Asymptotic representations of the reduced C*-algebra of a free group: an example

Abstract

We give an example of a non-trivial asymptotic representation of the reduced C * -algebra of a free group. This example allows us to evaluate the asymptotic tensor C * -norm of some elements in tensor product C * -algebras and to show semi-invertibility of the non-invertible extension of C * r (F2) considered by Haagerup and Thorbjørnsen.

All extensions of $${C^*_r\left(\mathbb F_n\right)}$$ are semi-invertible

“…Since the example in [9] was based on groups with property T, which is a property free groups do not have, it became interesting to decide if there are examples of extensions of C * r (F n ) which are not semiinvertible. Very recently V. Manuilov has shown that the extension constructed by U. Haagerup and S. Thorbjørnsen is not only semi-invertible; it can be made asymptotically split by addition of a split extension, [6]. Manuilovs method actually works to obtain the same conclusion for any quasi-diagonal extension of C * r (F n ) and he conjectured that all extensions of C * r (F n ) (by compact operators) are semi-invertible.…”

“…There is then an invertible extensionπ : C * r (F n ) → Q such that π ⊕π is strongly homotopic to a split extension. Proof As in the work of Manuilov [6], the main ingredient in the proof is a homotopy of representations of F n which was constructed by J. Cuntz based on work of Pimsner and Voiculescu [10]. It is described on p. 187 of [4].…”

“…By Fell's absorption principle, which was also heavily used in [6], the representation R ⊗ λ (n) of F n is equivalent to a multiple of the regular representation and hence h R⊗λ (n) factors through C * r (F n ). Thus ω 1 : C * r (F n ) → Q is (unitarily equivalent to) the direct sum of ω and the split extension q • h R⊗λ (n) : C * r (F n ) → Q.…”

Semi-invertible extensions of C⁎-algebras

“…Complementing on the cases covered by the results in [16,15,14,26] and [21] we shall show in this paper that all extensions in Ext(A, B) are semi-invertible when a) A is the reduced group C * -algebra C * r (G) and the group G is an amalgamated free product G = G 1 * F G 2 with F finite, G 2 is amenable and G 1 abelian, and b) A is the amalgamated free product of C * -algebras, A = A 1 * D A 2 , when D is nuclear and all extensions of A i by B are semi-invertible, i = 1, 2.…”

“…The proof of a) above is an elaboration of the ideas developed in [14,26] and [21]. In particular, the argument uses the notion of strong homotopy of extensions and depends on Lemma 4.3 in [15].…”