomologies are naturally isomorphic.