Cohomology in Banach algebras.

*(English)*Zbl 0256.18014
Mem. Am. Math. Soc. 127, 96 p. (1972).

It was about the end of the Second World War when G. Hochschild
introduced a cohomology theory of associative algebras, which now
carries his name ([G. Hochschild,
“On the cohomology groups of an associative algebra,” Ann.
Math. (2) 46, No. 1, 58–67 (1945; Zbl 0063.02029)] and
[G. Hochschild, “On the cohomology theory for
associative algebras,” Ann. Math. (2) 47, No. 3, 568–579 (1946; Zbl
0063.02030)]).
One can add functional analytic overtones to Hochschild’s ideas:
instead of general associative algebras, consider Banach algebras,
replace general modules by Banach modules, and suppose that all
cochains concerned are bounded. The first to do it – and to prove
substantial theorems about such a modified Hochschild cohomology
theory (for the Banach algebra of all continuous functions on a
compact Hausdorff space) – seems to have been H. Kamowitz
[“Cohomology groups of
commutative Banach algebras,” Trans. Am. Math. Soc.\
102, 352–372 (1962; Zbl 0114.31703)]. In the decade after the publication
of Kamowitz’s paper, the attempts to develop a homological algebra for
Banach algebras (and more general topological algebras) became more
frequent and more systematic. I would like to mention only the
contributions of A. Guichardet
[“Sur l’homologie et la cohomologie des algèbres de
Banach,” C. R. Acad. Sci., Paris, Sér. A 262, 38–41
(1966; Zbl 0131.13101)], A. Ya. Helemskiĭ
[“The homological dimension of normed
modules over Banach algebras” (in Russian), Mat. Sb., N.\
Ser. 81 (123) 430–444 (1970; Zbl 0189.44602)], and J. L. Taylor
[“Homology and
cohomology for topological algebras,” Adv. Math. 9, 137–182
(1972; Zbl 0271.46040)].

Among all those efforts, however, B. E. Johnson’s memoir under review stands out – at least in my opinion – due to the impact it has had in the four decades to follow. Johnson’s interest in the topic seems to have originated with his development of the theory of centralizers [B. E. Johnson, “An introduction to the theory of centralizers,” Proc. Lond. Math. Soc., III. Ser. 14, 299–320 (1964; Zbl 0143.36102)]. Roughly speaking, a (left) centralizer on a Banach algebra is a bounded left module homomorphism from the algebra into itself. If \(G\) is a locally compact group, then every left centralizer \(T\) of the group algebra \(L^1(G)\) is of the form \(Tf = f \ast \mu\) for \(f \in L^1(G)\), where \(\mu\) is a unique element of the measure algebra \(M(G)\): this is the statement of Wendel’s theorem. As it turns out, Wendel’s theorem can be viewed as a statement about the vanishing of Hochschild cohomology groups. Let \({\mathfrak A}\) be a Banach algebra, and let \(E\) be a Banach \({\mathfrak A}\)-bimodule. Then a derivation from \({\mathfrak A}\) into \(E\) is a bounded linear map \(D \!: {\mathfrak A} \to E\) satisfying the product rule \[ D(ab) = a \cdot Db + (Da) \cdot b \qquad (a,b \in {\mathfrak A}); \] if there is \(x \in E\) such that \(Da = a \cdot x - x \cdot a\) for \(a \in {\mathfrak A}\), we call \(D\) an inner derivation. The derivations from \({\mathfrak A}\) into \(E\) form a linear space, containing the inner derivations as a (not necessarily closed) subspace; the resulting quotient space is called the first Hochschild cohomology group of \({\mathfrak A}\) with coefficients in \(E\), which we denote by \(\mathcal{H}^1({\mathfrak A},E)\). Now, let \({\mathfrak A} = L^1(G)\) for a locally compact group \(G\), and let \(E := M(G)\) be turned into a Banach \({\mathfrak A}\)-bimodule as follows: for \(f \in L^1(G)\) and \(\mu \in M(G)\), set \(f \cdot \mu := f \ast \mu\) and \(\mu \cdot f := 0\). A derivation from \({\mathfrak A}\) into \(E\) is then nothing but a left centralizer of \(L^1(G)\), and Wendel’s theorem becomes the statement that \(\mathcal{H}^1({\mathfrak A},E) = \{ 0 \}\). Of course, a much more natural bimodule action of \(L^1(G)\) on \(M(G)\) is through convolution from both sides, which begets the question if \(\mathcal{H}^1(L^1(G), M(G)) = \{ 0 \}\) as well: it was this question that apparently motivated much of Johnson’s memoir. In Section 4, Johnson establishes that indeed \(\mathcal{H}^1(L^1(G), M(G)) = \{ 0 \}\) if \(G\) is amenable, has small invariant neighborhoods – which is the case, for instance, if \(G\) is discrete – , or is one of several groups of invertible \(N \times N\)-matrices. All these results are interesting (even impressive) in their own right, but it is the case of amenable groups that had the biggest impact on the theory of Banach algebras. Given a Banach algebra \({\mathfrak A}\) and a Banach \({\mathfrak A}\)-bimodule \(E\), the dual space \(E^\ast\) of \(E\) canonically becomes a Banach \({\mathfrak A}\)-bimodule through \[ \langle x, a \cdot \phi \rangle := \langle x \cdot a, \phi \rangle \;\text{ and } \;\langle x , \phi \cdot a \rangle := \langle a \cdot x, \phi \rangle \] for \(a \in {\mathfrak A}\), \(x \in E\) and \(\phi \in E^\ast\). Dual modules can be used to characterize the amenable locally compact groups in terms of Hochschild cohomology: Theorem 2.5 asserts that a locally compact group \(G\) is amenable if and only if \(\mathcal{H}^1(L^1(G),E^\ast) = \{ 0 \}\) for every Banach \(L^1(G)\)-bimodule \(E\). As \(M(G)\) is the dual space of the continuous functions on \(G\) vanishing at infinity, it yields immediately that \(\mathcal{H}^1(L^1(G), M(G)) = \{ 0 \}\) for amenable \(G\).

In retrospect, Theorem 2.5 is the central result of the memoir: it characterizes the amenable locally compact groups through a cohomological triviality condition for its group algebra. That triviality condition, however, can be formulated for any Banach algebra. Consequently, Johnson defines an arbitrary Banach algebra \({\mathfrak A}\) to be amenable if and only if \(\mathcal{H}^1({\mathfrak A},E^\ast) = \{ 0 \}\) for every Banach \({\mathfrak A}\)-bimodule \(E\). This defines a new class of Banach algebras, but how significant is it? What does amenability “mean” for \({\mathfrak A}\)? In Section 6, Johnson studies the amenability of the Banach algebra \(\mathcal{K}(E)\) of all compact operators on a Banach space \(E\), and shows that \(\mathcal{K}(E)\) is indeed amenable if \(E = \ell^p\) with \(p \in (1,\infty)\) or if \(E\) is the space of all continuous functions on the unit circle. He then goes on and introduces – in Section 7 – the notion of strong amenability for \(C^*\)-algebras and shows that all postliminal \(C^*\)-algebras are strongly amenable. Altogether, the memoir consists of ten sections, and the one that has stimulated me most – and still continues to stimulate me – is the last one: just two and a half pages long, it has the title “Suggestions for further research”. It is here where the impact of Johnson’s memoir on mathematics becomes most apparent.

In 10.2, the question is raised if every \(C^*\)-algebra is amenable; in particular, it is asked if the Banach algebra \(\mathcal{B}({\mathfrak H})\) of all bounded linear operators on an infinite-dimensional Hilbert space \({\mathfrak H}\) is amenable. Both question were settled within little more than a decade after they were posed. In [“On the cohomology of operator algebras,” J. Funct. Anal. 28, 248–253 (1978; Zbl 0408.46042)], A. Connes showed that a separable, amenable \(C^*\)-algebra had to be nuclear. In fact, the separability hypothesis turned out to be unnecessary, so that every amenable \(C^*\)-algebra must be nuclear. As had been discovered only a few years earlier by S. Wassermann [“On tensor products of certain group \(C^*\)-algebras,” J. Funct. Anal. 23, 239–254 (1976; Zbl 0358.46040)], nuclear von Neumann algebras are subhomogeneous, so that, in particular, \(\mathcal{B}({\mathfrak H})\) cannot be amenable if \(\dim {\mathfrak H} = \infty\). Finally, U. Haagerup proved that amenability not only implies nuclearity, but is equivalent to it [“All nuclear \(C^*\)-algebras are amenable,” Invent. Math. 74, 305–319 (1983; Zbl 0529.46041)]: the amenable \(C^*\)-algebras are precisely the nuclear ones.

Given that Hilbert spaces are the “nicest” of Banach spaces, the fact that \(\mathcal{B}({\mathfrak H})\) is not amenable for \(\dim {\mathfrak H} = \infty\) strongly suggests that \(\mathcal{B}(E)\), the Banach algebra of all bounded linear operators on a Banach space \(E\), can never be amenable unless \(\dim E < \infty\). In 10.4, it is asked if this is true. Somewhat frustratingly, progress towards a solution to that problem turned out to be extremely elusive. In [“Relative amenability and the non-amenability of \(B(\ell^1)\),” J. Aust. Math. Soc.\ 80, No. 3, 317–333 (2006; Zbl 1111.46028)], C. J. Read showed that \(\mathcal{B}(\ell^1)\) is not amenable: this was the first proof of the non-amenability of \(\mathcal{B}(\ell^p)\) for any value of \(p\) other than \(2\). Only quite recently, it was shown that \(\mathcal{B}(\ell^p)\) fails to be amenable for every \(p \in [1,\infty]\) [V. Runde, “\(\mathcal{B}(\ell ^p)\) is never amenable,” J. Am. Math. Soc. 23, No. 4, 1175–1185 (2010; Zbl 1221.47143)]. To everyone’s – or at least my – surprise, there are, however, infinite-dimensional Banach spaces \(E\) such that \(\mathcal{B}(E)\) is amenable: the space \(E\) constructed in [S. A. Argyros and R. G. Haydon, “A hereditarily indecomposable \(\mathcal{L}_\infty\)-space that solves the scalar-plus-compact problem,” Acta Math. 206, No. 1, 1–54 (2011; Zbl 1223.46007)] is such an example. The question if \(\mathcal{H}^1(L^1(G),M(G)) = \{ 0 \}\) for all locally compact groups \(G\), is stated in 10.8. After a hiatus of almost three decades, Johnson returned to the problem in [B. E. Johnson, “The derivation problem for group algebras of connected locally compact groups,” J. Lond. Math. Soc., II. Ser. 63, No. 2, 441–452 (2001; Zbl 1012.43001)] and proved that the answer is “yes” if \(G\) is connected, which subsumes the results about matrix groups from the memoir. Eventually, V. Losert gave an affirmative answer for general \(G\) [“The derivation problem for group algebras,” Ann. Math. (2) 168, No. 1, 221–246 (2008; Zbl 1171.43004)]; see also [U. Bader, T. Gelander, and N. Monod, “A fixed point theorem for \(L^1\) spaces,” arXiv:1012.1488] for a very simple proof.

I would like to conclude this review with two observations and an obvious question they prompt: MathSciNet lists 215 citations of Johnson’s memoir, the Science Citation Index has none. What does this tell us about the useful(less?)ness of certain scholarly beancounting toys?

Among all those efforts, however, B. E. Johnson’s memoir under review stands out – at least in my opinion – due to the impact it has had in the four decades to follow. Johnson’s interest in the topic seems to have originated with his development of the theory of centralizers [B. E. Johnson, “An introduction to the theory of centralizers,” Proc. Lond. Math. Soc., III. Ser. 14, 299–320 (1964; Zbl 0143.36102)]. Roughly speaking, a (left) centralizer on a Banach algebra is a bounded left module homomorphism from the algebra into itself. If \(G\) is a locally compact group, then every left centralizer \(T\) of the group algebra \(L^1(G)\) is of the form \(Tf = f \ast \mu\) for \(f \in L^1(G)\), where \(\mu\) is a unique element of the measure algebra \(M(G)\): this is the statement of Wendel’s theorem. As it turns out, Wendel’s theorem can be viewed as a statement about the vanishing of Hochschild cohomology groups. Let \({\mathfrak A}\) be a Banach algebra, and let \(E\) be a Banach \({\mathfrak A}\)-bimodule. Then a derivation from \({\mathfrak A}\) into \(E\) is a bounded linear map \(D \!: {\mathfrak A} \to E\) satisfying the product rule \[ D(ab) = a \cdot Db + (Da) \cdot b \qquad (a,b \in {\mathfrak A}); \] if there is \(x \in E\) such that \(Da = a \cdot x - x \cdot a\) for \(a \in {\mathfrak A}\), we call \(D\) an inner derivation. The derivations from \({\mathfrak A}\) into \(E\) form a linear space, containing the inner derivations as a (not necessarily closed) subspace; the resulting quotient space is called the first Hochschild cohomology group of \({\mathfrak A}\) with coefficients in \(E\), which we denote by \(\mathcal{H}^1({\mathfrak A},E)\). Now, let \({\mathfrak A} = L^1(G)\) for a locally compact group \(G\), and let \(E := M(G)\) be turned into a Banach \({\mathfrak A}\)-bimodule as follows: for \(f \in L^1(G)\) and \(\mu \in M(G)\), set \(f \cdot \mu := f \ast \mu\) and \(\mu \cdot f := 0\). A derivation from \({\mathfrak A}\) into \(E\) is then nothing but a left centralizer of \(L^1(G)\), and Wendel’s theorem becomes the statement that \(\mathcal{H}^1({\mathfrak A},E) = \{ 0 \}\). Of course, a much more natural bimodule action of \(L^1(G)\) on \(M(G)\) is through convolution from both sides, which begets the question if \(\mathcal{H}^1(L^1(G), M(G)) = \{ 0 \}\) as well: it was this question that apparently motivated much of Johnson’s memoir. In Section 4, Johnson establishes that indeed \(\mathcal{H}^1(L^1(G), M(G)) = \{ 0 \}\) if \(G\) is amenable, has small invariant neighborhoods – which is the case, for instance, if \(G\) is discrete – , or is one of several groups of invertible \(N \times N\)-matrices. All these results are interesting (even impressive) in their own right, but it is the case of amenable groups that had the biggest impact on the theory of Banach algebras. Given a Banach algebra \({\mathfrak A}\) and a Banach \({\mathfrak A}\)-bimodule \(E\), the dual space \(E^\ast\) of \(E\) canonically becomes a Banach \({\mathfrak A}\)-bimodule through \[ \langle x, a \cdot \phi \rangle := \langle x \cdot a, \phi \rangle \;\text{ and } \;\langle x , \phi \cdot a \rangle := \langle a \cdot x, \phi \rangle \] for \(a \in {\mathfrak A}\), \(x \in E\) and \(\phi \in E^\ast\). Dual modules can be used to characterize the amenable locally compact groups in terms of Hochschild cohomology: Theorem 2.5 asserts that a locally compact group \(G\) is amenable if and only if \(\mathcal{H}^1(L^1(G),E^\ast) = \{ 0 \}\) for every Banach \(L^1(G)\)-bimodule \(E\). As \(M(G)\) is the dual space of the continuous functions on \(G\) vanishing at infinity, it yields immediately that \(\mathcal{H}^1(L^1(G), M(G)) = \{ 0 \}\) for amenable \(G\).

In retrospect, Theorem 2.5 is the central result of the memoir: it characterizes the amenable locally compact groups through a cohomological triviality condition for its group algebra. That triviality condition, however, can be formulated for any Banach algebra. Consequently, Johnson defines an arbitrary Banach algebra \({\mathfrak A}\) to be amenable if and only if \(\mathcal{H}^1({\mathfrak A},E^\ast) = \{ 0 \}\) for every Banach \({\mathfrak A}\)-bimodule \(E\). This defines a new class of Banach algebras, but how significant is it? What does amenability “mean” for \({\mathfrak A}\)? In Section 6, Johnson studies the amenability of the Banach algebra \(\mathcal{K}(E)\) of all compact operators on a Banach space \(E\), and shows that \(\mathcal{K}(E)\) is indeed amenable if \(E = \ell^p\) with \(p \in (1,\infty)\) or if \(E\) is the space of all continuous functions on the unit circle. He then goes on and introduces – in Section 7 – the notion of strong amenability for \(C^*\)-algebras and shows that all postliminal \(C^*\)-algebras are strongly amenable. Altogether, the memoir consists of ten sections, and the one that has stimulated me most – and still continues to stimulate me – is the last one: just two and a half pages long, it has the title “Suggestions for further research”. It is here where the impact of Johnson’s memoir on mathematics becomes most apparent.

In 10.2, the question is raised if every \(C^*\)-algebra is amenable; in particular, it is asked if the Banach algebra \(\mathcal{B}({\mathfrak H})\) of all bounded linear operators on an infinite-dimensional Hilbert space \({\mathfrak H}\) is amenable. Both question were settled within little more than a decade after they were posed. In [“On the cohomology of operator algebras,” J. Funct. Anal. 28, 248–253 (1978; Zbl 0408.46042)], A. Connes showed that a separable, amenable \(C^*\)-algebra had to be nuclear. In fact, the separability hypothesis turned out to be unnecessary, so that every amenable \(C^*\)-algebra must be nuclear. As had been discovered only a few years earlier by S. Wassermann [“On tensor products of certain group \(C^*\)-algebras,” J. Funct. Anal. 23, 239–254 (1976; Zbl 0358.46040)], nuclear von Neumann algebras are subhomogeneous, so that, in particular, \(\mathcal{B}({\mathfrak H})\) cannot be amenable if \(\dim {\mathfrak H} = \infty\). Finally, U. Haagerup proved that amenability not only implies nuclearity, but is equivalent to it [“All nuclear \(C^*\)-algebras are amenable,” Invent. Math. 74, 305–319 (1983; Zbl 0529.46041)]: the amenable \(C^*\)-algebras are precisely the nuclear ones.

Given that Hilbert spaces are the “nicest” of Banach spaces, the fact that \(\mathcal{B}({\mathfrak H})\) is not amenable for \(\dim {\mathfrak H} = \infty\) strongly suggests that \(\mathcal{B}(E)\), the Banach algebra of all bounded linear operators on a Banach space \(E\), can never be amenable unless \(\dim E < \infty\). In 10.4, it is asked if this is true. Somewhat frustratingly, progress towards a solution to that problem turned out to be extremely elusive. In [“Relative amenability and the non-amenability of \(B(\ell^1)\),” J. Aust. Math. Soc.\ 80, No. 3, 317–333 (2006; Zbl 1111.46028)], C. J. Read showed that \(\mathcal{B}(\ell^1)\) is not amenable: this was the first proof of the non-amenability of \(\mathcal{B}(\ell^p)\) for any value of \(p\) other than \(2\). Only quite recently, it was shown that \(\mathcal{B}(\ell^p)\) fails to be amenable for every \(p \in [1,\infty]\) [V. Runde, “\(\mathcal{B}(\ell ^p)\) is never amenable,” J. Am. Math. Soc. 23, No. 4, 1175–1185 (2010; Zbl 1221.47143)]. To everyone’s – or at least my – surprise, there are, however, infinite-dimensional Banach spaces \(E\) such that \(\mathcal{B}(E)\) is amenable: the space \(E\) constructed in [S. A. Argyros and R. G. Haydon, “A hereditarily indecomposable \(\mathcal{L}_\infty\)-space that solves the scalar-plus-compact problem,” Acta Math. 206, No. 1, 1–54 (2011; Zbl 1223.46007)] is such an example. The question if \(\mathcal{H}^1(L^1(G),M(G)) = \{ 0 \}\) for all locally compact groups \(G\), is stated in 10.8. After a hiatus of almost three decades, Johnson returned to the problem in [B. E. Johnson, “The derivation problem for group algebras of connected locally compact groups,” J. Lond. Math. Soc., II. Ser. 63, No. 2, 441–452 (2001; Zbl 1012.43001)] and proved that the answer is “yes” if \(G\) is connected, which subsumes the results about matrix groups from the memoir. Eventually, V. Losert gave an affirmative answer for general \(G\) [“The derivation problem for group algebras,” Ann. Math. (2) 168, No. 1, 221–246 (2008; Zbl 1171.43004)]; see also [U. Bader, T. Gelander, and N. Monod, “A fixed point theorem for \(L^1\) spaces,” arXiv:1012.1488] for a very simple proof.

I would like to conclude this review with two observations and an obvious question they prompt: MathSciNet lists 215 citations of Johnson’s memoir, the Science Citation Index has none. What does this tell us about the useful(less?)ness of certain scholarly beancounting toys?

Reviewer: Volker Runde (Edmonton) (2012)

##### MSC:

46H99 | Topological algebras, normed rings and algebras, Banach algebras |

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |

22D15 | Group algebras of locally compact groups |

46H25 | Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) |

46L10 | General theory of von Neumann algebras |

43A20 | \(L^1\)-algebras on groups, semigroups, etc. |

43A07 | Means on groups, semigroups, etc.; amenable groups |