On the Lack of Inverses to C-extensions Related to Property T Groups

Using ideas of S. Wassermann on non-exact C∗-algebras and property T groups, we show that one of his examples of non-invertible C∗-extensions is not semiinvertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a C∗-extension which is not even invertible up to homotopy. Introduction The Brown–Douglas–Fillmore theory of C-extensions, [2], works nicely for nuclear Calgebras because an extension of a nuclear C-algebra is always invertible in the extension semi-group. As a steadily growing number of examples show, this is not the case for general extensions, cf. [1],[9],[18],[17],[19],[6],[13], etc. In contrast, besides all its other merits, the E-theory of Connes and Higson, [3], provides a framework which incorporates arbitrary extensions of C-algebras, and in previous work we have clarified in which way this happens, cf. [10],[11]. Specifically, in the E-theory setting the notion of triviality of extensions must we weakened, at least so far as to consider an extension of C-algebras 0 // B // E q // A // 0 (0.1) to be trivial when it is asymptotically split, by which we mean that there is an asymptotic homomorphism, [3], φ = (φt)t∈[0,∞) : A → E such that q ◦ φt = idA for each t ∈ [0,∞). When the quotient C-algebra A is a suspension, i.e. is of the form C0(R) ⊗ D, this is the only change which is needed to ensure that E-theory becomes a complete analogue of the BDF theory for nuclear C-algebras. Specifically, when the quotient C-algebra is a suspension and the ideal is stable, every extension is semi-invertible, by which we mean that it is invertible in the sense corresponding to the weakened notion of triviality, i.e. one can add an extension to it so that the result is asymptotically split. Furthermore a given extension will represent 0 in E-theory if and only if it can be made asymptotically split by adding an asymptotically split extension to it. One purpose of the present paper is to show by example that this nice situation does not persist when the quotient C-algebra is not a suspension. We will show that an extension considered by S. Wassermann in [19], and shown by him to be non-invertible, is not semi-invertible either. By slightly modifying Wassermann’s example, we obtain also an extension which is not even invertible up to homotopy, giving us the first example of a C-algebra for which the semi-group of homotopy classes of extensions by a stable C-algebra, in casu the algebra of compact operators, is not a group. The conclusion is that the E-theory approach to C-extensions does not completely save us from the unpleasantness of extensions without inverses. But unlike the BDF theory, as shown in [11], in E-theory they can be eliminated at the cost of a single suspension. 2000 Mathematics Subject Classification. Primary 19K33; Secondary 46L06, 46L80, 20F99.