Amenability and uniqueness for groupoids associated with inverse semigroups

We investigate recent uniqueness theorems for reduced \(C^*\)-algebras of Hausdorff étale groupoids in the context of inverse semigroups. In many cases the distinguished subalgebra is closely related to the structure of the inverse semigroup. In order to apply our results to full \(C^*\)-algebras, we also investigate amenability. More specifically, we obtain conditions that guarantee amenability of the universal groupoid for certain classes of inverse semigroups. These conditions also imply the existence of a conditional expectation onto a canonical subalgebra.     

Zappa–Szép product groupoids and $$C^*$$-blends

We study the external and internal Zappa–Szép product of topological groupoids. We show that under natural continuity assumptions the Zappa–Szép product groupoid is étale if and only if the individual groupoids are étale. In our main result we show that the \(C^*\)-algebra of a locally compact Hausdorff étale Zappa–Szép product groupoid is a \(C^*\)-blend, in the sense of Exel, of the individual groupoid \(C^*\)-algebras. We finish with some examples, including groupoids built from \(*\)-commuting endomorphisms, and skew product groupoids.     

Quasi hyperrigidity and weak peak points for non-commutative operator systems

In this article, we introduce the notions of weak boundary representation, quasi hyperrigidity and weak peak points in the non-commutative setting for operator systems in \(C^*\)-algebras. An analogue of Saskin’s theorem relating quasi hyperrigidity and weak Choquet boundary for particular classes of \(C^*\)-algebras is proved. We also show that, if an irreducible representation is a weak boundary representation and weak peak then it is a boundary representation. Several examples are provided to illustrate these notions.     

Hilbert $$C^*$$-modules as a subcategory of operator systems and injectivity

In this paper, we study a category whose objects are Hilbert \(C^*\)-modules and whose morphisms are completely semi-\(\phi \)-maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi-\(\phi \)-maps on Hilbert \(C^*\)-modules, leading to an analog of the Arveson’s extension theorem for completely semi-\(\phi \)-maps (in contrast with \(\phi \)-maps). This theorem together with previous results suggest that the completely semi-\(\phi \)-maps are proper generalizations of the completely positive maps.     

Polynomials in operator space theory: matrix ordering and algebraic aspects

We extend the \(\lambda \)-theory of operator spaces given in Defant and Wiesner (J. Funct. Anal. 266(9): 5493–5525, 2014), that generalizes the notion of the projective, Haagerup and Schur tensor norm for operator spaces to matrix ordered spaces and Banach \(*\)-algebras. Given matrix regular operator spaces and operator systems, we introduce cones related to \(\lambda \) for the algebraic operator space tensor product that respect the matricial structure of matrix regular operator spaces and operator systems, respectively. The ideal structure of \(\lambda \)-tensor product of \(C^*\)-algebras has also been discussed.     

A Gelfand duality for compact pospaces

It is well known that the category of compact Hausdorff spaces is dually equivalent to the category of commutative \(C^\star \)-algebras. More generally, this duality can be seen as a part of a square of dualities and equivalences between compact Hausdorff spaces, \(C^\star \)-algebras, compact regular frames and de Vries algebras. Three of these equivalences have been extended to equivalences between compact pospaces, stably compact frames and proximity frames, the fourth part of what will be a second square being lacking. We propose the category of bounded Archimedean \(\ell \)-semi-algebras to complete the second square of equivalences and to extend the category of \(C^\star \)-algebras.     

Pair frames in Hilbert $$\varvec{C^*}$$-modules

In this paper, we introduce pair frames in Hilbert \(C^*\)-modules and show that they share many useful properties with their corresponding notions in Hilbert spaces. We also obtain the necessary and sufficient conditions for a standard Bessel sequence to construct a pair frame and get the necessary and sufficient conditions for a Hilbert \(C^*\)-module to admit a pair frame with a symbol and two standard Bessel sequences. Moreover by generalizing some of the results obtained for Bessel multipliers in Hilbert \(C^*\)-modules to pair frames and considering the stability of pair frames under invertible operators, we construct new pair frames and show that pair frames are stable under small perturbations.     

Unitary Equivalence of Composition C$$^*$$-Algebras on the Hardy and Weighted Bergman Spaces

Let \(\varphi \) be an arbitrary linear-fractional self-map of the unit disk \({\mathbb {D}}\) and consider the composition operator \(C_{-1, \varphi }\) and the Toeplitz operator \(T_{-1,z}\) on the Hardy space \(H^2\) and the corresponding operators \(C_{\alpha , \varphi }\) and \(T_{\alpha , z}\) on the weighted Bergman spaces \(A^2_{\alpha }\) for \(\alpha >-1\). We prove that the unital C\(^*\)-algebra \(C^*(T_{\alpha , z}, C_{\alpha , \varphi })\) generated by \(T_{\alpha , z}\) and \(C_{\alpha , \varphi }\) is unitarily equivalent to \(C^*(T_{-1, z}, C_{-1, \varphi }),\) which extends a known result for automorphism-induced composition operators. For maps \(\varphi \) that are not automorphisms of \({\mathbb {D}}\), we show that \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })\) is unitarily equivalent to \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})\), where \({\mathcal {K}}_{\alpha }\) and \({\mathcal {K}}_{-1}\) denote the ideals of compact operators on \(A^2_{\alpha }\) and \(H^2\), respectively, and apply existing structure theorems for \(C^*(C_{-1, \varphi }, {\mathcal {K}}_{-1})/{\mathcal {K}}_{-1}\) to describe the structure of \(C^*(C_{\alpha , \varphi }, {\mathcal {K}}_{\alpha })/\mathcal {K_{\alpha }}\), up to isomorphism. We also establish a unitary equivalence between related weighted composition operators induced by maps \(\varphi \) that fix a point on the unit circle.     

Orthogonality in $$C^{*}$$-algebras

The aim in this paper is to study algebraic orthogonality between positive elements of a \(C^{*}\)-algebra in the context of geometric orthogonality. It has been shown that the algebraic orthogonality in certain classes of \(C^{*}\)-algebras is equivalent to geometric orthogonality when supported with some order-theoretic conditions. Further more, algebraic orthogonality between positive elements in a \(C^{*}\)-algebra is also characterized in terms of positive linear functionals.     

The Tracial Topological Rank of Extensions of $$C^*$$-Algebras

Let \( 0 \xrightarrow {}J \xrightarrow { } A \xrightarrow { }B \xrightarrow {} 0\) be an extension of \(C^*\)-algebras. Suppose that both J and B have tracial rank no more than one. It is shown that A has tracial topological rank no more than one whenever it is a quasidiagonal extension, and A has property \((P_1)\) if the extension is tracially quasidiagonal.     

On just infinite periodic locally soluble groups

We construct an uncountable family of periodic locally soluble groups which are hereditarily just infinite. We also show that the associated full \(C^*\)-algebra \(C^*(G)\) is just infinite for many groups G in this family.     

Orthogonal-gradings on $$H^*$$-algebras

We study set-gradings on proper \(H^*\)-algebras A, which are compatible with the involution and the inner product of A, that will be called orthogonal-gradings. If A is an arbitrary \(H^*\)-algebra with a fine grading, we obtain a (fine) orthogonal-graded version of the main structure theorem for proper arbitrary \(H^*\)-algebras. If A is associative, we show that any fine orthogonal-grading is either a group-grading or a (non-group grading) Peirce decomposition of A respect to a family of orthogonal projections. If A is alternative, we prove that any fine orthogonal-grading is either a fine orthogonal-grading of a (proper) associative \(H^*\)-algebra, or a \({\mathbb Z}_2^3\)-grading of the complex octonions \({\mathbb O}\) or a non-group grading which is a refinement of the Peirce decomposition of \({\mathbb O}\) respect to its family of orthogonal projections. Finally, we also show that any orthogonal-grading on the real octonion division algebra is necessarily a group-grading.     

Stability of an $$\varvec{}{}$$th root functional equation in $$\varvec{C}^{*}$$C∗-algebras: a fixed point approach

In this paper, we introduce an \(m{\text{th}}\) root functional equation. Using the fixed point approach, we prove the Hyers–Ulam stability of the \(m{\text{th}}\) root functional equation in \(C^{*}\)-algebras.     

On a tensor-analogue of the Schur product

We consider the tensorial Schur product \(R \circ ^\otimes S = [r_{ij} \otimes s_{ij}]\) for \(R \in M_n(\mathcal {A}), S\in M_n(\mathcal {B}),\) with \(\mathcal {A}, \mathcal {B}~\text{ unital }~ C^*\)-algebras, verify that such a ‘tensorial Schur product’ of positive operators is again positive, and then use this fact to prove (an apparently marginally more general version of) the classical result of Choi that a linear map \(\phi :M_n \rightarrow M_d\) is completely positive if and only if \([\phi (E_{ij})] \in M_n(M_d)^+\), where of course \(\{E_{ij}:1 \le i,j \le n\}\) denotes the usual system of matrix units in \(M_n (:= M_n(\mathbb C))\). We also discuss some other corollaries of the main result.     

Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Let \(L_t:=\Delta _t+Z_t\) for a \(C^{\infty }\)-vector field Z on a differentiable manifold M with boundary \(\partial M\), where \(\Delta _t\) is the Laplacian operator, induced by a time dependent metric \(g_t\) differentiable in \(t\in [0,T_\mathrm {c})\). We first establish the derivative formula for the associated reflecting diffusion semigroup generated by \(L_t\). Then, by using parallel displacement and reflection, we construct the couplings for the reflecting \(L_t\)-diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.     

Structure theorem of the generator of a norm continuous completely positive semigroup: an alternative proof using Bures distance

Here we present an alternative proof using Bures distance that the generator L of a norm continuous completely positive semigroup acting on a \(C^*\)-algebra \({\mathcal {B}}\subset \mathcal B(H)\) has the form \( L(b) = \Psi (b) + k^*b+bk\), \(b\in {\mathcal {B}}\) for some completely positive map \(\Psi :{\mathcal {B}}\rightarrow {\mathcal {B}}(H)\) and \(k\in {\mathcal {B}}(H)\).     

Popular edges and dominant matchings

Given a bipartite graph \(G = (A \cup B,E)\) with strict preference lists and given an edge \(e^* \in E\), we ask if there exists a popular matching in G that contains \(e^*\). We call this the popular edge problem. A matching M is popular if there is no matching \(M'\) such that the vertices that prefer \(M'\) to M outnumber those that prefer M to \(M'\). It is known that every stable matching is popular; however G may have no stable matching with the edge \(e^*\). In this paper we identify another natural subclass of popular matchings called “dominant matchings” and show that if there is a popular matching that contains the edge \(e^*\), then there is either a stable matching that contains \(e^*\) or a dominant matching that contains \(e^*\). This allows us to design a linear time algorithm for identifying the set of popular edges. When preference lists are complete, we show an \(O(n^3)\) algorithm to find a popular matching containing a given set of edges or report that none exists, where \(n = |A| + |B|\).     

A concave–convex problem with a variable operator

We study the following elliptic problem \(-A(u) = \lambda u^q\) with Dirichlet boundary conditions, where \(A(u) (x) = \Delta u (x) \chi _{D_1} (x)+ \Delta _p u(x) \chi _{D_2}(x)\) is the Laplacian in one part of the domain, \(D_1\), and the p-Laplacian (with \(p>2\)) in the rest of the domain, \(D_2 \). We show that this problem exhibits a concave–convex nature for \(1<q<p-1\). In fact, we prove that there exists a positive value \(\lambda ^*\) such that the problem has no positive solution for \(\lambda > \lambda ^*\) and a minimal positive solution for \(0<\lambda < \lambda ^*\). If in addition we assume that p is subcritical, that is, \(p<2N/(N-2)\) then there are at least two positive solutions for almost every \(0<\lambda < \lambda ^*\), the first one (that exists for all \(0<\lambda < \lambda ^*\)) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every \(0<\lambda < \lambda ^*\)) comes from an appropriate (and delicate) mountain pass argument.     

A note on the relationship between the Szlenk and $$w^*$$-dentability indices of arbitrary $$w^*$$-compact sets

We prove the optimal estimate between the Szlenk and \(w^*\)-dentability indices of an arbitrary \(w^*\)-compact subset of the dual of a Banach space. For a given \(w^*\)-compact, convex subset K of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of K. We give applications to the \(w^*\)-dentability index of a Banach space and of an operator.     

On the zero-stability of multistep methods on smooth nonuniform grids

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid \(\{t_n\}_{n=0}^N\) can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., \(t_n = {\varPhi }(\tau _n)\), where \(\tau _n = n/N\) and the map \({\varPhi }\) is monotonically increasing with \({\varPhi }(0)=0\) and \({\varPhi }(1)=1\). The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines \({\varPhi }\), and a tolerance requirement which determines N. Given any strongly stable multistep method, there is an \(N^*\) such that the method is zero stable for \(N>N^*\), provided that \({\varPhi }\in C^2[0,1]\). Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy \(h_n/h_{n-1} = 1 + {\mathrm {O}}(N^{-1})\) as \(N\rightarrow \infty \). The results are exemplified for BDF-type methods.