Bounded Hochschild cohomology of Banach algebras with a matrixlike structure
Author:
Niels Grønbæk
Journal:
Trans. Amer. Math. Soc. 358 (2006), 26512662
MSC (2000):
Primary 46M20; Secondary 47B07, 16E40
Published electronically:
January 24, 2006
MathSciNet review:
2204050
Fulltext PDF Free Access
Abstract 
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Abstract: Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrixlike structure on any twosided ideal in . In this setup we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong Hunitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional space, and is the Banach algebra of approximable operators on , then for all .
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 Ariel Blanco, On the weak amenability of and its relation with the approximation property, J. Funct. Anal. 203 (2003), 126. MR 1996866 (2004g:47092)
 [B2]
 , Weak amenability of and the geometry of , J. London Math. Soc. (2) 66 (2002), 721740. MR 1934302 (2003i:46044)
 [CS]
 E. Christensen and A. M. Sinclair, On the vanishing of for certain algebras, Pacific J. Math. 137 (1989), 5563. MR 0983328 (90c:46093)
 [DGG]
 H. G. Dales, F. Ghahramani, and N. Grønbæk, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1998), 1954. MR 1489459 (99g:46064)
 [G1]
 N. Grønbæk, Morita equivalence for selfinduced Banach algebras, Houston J. Math. 22 (1996), 109140. MR 1434388 (98c:46090)
 [G2]
 , Factorization and weak amenability of algebras of approximable operators, to appear in Math. Proc. R. Ir. Acad.
 [G3]
 , Selfinduced Banach algebras, Contemp. Math. 263 (2004), 129143. MR 2097956 (2005g:46093)
 [G4]
 , Amenability of weighted convolution algebras on locally compact groups, Trans. Amer. Math. Soc. 319 (1990), 765775. MR 0962282 (90j:43003)
 [J]
 B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972). MR 0374934 (51:11130)
 [JKR]
 B.E. Johnson, R. V. Kadison and J. R. Ringrose, Cohomology of operator algebras. III. Reduction to normal cohomology, Bull. Soc. Math. France 100 (1972), 7396. MR 0318908 (47:7454)
 [L]
 J.L. Loday, Cyclic Homology, Grundlehren der mathematischen Wissenschaften, SpringerVerlag, Berlin, Heidelberg, 1992. MR 1217970 (94a:19004)
 [W1]
 M. Wodzicki, The long exact sequence in cyclic homology associated with an extension of algebras, C. R. Acad. Sci. Paris Sér. AB 306 (1988), 399403. MR 0934604 (89i:18012)
 [W2]
 , Vanishing of cyclic homology of stable algebras, C. R. Acad. Sci. Paris Sér. I 307 (1988), 329334. MR 0958792 (89j:46069)
 [W3]
 , Homological properties of rings of functionalanalytic type, Proc. Natl. Acad. Sci. USA 87 (1990), 49104911. MR 1058786 (91j:19011)
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Additional Information
Niels Grønbæk
Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK2100 Copenhagen Ø, Denmark
Email:
gronbaek@math.ku.dk
DOI:
http://dx.doi.org/10.1090/S0002994706039134
PII:
S 00029947(06)039134
Keywords:
Bounded Hochschild cohomology,
Hunital,
simplicially trivial
Received by editor(s):
December 2, 2003
Received by editor(s) in revised form:
August 3, 2004
Published electronically:
January 24, 2006
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
