1. CMB Online first
|Correction to "Infinite Dimensional DeWitt Supergroups and Their Bodies"|
The Theorem below is a correction to Theorem 3.5 in the article entitled " Infinite Dimensional DeWitt Supergroups and Their Bodies" published in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283-288. Only part (iii) of that Theorem requires correction. The proof of Theorem 3.5 in the original article failed to separate the proof of (ii) from the proof of (iii). The proof of (ii) is complete once it is established that $ad_a$ is quasi-nilpotent for each $a$ since it immediately follows that $K$ is quasi-nilpotent. The proof of (iii) is not complete in the original article. The revision appears as the proof of (iii) of the revised Theorem below.
Keywords:super groups, body of super groups, Banach Lie groups
Categories:58B25, 17B65, 81R10, 57P99
2. CMB 2005 (vol 48 pp. 340)
|Short Geodesics of Unitaries in the $L^2$ Metric |
Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$
Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.
Categories:46L51, 58B10, 58B25