Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Author:
Niels Grønbæk

Journal:
Trans. Amer. Math. Soc. **358** (2006), 2651-2662

MSC (2000):
Primary 46M20; Secondary 47B07, 16E40

Published electronically:
January 24, 2006

MathSciNet review:
2204050

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality 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 .

**[B1]**A. Blanco,*On the weak amenability of 𝒜(𝒳) and its relation with the approximation property*, J. Funct. Anal.**203**(2003), no. 1, 1–26. MR**1996866**, 10.1016/S0022-1236(02)00050-2**[B2]**A. Blanco,*Weak amenability of 𝒜(ℰ) and the geometry of ℰ*, J. London Math. Soc. (2)**66**(2002), no. 3, 721–740. MR**1934302**, 10.1112/S0024610702003642**[CS]**Erik Christensen and Allan M. Sinclair,*On the vanishing of 𝐻ⁿ(𝒜,𝒜*) for certain 𝒞*-algebras*, Pacific J. Math.**137**(1989), no. 1, 55–63. MR**983328****[DGG]**H. G. Dales, F. Ghahramani, and N. Grønbæk,*Derivations into iterated duals of Banach algebras*, Studia Math.**128**(1998), no. 1, 19–54. MR**1489459****[G1]**Niels Grønbæk,*Morita equivalence for self-induced Banach algebras*, Houston J. Math.**22**(1996), no. 1, 109–140. MR**1434388****[G2]**-,*Factorization and weak amenability of algebras of approximable operators*, to appear in Math. Proc. R. Ir. Acad.**[G3]**Niels Grønbæk,*Self-induced Banach algebras*, Banach algebras and their applications, Contemp. Math., vol. 363, Amer. Math. Soc., Providence, RI, 2004, pp. 129–143. MR**2097956**, 10.1090/conm/363/06647**[G4]**Niels Grønbæk,*Amenability of weighted convolution algebras on locally compact groups*, Trans. Amer. Math. Soc.**319**(1990), no. 2, 765–775. MR**962282**, 10.1090/S0002-9947-1990-0962282-5**[J]**Barry Edward Johnson,*Cohomology in Banach algebras*, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 127. MR**0374934****[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), 73–96. MR**0318908****[L]**Jean-Louis Loday,*Cyclic homology*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by María O. Ronco. MR**1217970****[W1]**Mariusz Wodzicki,*The long exact sequence in cyclic homology associated with an extension of algebras*, C. R. Acad. Sci. Paris Sér. I Math.**306**(1988), no. 9, 399–403 (English, with French summary). MR**934604****[W2]**Mariusz Wodzicki,*Vanishing of cyclic homology of stable 𝐶*-algebras*, C. R. Acad. Sci. Paris Sér. I Math.**307**(1988), no. 7, 329–334 (English, with French summary). MR**958792****[W3]**Mariusz Wodzicki,*Homological properties of rings of functional-analytic type*, Proc. Nat. Acad. Sci. U.S.A.**87**(1990), no. 13, 4910–4911. MR**1058786**, 10.1073/pnas.87.13.4910

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
46M20,
47B07,
16E40

Retrieve articles in all journals with MSC (2000): 46M20, 47B07, 16E40

Additional Information

**Niels Grønbæk**

Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark

Email:
gronbaek@math.ku.dk

DOI:
http://dx.doi.org/10.1090/S0002-9947-06-03913-4

Keywords:
Bounded Hochschild cohomology,
H-unital,
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.