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Approximate and pseudo-amenability of various classes of Banach algebras. (English) Zbl 1179.46040
A Banach algebra $$A$$ is called approximately amenable if, for every Banach $$A$$-bimodule $$E$$ and every bounded derivation $$D: A \to E^*$$, there is a – not necessarily bounded – net $$(\varphi_\alpha )_\alpha$$ in $$E^*$$ such that $$Da= \lim_\alpha (a\cdot \varphi_\alpha- \varphi_\alpha \cdot a)$$ for each $$a\in A$$. (If we require the net $$(\varphi_\alpha)_\alpha$$ to be bounded, we obtain the usual Banach algebraic amenability in the sense of B. E. Johnson [“Cohomology in Banach algebras”, Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)].)
Let $$\widehat{\otimes}$$ denote the projective tensor product. We call a Banach algebra $$A$$ pseudo-amenable if there is a net $$(d_\alpha)_\alpha$$ in $$A \widehat{\otimes} A$$ such that
$a\cdot d_\alpha- d_\alpha\cdot a\to 0 \qquad (a\in A)$
and
$a m(d_\alpha)\to a \qquad (a \in A),$
where $$m: A\widehat{\otimes} A\to A$$ is the map induced by multiplication in $$A$$. (If $$(d_\alpha )_\alpha$$ is bounded, we obtain again B. E. Johnson’s concept of amenability [“Approximate diagonals and cohomology of certain annihilator Banach algebras”, Am. J. Math. 94, 685–698 (1972; Zbl 0246.46040)].)
These and other generalized notions of amenability for Banach algebras were introduced by the second and third named author of the paper under review and R. J. Loy in a series of papers that started with [“Generalized notions of amenability”, J. Funct. Anal. 208, No. 1, 229–260 (2004; Zbl 1045.46029)].
The present paper is yet another installment in that series. The authors study various generalized notions of amenability, relate them to each other, and investigate examples, such as Fourier algebras, $$\ell^1$$-algebras of semigroups, Segal algebras on locally compact groups, and the algebras $$\text{PF}_p(\Gamma)$$ of $$p$$-pseudofunctions for discrete groups $$\Gamma$$. Among the noteworthy results of the paper are that the Fourier algebra of the free group in two generators is not (operator) approximately amenable and that for $$\text{PF}_p(\Gamma)$$ amenability, pseudo-amenability, and approximate amenability are all equivalent to the amenability of $$\Gamma$$.

##### MSC:
 46H05 General theory of topological algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 43A20 $$L^1$$-algebras on groups, semigroups, etc. 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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##### References:
  Bade, W.G.; Curtis, P.C.; Dales, H.G., Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London math. soc. (3), 55, 2, 359-377, (1987) · Zbl 0634.46042  Bunce, J.W., Finite operators and amenable $$C^*$$-algebras, Proc. amer. math. soc., 56, 145-151, (1976) · Zbl 0343.46039  Burnham, J.T., Closed ideals in subalgebras of Banach algebras. I, Proc. amer. math. soc., 32, 551-555, (1972) · Zbl 0234.46050  Dales, H.G.; Ghahramani, F.; Grønbæk, N., Derivations into iterated duals of Banach algebras, Studia math., 128, 1, 19-54, (1998) · Zbl 0903.46045  Dales, H.G.; Loy, R.J.; Zhang, Y., Approximate amenability for Banach sequence algebras, Studia math., 177, 1, 81-96, (2006) · Zbl 1117.46030  De Cannière, J.; Haagerup, U., Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups, Amer. J. math., 107, 2, 455-500, (1985) · Zbl 0577.43002  Duncan, J.; Namioka, I., Amenability of inverse semigroups and their semigroup algebras, Proc. roy. soc. Edinburgh sect. A, 80, 3-4, 309-321, (1978) · Zbl 0393.22004  Effros, E.G.; Ruan, Z.-J., On approximation properties for operator spaces, Internat. J. math., 1, 2, 163-187, (1990) · Zbl 0747.46014  Ghahramani, F.; Lau, A.T.M., Weak amenability of certain classes of Banach algebras without bounded approximate identities, Math. proc. Cambridge philos. soc., 133, 2, 357-371, (2002) · Zbl 1010.46048  Ghahramani, F.; Lau, A.T.-M., Approximate weak amenability, derivations and Arens regularity of Segal algebras, Studia math., 169, 2, 189-205, (2005) · Zbl 1084.43002  Ghahramani, F.; Loy, R.J., Generalized notions of amenability, J. funct. anal., 208, 1, 229-260, (2004) · Zbl 1045.46029  Ghahramani, F.; Loy, R.J.; Zhang, Y., Generalized notions of amenability, II, J. funct. anal., 254, 7, 1776-1810, (2008) · Zbl 1146.46023  Ghahramani, F.; Stokke, R., Approximate and pseudo-amenability of the Fourier algebra, Indiana univ. math. J., 56, 2, 909-930, (2007) · Zbl 1148.43002  Ghahramani, F.; Zhang, Y., Pseudo-amenable and pseudo-contractible Banach algebras, Math. proc. Cambridge philos. soc., 142, 111-123, (2007) · Zbl 1118.46046  Gourdeau, F., Amenability of Lipschitz algebras, Math. proc. Cambridge philos. soc., 112, 3, 581-588, (1992) · Zbl 0782.46043  Herz, C., Harmonic synthesis for subgroups, Ann. inst. Fourier (Grenoble), 23, 3, 91-123, (1973) · Zbl 0257.43007  Johnson, B.E., Cohomology in Banach algebras, (1972), Amer. Math. Soc. Providence, RI · Zbl 0246.46040  Johnson, B.E., Permanent weak amenability of group algebras of free groups, Bull. London math. soc., 31, 5, 569-573, (1999) · Zbl 0951.46041  Kelley, J.L., General topology, (1955), D. Van Nostrand Company, Inc. Toronto · Zbl 0066.16604  Kotzmann, E.; Rindler, H., Segal algebras on non-abelian groups, Trans. amer. math. soc., 237, 271-281, (1978) · Zbl 0363.43004  Lashkarizadeh Bami, M.; Samea, H., Approximate amenability of certain semigroup algebras, Semigroup forum, 71, 2, 312-322, (2005) · Zbl 1086.43002  Leinert, M., A contribution to Segal algebras, Manuscripta math., 10, 297-306, (1973) · Zbl 0265.46048  Losert, V., The derivation problem for group algebras, Ann. of math. (2), 168, 1, 221-246, (2008) · Zbl 1171.43004  Ozawa, N., A note on non-amenability of $$\mathcal{B}(\ell^p)$$ for $$p = 1, 2$$, Internat. J. math., 15, 6, 557-565, (2004) · Zbl 1056.46046  Pirkovskii, A.Yu., Approximate characterizations of projectivity and injectivity for Banach modules, Math. proc. Cambridge philos. soc., 143, 2, 375-385, (2007) · Zbl 1135.46028  Ramagge, J.; Robertson, G.; Steger, T., A Haagerup inequality for $$\widetilde{A}_1 \times \widetilde{A}_1$$ and $$\widetilde{A}_2$$ buildings, Geom. funct. anal., 8, 4, 702-731, (1998) · Zbl 0906.43009  Reiter, H., $$L^1$$-algebras and Segal algebras, Lecture notes in math., vol. 231, (1971), Springer-Verlag Berlin · Zbl 0219.43003  Reiter, H.; Stegeman, J.D., Classical harmonic analysis and locally compact groups, London math. soc. monogr., vol. 22, (2000), Oxford Univ. Press New York · Zbl 0965.43001  Samei, E.; Stokke, R.; Spronk, N., Biflatness and pseudo-amenability of Segal algebras, (2008), preprint, see · Zbl 1201.43002  Willis, G.A., Approximate units in finite codimensional ideals of group algebras, J. London math. soc. (2), 26, 1, 143-154, (1982) · Zbl 0465.22004
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