부산대학교 도서관

Let $A$ and $B$ be two subalgebras of $\roman{B}(H)$, the space of bounded operators on a Hilbert space $H$. Denote by $(A)_1$ the unit ball of $A$ and by $(B)_1$ the unit ball of $B$ and let $d_{\roman{B}(H)}$ be the Hausdorff distance. R.~V. Kadison and D. Kastler [Amer. J. Math. {\bf 94} (1972), 38--54; MR0296713 (45 \#5772)] defined a distance $d_{H}$ between $A$ and $B$, by setting $d_{H}(A,B) = d_{\roman{B}(H)}( (A)_1 , (B)_1 )$. The Kadison-Kastler distance $d$ between two $C^*$-algebras $A$ and $B$ is defined by taking the infimum of $d_{H}(\pi(A),\rho(B))$ over all faithful unital $*$-representations $\pi \: A \to \roman{B}(H)$ and $\rho \: B \to \roman{B}(H)$. They conjectured that certain operator algebras should be stable for the distance $d$; close enough operator algebras have to be isomorphic. The notion of stability can be formulated in the following ways. \par A von Neumann algebra is said to be {\it strongly Kadison-Kastler stable} if for every $\epsilon >0$, there exists $\delta >0$ such that given any faithful unital normal representation $M \subset \roman{B}(H)$ and any von Neumann algebra $N \subset \roman{B}(H)$ containing the identity of $H$ and satisfying $d(M,N) < \delta$, there exists a unitary $u \in H$ such that $\|u-\roman{id}\| < \epsilon$ and $uMu^*=N$. \par A von Neumann algebra is said to be {\it Kadison-Kastler stable} if there exists $\delta >0$ such that given any faithful unital normal representation $M \subset \roman{B}(H)$ and any von Neumann algebra $N \subset \roman{B}(H)$ containing the identity of $H$ and satisfying $d(M,N) < \delta$, there exists a unitary $u \in H$ such that $uMu^*=N$. \par A von Neumann algebra is said to be {\it weakly Kadison-Kastler stable} if there exists $\delta >0$ such that given any faithful unital normal representation $M \subset \roman{B}(H)$ and any von Neumann algebra $N \subset \roman{B}(H)$ containing the identity of $H$ and satisfying $d(M,N) < \delta$, then $M$ and $N$ are $*$-isomorphic. \par The conjecture of Kadison and Kastler then states that von Neumann algebras are strongly stable and separable $C^*$-algebras are stable. E. Christensen et al. [Acta Math. {\bf 208} (2012), no.~1, 93--150; 2910797 ] proved that the conjecture holds true for separable nuclear $C^*$-algebras. Counter-examples were found in the non-separable setting [M.~D. Choi and E. Christensen, Bull. London Math. Soc. {\bf 15} (1983), no.~6, 604--610; MR0720750 (86a:46072)]. On the von Neumann algebra level, the conjecture was established in the injective case [see, for example, E. Christensen, Invent. Math. {\bf 43} (1977), no.~1, 1--13; MR0512367 (58 \#23628a)]. \par In this article, the authors give the first example of non-injective von Neumann algebras being strongly Kadison-Kastler stable. A crucial ingredient of the proof is the vanishing of certain bounded cohomology groups.