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We continue our investigation, from cite{dh}, of the ring-theoretic infiniteness properties of ultrapowers of Banach algebras, studying in this paper the notion of being purely infinite. It is well known that a $C^*$-algebra is purely infinite if and only if any of its ultrapower is. We find examples of Banach algebras, as algebras of operators on Banach spaces, which do have purely infinite ultrapowers. Our main contribution is the construction of a Cuntz-like Banach $*$-algebra which is purely infinite, but does not have purely infinite ultrapowers. Our proof of being purely infinite is combinatorial, but direct, and so differs from the proof for the Cuntz algebra. We use an indirect method (and not directly computing norm estimates) to show that this algebra does not have purely infinite ultrapowers.
Let $G$ be a Hausdorff, etale groupoid that is minimal and topologically principal. We show that $C^*_r(G)$ is purely infinite simple if and only if all the nonzero positive elements of $C_0(G^0)$ are infinite in $C_r^*(G)$. If $G$ is a Hausdorff, am
From a suitable groupoid G, we show how to construct an amenable principal groupoid whose C*-algebra is a Kirchberg algebra which is KK-equivalent to C*(G). Using this construction, we show by example that many UCT Kirchberg algebras can be realised
[REVISED VERSION] The aim of this paper is to state a sharp version of the Konig supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz Joh
We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano-Poincare-Miranda theorem to infinite dimensional Banach spaces. We also establish a result regarding the existence
In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint limiting distr