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.

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.

Reviewer: S.Berger (Novosibirsk)

##### 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) |