For any unital separable simple infinite-dimensional nuclear$C$^{∗}-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$ -absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear$C$^{∗}-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$ -absorbing.

We characterize mappings$S$_{$i$}and$T$_{$i$}, not necessarily linear, from sets $\mathcal {J}_{i}$ ,$i$=1,2, onto multiplicative subsets of function algebras, subject to the following conditions on the peripheral spectra of their products:$σ$_{$π$}($S$_{1}($a$)$S$_{2}($b$))⊂$σ$_{$π$}($T$_{1}($a$)$T$_{2}($b$)) and$σ$_{$π$}($S$_{1}($a$)$S$_{2}($b$))∩$σ$_{$π$}($T$_{1}($a$)$T$_{2}($b$))≠∅, $a\in \mathcal {J}_{1}$ , $b\in \mathcal {J}_{2}$ . As a direct consequence we obtain a large number of previous results about mappings subject to various spectral conditions.

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries (mn−−−√log(mn))−1 for m,n≥2. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with [0,π]. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.

We study the control systems governed by abstract Volterra equations without uniqueness in a Banach space. By using the technique of the theory of condensing maps and multivalued analysis tools, we obtain the existence result, investigate the topological structure of the solution set, and prove the invariance of a reachability set of the control system under nonlinear perturbations. Examples concerning fractional order differential equations and first order evolution equations with multiple delays are proposed to demonstrate the applications to approximate controllability results.

Let 1 p+. We show that the positive part of the closed unit ball of a non-commutative L p-space, as a metric space, is a complete Jordan-invariant for the underlying von Neumann algebra.

Let $ \gamma = \frac{1}{2}\left( {1 + \sqrt {5} } \right) $ denote the golden ratio. H. Davenport and W. M. Schmidt showed in 1969 that, for each non-quadratic irrational real number ξ, there exists a constant$c$> 0 with the property that, for arbitrarily large values of$X$, the inequalities $ \left| {{x_0}} \right| \leqslant X,\,\,\,\left| {{x_0}\xi - {x_1}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}}\,\,\,{\text{and}}\,\,\,\left| {{x_0}{\xi^2} - {x_2}} \right| \leqslant c{X^{{{{ - 1}} \left/ {\gamma } \right.}}} $ admit no non-zero solution $ \left( {{x_0},{x_1},{x_2}} \right) \in {\mathbb{Z}^3} $ . Their result is best possible in the sense that, conversely, there are countably many non-quadratic irrational real numbers ξ such that, for a larger value of c, the same inequalities admit a non-zero integer solution for each$X$≥ 1. Such$extremal$numbers are transcendental and their set is stable under the action of $ {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) $ on $ \mathbb{R}\backslash \mathbb{Q} $ by linear fractional transformations. In this paper, it is shown that there exist extremal numbers ξ for which the Lagrange constant ν(ξ) = liminf_{$q$→∞}$q$||$q$ξ|| is $ \frac{1}{3} $ , the largest possible value for a non-quadratic number, and that there is a natural bijection between the $ {\text{G}}{{\text{L}}_2}\left( \mathbb{Z} \right) $ -equivalence classes of such numbers and the non-trivial solutions of Markoff’s equation.

We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the stretch factor approaches 1 as the “radius” approaches infinity. In particular, the result implies that among all p-ellipses (or Lamé curves), the p-circle encloses the most first-quadrant lattice points as the radius approaches infinity, for 1<p<∞. The case p=2 corresponds to minimization of high eigenvalues of the Dirichlet Laplacian on rectangles, and so our work generalizes a result of Antunes and Freitas. Similarly, we generalize a Neumann eigenvalue maximization result of van den Berg, Bucur and Gittins. Further, Ariturk and Laugesen recently handled 0<p<1 by building on our results here. The case p=1 remains open, and is closely related to minimizing energy levels of harmonic oscillators: which right triangles in the first quadrant with two sides along the axes will enclose the most lattice points, as the area tends to infinity?

The analyticity of the Stokes semigroup with the Dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the Helmholtz decomposition. The proof is based on a blow-up argument. This is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years.

The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the HausdorffYoung inequality, Young's inequality, the HirschmanBeckner uncertainty principle, the DonohoStark uncertainty principle. We characterize the minimizers of the uncertainty principles and then we prove Hardy's uncertainty principle by using minimizers. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's <i></i>-lattices, modular tensor categories, etc.

In this paper we consider various aspects of generalized invertibility of the operator matrixacting on a Banach space$X$⊕$Y$.