Bounded Hochschild cohomology of Banach algebras with a matrix-like structure
HTML articles powered by AMS MathViewer
- by Niels Grønbæk PDF
- Trans. Amer. Math. Soc. 358 (2006), 2651-2662 Request permission
Abstract:
Let $\mathfrak {B}$ be a unital Banach algebra. A projection in $\mathfrak {B}$ which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal $\mathfrak {A}$ in $\mathfrak {B}$. In this set-up we prove a theorem to the effect that the bounded cohomology $\mathcal {H}^{n}(\mathfrak {A}, \mathfrak {A}^{*})$ vanishes for all $n\geq 1$. The hypotheses of this theorem involve (i) strong H-unitality of $\mathfrak {A}$, (ii) a growth condition on diagonal matrices in $\mathfrak {A}$, and (iii) an extension of $\mathfrak {A}$ in $\mathfrak {B}$ by an amenable Banach algebra. As a corollary we show that if $X$ is an infinite dimensional Banach space with the bounded approximation property, $L_{1}(\mu ,\Omega )$ is an infinite dimensional $L_{1}$-space, and $\mathfrak {A}$ is the Banach algebra of approximable operators on $L_{p}(X,\mu ,\Omega )\;(1\leq p<\infty )$, then $\mathcal {H}^{n}(\mathfrak {A},\mathfrak {A}^{*})=(0)$ for all $n\geq 0$.References
- A. Blanco, On the weak amenability of $\scr A(X)$ and its relation with the approximation property, J. Funct. Anal. 203 (2003), no. 1, 1–26. MR 1996866, DOI 10.1016/S0022-1236(02)00050-2
- A. Blanco, Weak amenability of $\scr A(E)$ and the geometry of $E$, J. London Math. Soc. (2) 66 (2002), no. 3, 721–740. MR 1934302, DOI 10.1112/S0024610702003642
- Erik Christensen and Allan M. Sinclair, On the vanishing of $H^n({\scr A},{\scr A}^*)$ for certain $C^*$-algebras, Pacific J. Math. 137 (1989), no. 1, 55–63. MR 983328
- 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
- Niels Grønbæk, Morita equivalence for self-induced Banach algebras, Houston J. Math. 22 (1996), no. 1, 109–140. MR 1434388
- —, Factorization and weak amenability of algebras of approximable operators, to appear in Math. Proc. R. Ir. Acad.
- 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, DOI 10.1090/conm/363/06647
- Niels Grønbæk, Amenability of weighted convolution algebras on locally compact groups, Trans. Amer. Math. Soc. 319 (1990), no. 2, 765–775. MR 962282, DOI 10.1090/S0002-9947-1990-0962282-5
- Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
- 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 318908
- 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, DOI 10.1007/978-3-662-21739-9
- 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
- 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
- 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, DOI 10.1073/pnas.87.13.4910
Additional Information
- Niels Grønbæk
- Affiliation: Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark
- Email: gronbaek@math.ku.dk
- Received by editor(s): December 2, 2003
- Received by editor(s) in revised form: August 3, 2004
- Published electronically: January 24, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 2651-2662
- MSC (2000): Primary 46M20; Secondary 47B07, 16E40
- DOI: https://doi.org/10.1090/S0002-9947-06-03913-4
- MathSciNet review: 2204050