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The homological essence of Connes amenability: Injectivity of the predual bimodule. (English. Russian original) Zbl 0721.46041
Math. USSR, Sb. 68, No. 2, 555-566 (1990); translation from Mat. Sb. 180, No. 12, 1680-1690 (1989).
As is known, the notion of amenability of Banach algebra was recorded from that of LCA-groups (amenability of $$L^ 1(G))$$ [see B. E. Johnson, Cohomology in Banach algebras, Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)]. According to an initial definition due to B. E. Johnson in his memoir of 1972, a unital Banach algebra A is said to be amenable if every derivation of A into a dual Banach A-bimodule is inner. B. E. Johnson described such an algebra in terms of virtual diagonal and U. Haagerup [Invent. Math. 74, 305-319 (1983; Zbl 0529.46041)] proved that any nuclear $$C^*$$-algebra A is amenable by constructing a virtual diagonal for A [see also E. G. Effros, J. Funct. Anal. 78, 137-153 (1988; Zbl 0655.46053)] for alternate proof and criteria of Connes amenability (see definition below) in the class of von Neumann algebras]. A. Connes proved earlier that amenable $$C^*$$-algebra is nuclear [J. Funct. Anal. 28, 248-253 (1978; Zbl 0408.46042)]; and A. Connes and U. Haagerup used the normal cohomology as the Johnson-Kadison-Ringrose variant of Hochschild-Kamowitz cohomology groups.
Some remarks on the terminology of the paper, which is not apparently conventional. The author of this paper deals with two variants of amenability: Johnson amenability for Banach algebras (in the author’s terminology, an ordinary variant) and A. Connes amenability $$(=amenable$$ as von Neumann algebras in Connes terminology) [Ann. Math., II. Ser. 104, 73-115 (1976; Zbl 0343.46042)] for operator $$C^*$$- algebras; the latter is also referred to as normal amenability.
As for the precise definitions, a Banach algebra (an operator $$C^*$$- algebra) A is said to be Johnson (Connes) amenable if $$H^ 1(A,X)=0$$ for all dual A-bimodules $$(H^ 1_ w(A,X)=0$$ for all normal A-bimodules). The cohomology groups $$H^ n(A,X)$$ are ordinary n-dimensional cohomology groups of A with coefficients in X corresponding to the standard cohomological complex of Banach spaces (n-linear continuous operators are n-cochains). If n-cochains are normal $$(=ultraweak$$-weak$${}^*$$ continuous w.r.t. each argument), the normal cohomology groups $$H^ n_ w(A,X)$$ of A with coefficients in normal module X, arise.
The class of amenable Banach algebras was studied by various methods including the approaches based upon a suitable variant of cohomology groups of the algebras with corresponding topological requirements on cochains simultaneously with the “inner” methods such as Johnson virtual diagonal. It is well-known fact that $$H^ n(A,X)\cong_{TOP}Ext^ n(A_+,X)$$ for all $$n\geq 0$$, a Banach algebra A, A-bimodule X and trivial left A-module $$A_+$$ (with natural actions). The author proves the following general result for normal A-bimodule:
Let A be an operator $$C^*$$-algebra in a Hilbert space H, $$X=(X_*)^*$$ be a normal A-bimodule $$(X_*$$ is predual to X). Then, for all $$n\geq 0$$, $H^ n_ w(A,X)=Ext^ n(X_*,\bar A_*),$ where $$\bar A_*$$ denotes the predual Banach space for w-closure of the algebra A in the algebra of all operators in H. Moreover, as the author proves, we may put $$A^*$$ instead of $$\bar A_*$$ for dual A-bimodule $$(X_*)^*$$ with at least one of multiplication operator normal. Also, as a consequence, $$H^ n_ w(A,X)=H^ n(A,X)$$ for any normal A- bimodule X and $$n\geq 0$$ [cf. B. E. Johnson, R. V. Radison and J. R. Ringrose, Bull. Soc. Math. France 100, 73-96 (1972; Zbl 0234.46066)]. So Johnson amenability for an operator $$C^*$$-algebra implies Connes amenability.
Also, the following theorem is proved:
The following properties of the operator $$C^*$$-algebra are equivalent
1) A is Connes amenable.
2) $$H^ n_ w(A,X)=0$$ for any normal A-bimodule X and $$n>0.$$
3) The A-bimodule $$\bar A_*$$ is injective.
Of course, the author concentrates this attention on implication $$3\to 1$$. The essential tools for the proof are some results from the paper of E. G. Effros mentioned above and Effros’ reduced bilinear functional for normal state on von Neumann algebras [the latter was introduced by E. G. Effros inconnection with a variant of Haagerup-Pisier-Grothendieck inequality: U. Haagerup, Adv. Math. 56, 93-116 (1985; Zbl 0593.46052)].
The results formulated above enable the author to trace the homological phenomenon of Connes amenability.

##### MSC:
 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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