Approximate semi-amenability of Banach algebras
- PDF / 3,576,056 Bytes
- 27 Pages / 439.37 x 666.142 pts Page_size
- 17 Downloads / 185 Views
Approximate semi‑amenability of Banach algebras F. Ghahramani1 · R. J. Loy2 Received: 2 October 2019 / Accepted: 6 January 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In recent work of the authors the notion of a derivation being approximately semiinner arose as a tool for investigating (approximate) amenability questions for Banach algebras. Here we investigate this property in its own right, together with the consequent one of approximately semi-amenability. Under certain hypotheses regarding approximate identities this new notion is the same as approximate amenability, but more generally it covers some important classes of algebras which are not approximately amenable, in particular Segal algebras on amenable SIN-groups. Keywords Approximately semi-inner · Amenable Banach algebra · Approximate identity · Segal algebra
1 Introduction The concept of amenability for a Banach algebra, introduced by Johnson in [20], has proved to be of enormous importance in Banach algebra theory. In [12], and subsequently in [17], several modifications of this notion were introduced, in particular that of approximate amenability; and much work has been done in the last 10 years or so, [6, 7, 9, 10, 14–16], for example. See also [33] for a recent survey. More recently, the present authors [13] investigated the situation for tensor products, and, en passant, introduced the notions to be considered here. Let A be an algebra, X an A-bimodule. A derivation is a linear map D ∶ A → X such that Communicated by Anthony To-Ming Lau. * F. Ghahramani [email protected] R. J. Loy [email protected] 1
Department of Mathematics, University of Manitoba, Winnipeg R3T 2N2, Canada
2
Mathematical Sciences Institute, Australian National University, Hanna Neumann Building 145, Science Road, Canberra, ACT 2601, Australia
13
Vol.:(0123456789)
F. Ghahramani, R. J. Loy
D(ab) = a ⋅ D(b) + D(a) ⋅ b
(a, b ∈ A).
For 𝜉 ∈ X , set ad𝜉 ∶ a ↦ a ⋅ 𝜉 − 𝜉 ⋅ a, A → X . Then ad𝜉 is a derivation; these are the inner derivations. Let A be a Banach algebra, X be a Banach A-bimodule. A continuous derivation D ∶ A → X is approximately inner if there is a net (𝜉i ) in X such that
D(a) = lim(a ⋅ 𝜉i − 𝜉i ⋅ a) (a ∈ A), i
so that D = limi adxi in the strong-operator topology of B(A, X). Definition 1 [12, 17] Let A be a Banach algebra. Then A is approximately amenable (resp. approximately contractible) if, for each Banach A-bimodule X, every continuous derivation D ∶ A → X ∗ (resp. D ∶ A → X ), is approximately inner. If it is always possible to choose the approximating net (ad𝜉i ) to be bounded (with the bound dependent only on D) then A is boundedly approximately amenable (resp. boundedly approximately contractible). Of course A is amenable (resp. contractible) if every continuous derivation D ∶ A → X ∗ (resp. D ∶ A → X ), is inner, for every Banach A-bimodule X. During final preparations of this paper, the article [22] appeared on the arXiv. The authors there refer to [13] for the introduction of the
Data Loading...
