Let A be a C*-algebra, L a closed left ideal of A and p the closed projection related to L. We show that for an xp in A**p ( A**/L**) if pAxp pAp and px*xp pAp then xp Ap ( A/L). The proof goes by interpreting elements of A**p (resp. Ap) as admissible (resp. continuous admissible) vector sections over the base space F(p) = { A* : 0, (p) = 1} in the notions developed by Diximier and Douady, Fell, and Tomita. We consider that our results complement both Kadison function representation and Takesaki duality theorem.

We prove that every lattice in a product of higher-rank simple Lie groups or higher-rank simple algebraic groups over local fields has Vincent Lafforgue’s strong property (T). Over non-Archimedean local fields, we also prove that they have strong Banach property (T) with respect to all Banach spaces with non-trivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-co-compact lattices, such as SLn(Z) for n⩾3. To do so, we introduce a stronger form of strong property (T) which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher-rank groups have this property and that this property passes to undistorted lattices. Full Text (PDF format)

Let$f$be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, $${f \in \mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$ , $${\# A< \infty}$$ . J. Nuttall has put forward the important relation between the$maximal domain$of$f$where the function has a single-valued branch and the$domain of convergence$of the diagonal Padé approximants for$f$. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of$f$in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function$f$single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has$minimal logarithmic capacity$among all other systems converting the function$f$to a single-valued branch. Thus the domain of convergence corresponds to the$maximal$(in the sense of$minimal$boundary) domain of single-valued holomorphy for the analytic function $${f\in\mathcal{A}(\bar{\mathbb{C}} \setminus A)}$$ . The complete proof of Nuttall’s conjecture (even in a more general setting where the set$A$has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive$strong asymptotics$for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that$A$is a finite set of branch points of$f$which have the$algebro-logarithmic character$and which are placed in a$generic position$. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.

Let A 1, A 2 be standard operator algebras on complex Banach spaces X 1, X 2, respectively. For k 2, let (i 1,, i m) be a sequence with terms chosen from {1,, k}, and define the generalized Jordan product T 1 T k= T i 1 T i m+ T i m T i 1 on elements in A i. This includes the usual Jordan product A 1 A 2= A 1 A 2+ A 2 A 1, and the triple {A 1, A 2, A 3}= A 1 A 2 A 3+ A 3 A 2 A 1. Assume that at least one of the terms in (i 1,, i m) appears exactly once. Let a map : A 1 A 2 satisfy that ( (A 1) (A k))= (A 1 A k) whenever any one of A 1,, A k has rank at most one. It is shown in this paper that if the range of contains all operators of rank at most three, then must be a Jordan isomorphism multiplied by an m th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained.

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