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 semi-invertible. 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.

Keywords:

$C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopy $C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopy

MSC Classifications:

19K33, 46L06, 46L80, 20F99 show english descriptions EXT and $K$-homology [See also 55N22]
Tensor products of $C^*$-algebras
$K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
None of the above, but in this section 19K33 - EXT and $K$-homology [See also 55N22]
46L06 - Tensor products of $C^*$-algebras
46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
20F99 - None of the above, but in this section

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