Some consequences of von Neumann algebra uniqueness (Q1939358)

The authors derive some consequences of the von Neumann algebra uniqueness theorem of \textit{A. Ciuperca} et al. [Adv. Math. 240, 325--345 (2013; Zbl 1292.46036)]. In particular: (1) Let \(\pi_1\), \(\pi_2\) be representations of a separable simple nuclear \(C^*\)-algebra \(A\) on a separable Hilbert space. It is shown that \(\pi_1\) and \(\pi_2\) are algebraically equivalent iff there is an automorphism \(\alpha\) of \(A\) such that \(\pi_1\circ\alpha\) and \(\pi_2\) are quasi-equivalent. This answers a question raised by \textit{H. Futamura} et al. [J. Funct. Anal. 197, No. 2, 560--575 (2003; Zbl 1034.46057)]. (2) Recall that a topological group \(G\) has the Kirchberg property if every homomorphism from \(\mathbb F_\infty\times\mathbb F_\infty\) to \(G\) can be approximated, in the pointwise topology, by homomorphisms with precompact image. For a \(C^*\)-algebra \(A\), let \(M(A\otimes\mathbb K)\) denote the stable multiplier algebra of \(A\), and let \(U(A)\) denote the unitary group of \(A\). It is shown that the following statements are equivalent: {\parindent=4mm \begin{itemize}\item[{\(\bullet\)}] The Connes embedding problem has an affirmative answer. \item[{\(\bullet\)}] For every unital separable nuclear \(C^*\)-algebra \(A\), \(U(M(A\otimes\mathbb K))\) has the Kirchberg property. \item[{\(\bullet\)}] There exists a unital separable nuclear \(C^*\)-algebra \(A\) such that \(U(M(A\otimes\mathbb K))\) has the Kirchberg property. \end{itemize}}