International Journal of Mathematics

Let (M,g,Q) be a simply connected, complete, quaternionic Kähler manifold without flat de Rham factor. Then any 1-parameter group of transformations of M which preserve the quaternionic structure Q preserves also the metric g. Moreover, if (M,g) is irreducible then the quaternionic Kähler metric g on (M,Q) is unique up to a homothety.

A sub-Riemannian manifold is a smooth manifold which carries a distribution equipped with a metric. We study the holonomy and the horizontal holonomy (i.e. holonomy spanned by loops everywhere tangent to the distribution) of sub-Riemannian manifolds of contact type relative to an adapted connection. In particular, we obtain an Ambrose–Singer type theorem for the horizontal holonomy and we classify the holonomy irreducible sub-Riemannian symmetric spaces (i.e. homogeneous sub-Riemannian manifolds admitting an involutive isometry whose restriction to the distribution is a central symmetry).

This paper provides the optimal exponential decay rate of the lower bound for the first positive eigenvalue of the Laplacian operator on a compact Riemannian manifold with a negative lower bound on the Ricci curvature and with large diameter. For manifolds with boundary, suitable convexity conditions are assumed.

We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.

We compute the real rank and the stable rank of full group C*-algebras. Main result is (i) rr(C*(F_n)) = ∞, (ii) sr(C*(G₁ * G₂)) = ∞(|G₁| ≥ 2, |G₂| ≥ 2 and |G₁| + |G₂| ≥ 5), (iii) sr(C*(G₁ * G₂)) = 1(|G₁| = |G₂| = 2), where F_n is the free group with n generators, G₁ and G₂ are finite groups and |G| means the order of the group G.

In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.

We prove that two subfactors N ⊂ M and P ⊂ Q arising from a non-degenerate commuting square have 3-dimensional topological quantum field theories (based on triangulations) complex conjugate to each other. Applying the asymptotic inclusion construction to each subfactor, we have new subfactors M ∨ (M′ ∩ M_∞) ⊂ M_∞ and Q ∨ (Q′ ∩ Q_∞) ⊂ Q_∞, then we also prove that the tensor categories of the M_∞-M_∞ bimodules and the Q_∞-Q_∞ bimodules are isomorphic to each other, if the two fusion graphs are connected. These results are based on our previous work and give a finer answer to a question raised by V. F. R. Jones in June, 1995.