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Axiomatic Fredholm theory in unital C*-algebras was established in [D. Kečkić, Z. Lazović, Acta Sci. Math., 83(3-4):629-655, 2017]. Following the pure algebraic approach by Keckic and Lazovic, in the author’s paper [S. Ivkovic, Banach J. Math. Anal., 17:51, 2023] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*- algebras. However, recently, in [S. Ivković, arXiv:2306.01133] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and is equivalent to the algebraic approach by Kečkić and Lazović. In this paper, we extend further that new Hilbert-module approach from Fredholm theory to semi-Fredholm and semiWeyl theory in unital C*-algebras. Hence, we provide new proofs to the results in [S. Ivkovic, Banach J. Math. Anal., 17:51, 2023]. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W = Der (k [ t , t − 1 ]) and the one-sided Witt algebra W ≥ − 1 = Der (k [ t ]). In the first part of the paper, we consider finite codimension subalgebras of W ≥ − 1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to Ext U (L) 1 (M , L) , where L is a subalgebra of W ≥ − 1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W ≥ − 1 extends to an automorphism of W ≥ − 1. The second part of the paper is devoted to explaining the observed rigidity. We define a notion of "completely non-split extension" and prove that W ≥ − 1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W ≥ − 1 as abstract Lie algebras, they remember that they are contained in W ≥ − 1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations. Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of all free $ U(\mathbb {C}L_0 \oplus \mathbb {C}Y_0\oplus \mathbb {C}M_0) $ U (C L 0 ⊕ C Y 0 ⊕ C M 0) -modules of rank 1 over a Schrödinger-Virasoro type algebra $ \mathfrak {tsv} $ tsv . [ABSTRACT FROM AUTHOR]

Let n ≥ 2 be a fixed integer and A be a C ∗ -algebra. A permuting n-linear map G : A n → A is known to be symmetric generalized n-derivation if there exists a symmetric n-derivation D : A n → A such that G ς 1 , ς 2 , ... , ς i ς i ′ , ... , ς n = G ς 1 , ς 2 , ... , ς i , ... , ς n ς i ′ + ς i D (ς 1 , ς 2 , ... , ς i ′ , ... , ς n) holds ∀ ς i , ς i ′ ∈ A . In this paper, we investigate the structure of C ∗ -algebras involving generalized linear n-derivations. Moreover, we describe the forms of traces of linear n-derivations satisfying certain functional identity. [ABSTRACT FROM AUTHOR]

The present paper is devoted to studying local derivations on the Lie algebra $ W(2,2) $ W (2 , 2) which has some outer derivations. Using some linear algebra methods in [1] and a key construction for $ W(2,2) $ W (2 , 2) , we prove that every local derivation on $ W(2, 2) $ W (2 , 2) is a derivation. As an application, we determine all local derivations on the deformed $ \mathfrak {bms}_3 $ b m s 3 algebra. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {L}(\mathscr {H}) $ L (H) denote the $ C^* $ C ∗ -algebra of adjointable operators on a Hilbert $ C^* $ C ∗ -module $ \mathscr {H} $ H . In this paper, we introduce the generalized Cauchy–Schwarz inequality for operators in $ \mathscr {L}(\mathscr {H}) $ L (H). More precisely, an operator $ A\in \mathscr {L}(\mathscr {H}) $ A ∈ L (H) is said to satisfy the generalized Cauchy–Schwarz inequality if there exists $ \nu \in (0, 1) $ ν ∈ (0 , 1) such that \[ \|\langle Ax, y\rangle\|\leq (\|Ax\|\|y\|)^{\nu}(\|Ay\|\|x\|)^{1 - \nu} \quad (x, y \in \mathscr{H}). \] ‖ ⟨ A x , y ⟩ ‖ ≤ (‖ A x ‖ ‖ y ‖) ν (‖ A y ‖ ‖ x ‖) 1 − ν (x , y ∈ H). We investigate various properties of operators which satisfy the generalized Cauchy–Schwarz inequality. In particular, we prove that if A satisfies the generalized Cauchy–Schwarz inequality such that A has the polar decomposition, then A is paranormal. In addition, we show that if for A the equality holds in the generalized Cauchy–Schwarz inequality, then A is cohyponormal. Among other things, when A has the polar decomposition, we prove that A is semi-hyponormal if and only if $ \left \|\langle Ax, y\rangle \right \| \leq \left \|{|A|}^{1/2}x\right \|\left \|{|A|}^{1/2}y\right \| $ ‖ ⟨ A x , y ⟩ ‖ ≤ ‖ | A | 1 / 2 x ‖ ‖ | A | 1 / 2 y ‖ for all $ x, y \in \mathscr {H} $ x , y ∈ H . [ABSTRACT FROM AUTHOR]

In this paper we study Cuntz–Pimsner algebras associated to 퐶*-correspondences over commutative \mathrm {C}^*-algebras from the point of view of the \mathrm {C}^*-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite dimensional infinite compact metric space X twisted by a vector bundle, the resulting Cuntz–Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these \mathrm {C}^*-algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz–Pimsner algebra has stable rank one. Otherwise, it is purely infinite. For a Cuntz–Pimsner algebra of a minimal homeomorphism of an infinite compact metric space X twisted by a line bundle over X, we introduce orbit-breaking subalgebras. With no assumptions on the dimension of X, we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of X is finite, they are furthermore \mathcal {Z}-stable and hence classified by the Elliott invariant. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of a B-filter in a B-algebra (X, ∗, 0) and presents characteristics of its properties: for any a ∈ X, the set ⟨a⟩B = {ak: k ∈ Z} forms a B-ideal and B-filter of X. Moreover, this paper showns some properties of exponents on B-algebra. [ABSTRACT FROM AUTHOR]

Voiculescu discovered asymptotic freeness of independent Haar-distributed unitary matrices. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this paper, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a nontrivial signature ρ , i.e. two finite sequences of non-increasing natural numbers, and for n large enough, consider the irreducible representation V n , ρ of U n associated with the signature ρ . We consider the quotient U n , ρ of U n viewed as a matrix subgroup of U (V n , ρ) , and show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. Thanks to classical results in representation theory, this result is closely related to strong asymptotic freeness for tensors, which we establish as a preliminary. To achieve this result, we need to develop four new tools, each of independent theoretical interest: (i) a centered Weingarten calculus and uniform estimates thereof, (ii) a systematic and uniform comparison of Gaussian moments and unitary moments of matrices, (iii) a generalized and simplified operator-valued non-backtracking theory in a general C ∗ -algebra, and finally, (iv) combinatorics of tensor moment matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study representations of non-finitely graded Lie algebras 풲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} related to Virasoro algebra, where ϵ = ± 1 {\epsilon=\pm 1} . Precisely speaking, we completely classify the free 풰 ⁢ ( 픥 ) {\mathcal{U}(\mathfrak{h})} -modules of rank one over 풲 ⁢ ( ϵ ) {\mathcal{W}(\epsilon)} , and find that these module structures are rather different from those of other graded Lie algebras. We also determine the simplicity and isomorphism classes of these modules. [ABSTRACT FROM AUTHOR]

One of the questions investigated in deformation theory is to determine to which algebras can a given associative algebra be deformed. In this paper we investigate a different but related question, namely: for a given associative finite-dimensional |${\mathbb{C}}$| -algebra |$A$|⁠ , find algebras |$N$|⁠ , which can be deformed to |$A$|⁠. We develop a simple method that produces associative and flat deformations to investigate this question. As an application of this method we answer a question of Michael Wemyss about deformations of contraction algebras. [ABSTRACT FROM AUTHOR]

In this paper, we consider the C∗-algebra generated by finitely many annihilation operators acting on the weakly monotone Fock space, and we call it weakly monotone C∗-algebra. We give an abstract presentation for this algebra, showing that it is isomorphic to a suitable quotient of a Cuntz–Krieger C∗-algebra 풪A corresponding to a suitable matrix A. Furthermore, we show that the diagonal subalgebra of the weakly monotone C∗-algebra is a MASA and we give a detailed description of its Gelfand spectrum. [ABSTRACT FROM AUTHOR]

A major open problem of AF-embedding is whether every separable exact quasidiagonal C ∗ -algebra can be embedded into an AF-algebra. In this paper we characterize AF-embeddable C ∗ -algebras by representations to observe their similarity to the separable exact quasidiagonal C ∗ -algebras. As an application, we show that every separable exact quasidiagonal C ∗ -algebra is AF-embeddable if and only if every faithful essential representation of a separable exact quasidiagonal C ∗ -algebra is a certain kind of ∗ -representation. We also show that a separable C ∗ -algebra is AF-embeddable if and only if it can be embedded into a particular C ∗ -algebra. [ABSTRACT FROM AUTHOR]

Inspired by recent developments in the theory of Banach and operator algebras of locally compact groups, we construct several new classes of bifunctors (A,B)\mapsto A\otimes _{\alpha } B, where A\otimes _\alpha B is a cross norm completion of A\odot B for each pair of C*-algebras A and B. For the first class of bifunctors considered (A,B)\mapsto A{\otimes _p} B (1\leq p\leq \infty), A{\otimes _p} B is a Banach algebra cross-norm completion of A\odot B constructed in a fashion similar to p-pseudofunctions \text {PF}^*_p(G) of a locally compact group. Taking a cue from the recently introduced symmetrized p-pseudofunctions due to Liao and Yu and later by the second and the third named authors, we also consider {\otimes _{p,q}} for Hölder conjugate p,q\in [1,\infty ] – a Banach *-algebra analogue of the tensor product {\otimes _{p,q}}. By taking enveloping C*-algebras of A{\otimes _{p,q}} B, we arrive at a third bifunctor (A,B)\mapsto A{\otimes _{\mathrm C^*_{p,q}}} B where the resulting algebra A{\otimes _{\mathrm C^*_{p,q}}} B is a C*-algebra. For G_1 and G_2 belonging to a large class of discrete groups, we show that the tensor products \mathrm C^*_{\mathrm r}(G_1){\otimes _{\mathrm C^*_{p,q}}}\mathrm C^*_{\mathrm r}(G_2) coincide with a Brown-Guentner type C*-completion of \mathrm \ell ^1(G_1\times G_2) and conclude that if 2\leq p'

This is the second of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to C*-algebraic Morita equivalence, and the verification of the Connes–Kasparov conjecture in operator K-theory for these groups. In Part I we presented the Morita equivalence and the Connes–Kasparov morphism. In this part we shall compute the morphism using David Vogan's description of the tempered dual. In fact we shall go further by giving a complete representation-theoretic description and parametrization, in Vogan's terms, of the essential components of the tempered dual, which carry the K-theory of the tempered dual. [ABSTRACT FROM AUTHOR]

Let A be a unital C ∗ -algebra with unit 1 A and let a ∈ A be a positive and invertible element. Suppose that S (A) is the set of all states on A and let S a (A) = f f (a) : f ∈ S (A) , f (a) ≠ 0. The norm ‖ x ‖ a for every x ∈ A is defined by ‖ x ‖ a = sup φ ∈ S a (A) φ (x ∗ a x). In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the norm ‖ · ‖ a in A , namely a-Birkhoff–James orthogonality. The characterization of a-Birkhoff–James orthogonality in A by means of the elements of generalized state space S a (A) is provided. As an application, a characterization for the best approximation to elements of A in a subspace B with respect to ‖ · ‖ a is obtained. Moreover, a formula for the distance of an element of A to the subspace B = C 1 A is given. We also study the strong version of a-Birkhoff–James orthogonality in A. The classes of C ∗ -algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong a-Birkhoff–James orthogonality and A -valued inner product orthogonality in A implies that the center of A is trivial. Finally, we show that if the (strong) a-Birkhoff–James orthogonality is right-additive (left-additive) in A , then the center of A is trivial. [ABSTRACT FROM AUTHOR]

In this paper, we exhibit two unital, separable, nuclear C∗${\rm C}^*$‐algebras of stable rank one and real rank zero with the same ordered scaled total K‐theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K‐theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of C∗${\rm C}^*$‐algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the C∗${\rm C}^*$‐algebras of stable rank one and real rank zero. [ABSTRACT FROM AUTHOR]

In this paper we consider certain algebraic properties concerning variants of the Heron mean on positive definite cones of general C ⁎ -algebras. Those variants are the Kubo-Ando type Heron mean and the Wasserstein mean. The main part of the investigation concerns associativity properties. We present a number of results that show how far operations related to those two kinds of means are from being associative. Many of our results can also be viewed as characterizations of central positive definite elements or as characterizations of commutative C ⁎ -algebras. [ABSTRACT FROM AUTHOR]

Recently, Ludewig and Thiang introduced a notion of a uniformly localized Wannier basis with localization centers in an arbitrary uniformly discrete subset D in a complete Riemannian manifold X. They show that, under certain geometric conditions on X, the class of the orthogonal projection onto the span of such a Wannier basis in the K-theory of the Roe algebra C*(X) is trivial. In this paper, we clarify the geometric conditions on X, which guarantee triviality of the K-theory class of any Wannier projection. We show that this property is equivalent to triviality of the unit of the uniform Roe algebra of D in the K-theory of its Roe algebra, and provide a geometric criterion for that. As a consequence, we prove triviality of the K-theory class of any Wannier projection on a connected proper measure space X of bounded geometry with a uniformly discrete set of localization centers. [ABSTRACT FROM AUTHOR]

An action of a compact, in particular finite group on a C*-algebra is called properly outer if no automorphism of the group that is distinct from identity is implemented by a unitary element of the algebra of local multipliers of the C*-algebra. In this paper, I define the notion of strictly outer action (similar to the definition for von Neumann factors in [S. Vaes, The unitary implementation of a locally compact group action, J. Funct. Anal. 180 (2001) 426–480]) and prove that for finite groups and prime C*-algebras, it is equivalent to the proper outerness of the action. For finite abelian groups this is equivalent to other relevant properties of the action. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the linear and the nonlinear Rayleigh–Taylor instability for the gravity-driven incompressible Navier–Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile ρ 0 (x 2) in a slab domain 2 π L T × (- 1 , 1) ( L > 0 , T is the usual 1D torus). The linear instability study of the viscous Rayleigh–Taylor model amounts to the study of the following ordinary differential equation on the finite interval (- 1 , 1) , 0.1 - λ 2 [ ρ 0 k 2 ϕ - (ρ 0 ϕ ′) ′ ] = λ μ (ϕ (4) - 2 k 2 ϕ ′ ′ + k 4 ϕ) - g k 2 ρ 0 ′ ϕ , with the boundary conditions 0.2 ϕ (- 1) = ϕ (1) = 0 , μ ϕ ′ ′ (1) = ξ + ϕ ′ (1) , μ ϕ ′ ′ (- 1) = - ξ - ϕ ′ (- 1) , where λ > 0 is the growth rate in time, g > 0 is the gravity constant, k is the wave number and two Navier-slip coefficients ξ ± are nonnegative constants. For each k ∈ L - 1 Z \ { 0 } , we define a k-supercritical regime of viscosity coefficient, i.e. μ > μ c (k , Ξ) with Ξ = (ξ + , ξ -) to describe a spectral analysis by adapting an operator method of Lafitte and Nguyễn (Water Waves 4:259–305, 2022) and to prove that there are infinite nontrivial solutions (λ n , ϕ n) n ≥ 1 of (0.1)–(0.2) with λ n → 0 as n → ∞ and ϕ n ∈ H 4 ((- 1 , 1)) . Hence, we prove the linear Rayleigh–Taylor instability for any viscosity coefficient μ > 0 . As a by-product, based on the existence of infinitely many normal modes of the linearized problem, we construct a wide class of initial data to the nonlinear equations, being inspired by the previous framework of Guo–Strauss (Commun Pure Appl Math 48:861–894, 1995) and of Grenier (Commun Pure Appl Math 53:1067–1091, 2000) with a refinement, to prove the nonlinear Rayleigh–Taylor instability in a high regime of viscosity coefficient, namely μ > 3 sup k ∈ L - 1 Z \ { 0 } μ c (k , Ξ) . [ABSTRACT FROM AUTHOR]

In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

We consider the semidirect product Z ⋊ φ Z × of the additive group Z of all integers and the multiplicative semigroup Z × of integers without zero relative to a semigroup homomorphism φ from Z × to the endomorphism semigroup of Z. It is shown that this semidirect product is a normal extension of the semigroup Z × N by the dihedral group, where N is the multiplicative semigroup of all natural numbers. Further, we study the structure of C ∗ -algebras associated with this extension. In particular, we prove that the reduced semigroup C ∗ -algebra of the semigroup Z ⋊ φ Z × is topologically graded over the dihedral group. As a consequence, there exists a structure of a free Banach module over the reduced semigroup C ∗ -algebra of Z × N in the underlying Banach space of the reduced semigroup C ∗ -algebra of Z ⋊ φ Z × . [ABSTRACT FROM AUTHOR]

In this paper, we study injective and nuclear real W*- and C*-algebras. The connection of these concepts with similar concepts of enveloping W*- and C*-algebras is considered. The equivalence of the concepts of injectivity and nuclearity for real C*-algebras is shown. As a consequence, nuclear real factors of types II1, II∞, III1, III0, and IIIλ (0 < λ < 1) are completely described. [ABSTRACT FROM AUTHOR]

In this paper, all Lie bialgebra structures on the derivation Lie algebra W over a rank d ≥ 3 quantum torus associated to q are considered, where q is a d × d matrix with all the entries being roots of unity. They are shown to be triangular coboundary. As a byproduct, it is also proved that the first cohomology group H1 (W, W ⊗ W) is trivial. [ABSTRACT FROM AUTHOR]

This study introduces an ample Hausdorff groupoid Aˆ ⋊ R extracted from an ample Hausdorff groupoid G and a unital commutative ring R; a Hausdorff groupoid D which is the discrete twist over Aˆ ⋊ R. In the groupoid C*-algebra perspective, when R = C there is an isomorphism between the non-twisted groupoid C*-algebra (C∗(G)) and the twisted groupoid C*-algebra (C∗(Aˆ ⋊ R;D)). However, in this paper, in a purely algebraic setting, the non-twisted Steinberg algebra (AR(G)) and the twisted Steinberg algebra (AR(D;Aˆ⋊R)) are non-isomorphic. [ABSTRACT FROM AUTHOR]

Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F -vector space L. In this paper we will provide a condition for a Z 2 -grading on E to behave like the natural Z 2 -grading E c a n. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [10]. The conjecture poses that every Z 2 -grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of E c a n by means of its Z 2 -graded polynomial identities. Furthermore we construct a Z 2 -grading on E that gives a negative answer to the conjecture. [ABSTRACT FROM AUTHOR]

Given a vertex operator algebra V , one can attach a graded Poisson algebra called the C 2 -algebra R (V). The associate Poisson scheme provides an important invariant for V and has been studied by Arakawa as the associated variety. In this article, we define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties and then construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational C 2 -cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. In this paper, we study R (V) as a commutative algebra only and do not use the property of its Poisson structure, which is expected to provide more refined invariants. The goal of this work is to study the cohomological supports of modules for vertex algebras as the cohomological support varieties for finite groups and restricted Lie algebras. [ABSTRACT FROM AUTHOR]

We extend Nekrashevych's KK-duality for C^*-algebras of regular, recurrent, contracting self-similar group actions to regular, contracting self-similar groupoid actions on a graph, removing the recurrence condition entirely and generalising from a finite alphabet to a finite graph. More precisely, given a regular and contracting self-similar groupoid (G,E) acting faithfully on a finite directed graph E, we associate two C^*-algebras, \mathcal {O}(G,E) and \widehat {\mathcal {O}}(G,E), to it and prove that they are strongly Morita equivalent to the stable and unstable Ruelle C*-algebras of a Smale space arising from a Wieler solenoid of the self-similar limit space. That these algebras are Spanier-Whitehead dual in KK-theory follows from the general result for Ruelle algebras of irreducible Smale spaces proved by Kaminker, Putnam, and the last author. [ABSTRACT FROM AUTHOR]

Suppose that B is a separable stable C^*-algebra with real rank zero, stable rank one and (\mathrm {K}_0(B), \mathrm {K}_0^+(B)) is weakly unperforated in the sense of Elliott [Internat. J. Math. 1 (1990), no. 4, pp. 361–380]. Let A be a unital simple separable nuclear \mathrm {C}^*-algebra. We show that B has the corona factorization property and any unital extension of A by B is absorbing. [ABSTRACT FROM AUTHOR]

The behavior of discrete-event systems, in which the individual components move from event to event rather than varying continuously through time, is often described by systems of linear equations in max-min algebra, in which classical addition and multiplication are replaced by ⊕ and ⊗, representing maximum and minimum, respectively. Max-min equations have found a broad area of applications in causal models, which emphasize relationships between input and output variables. Many practical situations can be described using max-min systems of linear equations. We shall deal with a two-sided max-min system of linear equations with unknown column vector x of the form A⊗x⊕c=B⊗x⊕d, where A, B are given square matrices, c, d are column vectors and operations ⊕ and ⊗ are extended to matrices and vectors in the same way as in the classical algebra. We give an equivalent condition for its solvability. For a given max-min objective function f, we consider optimization problem of type f⊤⊗x→max or min constraint to A⊗x⊕c=B⊗x⊕d. We solve the equation in the form f(x)=v on the set of solutions of the equation A⊗x⊕c=B⊗x⊕d and extend the problem to the case of an interval function f and an interval value v. We define several types of the reachability of the interval value v by the interval function f and provide equivalent conditions for them. [ABSTRACT FROM AUTHOR]

We prove that an action ρ:A→M(C0(G)⊗A) of a locally compact quantum group on a C∗-algebra has a universal equivariant compactification, and prove a number of other category-theoretic results on G-equivariant compactifications: that the categories compactifications of ρ and A respectively are locally presentable (hence complete and cocomplete), that the forgetful functor between them is a colimit-creating left adjoint, and that epimorphisms therein are surjective and injections are regular monomorphisms. When G is regular coamenable we also show that the forgetful functor from unital G-C∗-algebras to unital C∗-algebras creates finite limits and is comonadic, and that the monomorphisms in the former category are injective. [ABSTRACT FROM AUTHOR]

It is well known that any Lie algebra can be embedded into an associative algebra. We prove that any metabelian Lie algebra can be embedded into an algebra in the subvariety of perm algebras, i.e. associative algebras with the identity a b c − a c b = 0. In addition, a technical method to construct the universal enveloping perm algebra for a metabelian Lie algebra is given. [ABSTRACT FROM AUTHOR]

For each odd integer n \geq 3, we construct a rank-3 graph \Lambda _n with involution \gamma _n whose real C^*-algebra C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n) is stably isomorphic to the exotic Cuntz algebra \mathcal E_n. This construction is optimal, as we prove that a rank-2 graph with involution (\Lambda,\gamma) can never satisfy C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda, \gamma)\sim _{ME} \mathcal E_n, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math. 10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution (\Lambda, \gamma) whose real C^*-algebra C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda, \gamma) is stably isomorphic to the suspension S \mathbb {R}. In the Appendix, we show that the i-fold suspension S^i \mathbb {R} is stably isomorphic to a graph algebra iff -2 \leq i \leq 1. [ABSTRACT FROM AUTHOR]

It was demonstrated recently that the W 1 + ∞ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized W ˜ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the W ˜ algebras studied earlier. We suggest a definition of the generalized W ˜ algebra as differential operators in variables p k basing on the matrix realization of the W 1 + ∞ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the W 1 + ∞ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with W ˜ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric τ -functions corresponding to the completed cycles. [ABSTRACT FROM AUTHOR]

We examine inclusions of C*-algebras of the form AH⊆A⋊r​G, where G and H are groups acting on a unital simple C*-algebra A by outer automorphisms and H is finite. It follows from a theorem of Izumi that AH⊆A is C*-irreducible, in the sense that all intermediate C*-algebras are simple. We show that AH⊆A⋊r​G is C*-irreducible for all G and H as above if and only if G and H have trivial intersection in the outer automorphisms of A, and we give a~Galois type classification of all intermediate C*-algebras in the case when H is abelian and the two actions of G and H on A commute. We illustrate these results with examples of outer group actions on the irrational rotation C*-algebras. We exhibit, among other examples, C*-irreducible inclusions of AF-algebras that have intermediate C*-algebras that are not AF-algebras; in fact, the irrational rotation C*-algebra appears as an intermediate C*-algebra. [ABSTRACT FROM AUTHOR]

Let f, g be two continuous non-negative real-valued functions defined on the nonnegative half-line [0,∞) that satisfy the condition f(t)g(t) = t, for all t ≥ 0, and let P and Q denote two positive elements in an unital C*-algebra A . We shall show that the following model of inequality holds: ∀X ∈ A, ||f(P)Xg(Q)+g(P)Xf(Q)||≥2||P1/2 X Q1/2|| Through this model, we shall establish the universality of the Heinz operator norm inequality and related inequalities within the broad spectrum of any abstract unital C*-algebra. [ABSTRACT FROM AUTHOR]

The tracial cyclic Rokhlin property for automorphisms of simple not necessarily unital C*-algebras is investigated. We show that the tracial cyclic Rokhlin property is preserved by going to certain restrictions to subalgebras and taking direct limit or tensor products of actions. We also show that under certain conditions properties such as real rank zero, the tracial rank zero, stable rank one, (tracial) Z-stability, Property (SP), strict comparison on projections are passed to crossed products under automorphisms with the tracial cyclic Rokhlin property. [ABSTRACT FROM AUTHOR]

We show that full Hilbert C*-modules over a commutative unital C*-algebra A have fixed point property for nonexpansive mappings if and only if A is finite dimensional. We also show that the same is true for every Hilbert C*-module with unit vectors over an arbitrary unital C*-algebra. In particular, a classification of full Hilbert C*-modules with unit vectors over unital C*-algebras is given via fixed point property for nonexpansive mappings. [ABSTRACT FROM AUTHOR]

We consider Deaconu--Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted C*-algebra of such a groupoid determined by a continuous circle-valued groupoid 2- cocycle. When the groupoid is not minimal, this C*-algebra is never simple, so we focus on minimal groupoids. We describe an action of the quotient of the groupoid by the interior of its isotropy on the spectrum of the twisted C*-algebra of the interior of the isotropy. We prove that the twisted groupoid C*-algebra is simple if and only if this action is minimal. We describe applications to crossed products of topological-graph C*-algebras by quasi-free actions. [ABSTRACT FROM AUTHOR]

We say that a C*-algebra is nowhere scattered if none of its quotients contains a minimal open projection. We characterize this property in various ways, by topological properties of the spectrum, by divisibility properties in the Cuntz semigroup, by the existence of Haar unitaries for states, and by the absence of nonzero ideal-quotients that are elementary, scattered or type I. Under the additional assumption of real rank zero or stable rank one, we show that nowhere scatteredness implies even stronger divisibility properties of the Cuntz semigroup. [ABSTRACT FROM AUTHOR]

We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones--Wenzl projections in Temperley--Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C*-algebras that are KK-equivalent to C and obtain a complete list of generators and relations for them. We also show that their gaugeinvariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz--Pimsner algebras under monoidal equivalence of quantum symmetry groups. [ABSTRACT FROM AUTHOR]