Search

This paper is a short account of some of the scientific achievements of Wilhelm von Wldenfels with particular attention to the contributions he gave to quantum probability, a field in which he was one of the pioneers. [ABSTRACT FROM AUTHOR]

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 construct a symbol calculus yielding short exact sequences for the dual Toeplitz algebra generated by all bounded dual Toeplitz operators on the Hardy space associated with the polydisk D n in the unitary space C n , that have been introduced and well studied in our earlier paper (Benaissa and Guediri in Taiwan J Math 19: 31–49, 2015), as well as for the C*-subalgebra generated by dual Toeplitz operators with symbols continuous on the associated hypertorus T n . [ABSTRACT FROM AUTHOR]

In this paper, we show that every continuous linear map between unital C*-algebras is skew Lie triple derivable at the identity is a C*-derivation and that every continuous linear map between unital C*-algebras which is a skew Lie triple homomorphism at the identity is a Jordan *-homomorphism. [ABSTRACT FROM AUTHOR]

Error-correcting codes are known to define chiral 2d lattice CFTs where all the U(1) symmetries are enhanced to SU(2). In this paper, we extend this construction to a broader class of length-n codes which define full (non-chiral) CFTs with SU(2)n symmetry, where n = c + c ¯ . We show that codes give a natural discrete ensemble of 2d theories in which one can compute averaged observables. The partition functions obtained from averaging over all codes weighted equally is found to be given by the sum over modular images of the vacuum character of the full extended symmetry group, and in this case the number of modular images is finite. This averaged partition function has a large gap, scaling linearly with n, in primaries of the full SU(2)n symmetry group. Using the sum over modular images, we conjecture the form of the genus-2 partition function. This exhibits the connected contributions to disconnected boundaries characteristic of wormhole solutions in a bulk dual. [ABSTRACT FROM AUTHOR]

This paper provides a constructive method using unitary diagonalizable elements to obtain all hermitian matrices A in M_n(\Bbb C) such that \begin{equation*} \|A\|=\min _{B\in \mathcal {B}}\|A+B\|, \end{equation*} where \mathcal {B} is a C*-subalgebra of M_n(\Bbb C), \|\cdot \| denotes the operator norm. Such an A is called \mathcal {B}-minimal. Moreover, for a C*-subalgebra \mathcal {B} determined by a conditional expectation from M_n(\Bbb C) onto it, this paper constructs \bigoplus _{i=1}^k\mathcal {B}-minimal hermitian matrices in M_{kn}(\Bbb C) through \mathcal {B}-minimal hermitian matrices in M_n(\Bbb C), and gets a dominated condition that the matrix \hat {A}\!=\!\operatorname {diag}(A_1,A_2,\cdots, A_k) is \bigoplus _{i=1}^k\mathcal {B}-minimal if and only if \|\hat {A}\|\leq \|A_s\| for some s\in \{1,2,\cdots,k\} and A_s is \mathcal {B}-minimal, where A_i(1\leq i\leq k) are hermitian matrices in M_n(\Bbb C). [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]

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of W (n) under some conditions which is called B -subalgebras (Definition 3.1). We classify the conjugacy classes of B -subalgebras for W (n) and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of W (n). [ABSTRACT FROM AUTHOR]

The projective representation of groups was introduced in 1904 by Issai Schur in his paper [6]. It differs from the normal representation of groups (introduced by his tutor Ferdinand Georg Frobenius at the suggestion of Richard Dedekind) by a twisting factor, which we call Schur function in this paper and which is called sometimes multipliers or normalized factor set in the literature (other names are also used). It starts with a group T and a Schur function f for T. This is a scalar valued function on T × T satisfying the conditions f(1, 1) = 1 and |f(s, t)| = 1, f(r, s)f(rs, t) = f(r, st)f(s, t) for all r, s, t ∈ T. The projective representation of T twisted by f is a unital C*-subalgebra of the C*-algebra L(l²(T)) of operators on the Hilbert space l²(T). This representation can be used in order to construct many examples of C*-algebras (see e.g. [1, Chapter 7]). By replacing the scalars R or C with an arbitrary unital (real or complex) C*-algebra E, the field of applications is enhanced in an essential way. In this case, l²(T) is replaced by the Hilbert right E-module ⵀt∈T E ≈ E ⊗ l²(T) and L(l²(T)) is replaced by LE(E ⊗ l²(T)), the C*-algebra of adjointable operators of L(E ⊗ l²(T)). The projective representation of groups, which we present in this paper, has some similarities with the construction of cross products with discrete groups. It opens the way to create many K-theories. In a first section, we introduce some results which are needed for this construction, which is developed in the second section. In the third section, we present examples of C*-algebras obtained by this method. Examples of a special kind (the Clifford algebras) are presented in the last section. [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]

Perfect C∗-algebras were introduced by Akeman and Shultz in [Perfect C*-algebras, Mem. Amer. Math. Soc. 55(326) (1985)] and they form a certain subclass of C*-algebras determined by their pure states, and for which the general Stone–Weierstrass conjecture is true. In this paper, we introduce the notion of perfect JC-algebras, and we use the strong relationship between a JC-algebra A and its universal enveloping C∗-algebra C ∗ (A) , to establish that if C ∗ (A) is perfect and A is of complex type, then A is perfect. It is also shown that every scattered JC-algebra of complex type is perfect, and the same conclusion holds for every JC-algebra of complex type whose primitive spectrum is Hausdorff. [ABSTRACT FROM AUTHOR]

Let \mathcal {H} be a separable Hilbert space and let B(\mathcal {H}) be the *-algebra of all bounded linear operators on \mathcal {H}. In the present paper, we prove that a positive/regular operator from L_1(0,1) into an arbitrary separable operator ideal in B(\mathcal {H}) is necessarily Dunford–Pettis, extending and strengthening results due to Gretsky and Ostroy [Glasgow Math. J. 28 (1986), pp. 113–114], and Holub [Proc. Amer. Math. Soc. 104 (1988), pp. 89–95]. Consequently, for an arbitrary atomless von Neumann algebra \mathcal {M} and an arbitrary KB -ideal C_E in B(\mathcal {H}), the predual \mathcal {M}_* of \mathcal {M} is not isomorphic to any subspace of C_E. This observation complements several earlier results. [ABSTRACT FROM AUTHOR]

In this paper we provide some properties and applications of MV-algebras. We prove that a Fibonacci stationary sequence in an MV-algebra gives us an idempotent element. Moreover, taking into account of the representation of a finite MV-algebra, by using Boolean elements of this algebra, we prove that a Fibonacci sequence in an MV-algebra is always stationary. This result is interesting comparing with the behavior of such a sequence on the group (Zn, +), where the Fibonacci sequences are periodic, with the period given by the Pisano period. We also give some examples of finite MV-algebras and the number of their idempotent elements. As an application in Coding Theory, to a Boolean algebra it is attached a binary block code and it is proved that, under some conditions, the converse is also true. [ABSTRACT FROM AUTHOR]

The representations of a k-graph C*-algebra C*(Λ) which arise from Λ-semibranching function systems are closely linked to the dynamics of the k-graph Λ. In this paper, we undertake a systematic analysis of the question of irreducibility for these representations. We provide a variety of necessary and sufficient conditions for irreducibility, as well as a number of examples indicating the optimality of our results. We also explore the relationship between irreducible Λ-semibranching representations and purely atomic representations of C*(Λ). Throughout the paper, we work in the setting of row-finite source-free k-graphs; this paper constitutes the first analysis of Λ-semibranching representations at this level of generality. [ABSTRACT FROM AUTHOR]

We prove that the localization Cw of the monoidal category Cw is rigid, and the category Cw,v admits a localization via a real commuting family of central objects. For a quiver Hecke algebra R and an element w in the Weyl group, the subcategory Cw of the category R-gmod of finite-dimensional graded R-modules categorifies the quantum unipotent coordinate ring Aq(n(w)). In the previous paper, we constructed a monoidal category Cw such that it contains Cw and the objects {(M(wΛi,Λi) i ∈ 1} corresponding to the frozen variables are invertible. In this paper, we show that there is a monoidal equivalence between the category Cw and (Cw-1)rev. Together with the already known left-rigidity of Cw, it follows that the monoidal category Cw is rigid. If v ≤ w in the Bruhat order, there is a subcategory Cw,v of Cw of that categorifies the doubly-invariant algebra N'(w)ℂ[N]N(v). We prove that the family [M(wΛi,vΛi))i∈I of simple R-module forms a real commuting family of graded central objects in the category Cw,v so that there is a localization Cw,v of Cw,v in which M(wΛi,vΛi) are invertible. Since the localization of the algebra N'(w)ℂ[N]N(v) by the family of the isomorphism classes of M(wΛi,vΛi) is isomorphic to the coordinate ring ℂ[Rw,v] of the open Richardson variety associated with w and v, the localization Cw,v categorifies the coordinate ring ℂ[Rw,v]. [ABSTRACT FROM AUTHOR]

Let G be a locally compact étale groupoid and L (L 2 (G)) be the C ∗ -algebra of adjointable operators on the Hilbert C ∗ -module L 2 (G) . In this paper, we discover a notion called quasi-locality for operators in L (L 2 (G)) , generalising the metric space case introduced by Roe. Our main result shows that when G is additionally σ -compact and amenable, an equivariant operator in L (L 2 (G)) belongs to the reduced groupoid C ∗ -algebra C r ∗ (G) if and only if it is quasi-local. This provides a practical approach to describe elements in C r ∗ (G) using coarse geometry. Our main tool is a description for operators in L (L 2 (G)) via their slices with the same philosophy to the computer tomography. As applications, we recover a result by Špakula and the second-named author in the metric space case, and deduce new characterisations for reduced crossed products and uniform Roe algebras for groupoids. [ABSTRACT FROM AUTHOR]

The key motive of this paper is to study symmetric additive mappings and discuss their applications. The study of these symmetric mappings makes it possible to characterize symmetric n-derivations and describe the structure of the quotient ring S / P , where S is any ring and P is a prime ideal of S. The symmetricity of additive mappings allows us to transfer ring theory results to functional analyses, particularly to C ∗ -algebras. Precisely, we describe the structures of C ∗ -algebras via symmetric additive mappings. [ABSTRACT FROM AUTHOR]

Let A be an infinite dimensional simple unital C*-algebra and B be a centrally large subalgebra of A. Archey and Phillips [J. Operator Theory 83 (2020), pp. 353–389] showed that A has real rank zero when B has real rank zero and stable rank one. In this paper, we have removed the restriction of B has stable rank one. Using matrix decomposition by some suitable projections, we have shown that A has real rank zero if B has real rank zero. [ABSTRACT FROM AUTHOR]

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra W 0 (p) A n , 1 ≤ n ≤ 3 , and from Ẑ -invariants of three-manifolds associated with gauge group SU(3). [ABSTRACT FROM AUTHOR]

In this paper, we realize C ∗ -algebras of generalized Boolean dynamical systems as partial crossed products. Reciprocally, we give some sufficient conditions for a partial crossed product to be isomorphic to a C ∗ -algebra of a generalized Boolean dynamical system. As an application, we show that gauge-invariant ideals of C ∗ -algebras of generalized Boolean dynamical systems are themselves C ∗ -algebras of generalized Boolean dynamical systems. [ABSTRACT FROM AUTHOR]

The study of algebraic structures endowed with the concept of symmetry is made possible by the link between Lie algebras and symmetric monoidal categories. This relationship between Lie algebras and symmetric monoidal categories is useful and has resulted in many areas, including algebraic topology, representation theory, and quantum physics. In this paper, we present analogous definitions for Lie algebras within the framework of whiskered structures, bimorphisms, crossed complexes, crossed differential graded algebras, and tensor products. These definitions, given for groupoids in existing literature, have been adapted to establish a direct correspondence between these algebraic structures and Lie algebras. We show that a 2-truncation of the crossed differential graded Lie algebra, obtained from our adapted definitions, gives rise to a braided crossed module of Lie algebras. We also construct a functor to simplicial Lie algebras, enabling a systematic mapping between different Lie algebraic categories, which supports the validity of our adapted definitions and establishes their compatibility with established categories. [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 paper studies Rota-Baxter (RB) operators in complex semi-simple algebras A. They are certain C-linear endomorphisms of A, when considered the latter as a C-vector space. Properties as the nilpotency or the spectrum of such operator R are studied. Some examples are given when A is a Clifford algebra. [ABSTRACT FROM AUTHOR]

Let B(H ) denote the C∗ -algebra of all bounded linear operators acting on a complex Hilbert space H. For A,B ∈ B(H ), define the bimultiplication operator M2,A,B on the class of Hilbert-Schmidt operators by M2,A,B(X) = AXB. It is known that if either A or B is hyponormal, then /W(M2,A,B) = /co W(A)W(B), where the bar and co stand for the closure and the convex hull, respectively and W(·) denotes the numerical range. In this paper, we give some conditions satisfied by A and B to have the following equality W0(M2,A,B) = co W0(A)W0(B), where W0(·) denotes the maximal numerical range. [ABSTRACT FROM AUTHOR]

This paper introduces a (universal) C*-algebra of continuous functions vanishing at infinity on the n -dimensional quantum complex space. To this end, the well-behaved Hilbert space representations of the defining relations are classified. Then these representations are realized by multiplication operators on an L 2 -space. The C*-algebra of continuous functions vanishing at infinity is defined by considering an *-algebra such that its classical counterpart separates the points of the n -dimensional complex space and by taking the operator norm closure of a universal representation of this algebra. [ABSTRACT FROM AUTHOR]

Let and ′ be two alternative ∗-algebras with identities 1 and 1 ′ , respectively, and e 1 and e 2 = 1 − e 1 nontrivial symmetric idempotents in . In this paper, we study the characterization of multiplicative ∗-Lie-type maps. As application, we get a result on alternative W ∗ -algebras. [ABSTRACT FROM AUTHOR]

Generally, the field of fixed point theory has attracted the attention of researchers in different fields of science and engineering due to its use in proving the existence and uniqueness of solutions of real-world dynamic models. C*- algebra is being continually used to explain a physical system in quantum field theory and statistical mechanics and has subsequently become an important area of research. The concept of a C*-algebra-valued metric space was introduced in 2014 to generalize the concept of metric space. In fact, It is a generalization by replacing the set of real numbers with a C*-algebra. After that, this line of research continued, where several fixed point results have been obtained in the framework of C*-algebra valued metric, as well as (more general) C*-algebra-valued b-metric spaces and C*-algebra-valued extended b-metric spaces. Very recently, based on the concept and properties of C* -algebras, we have studied the quasi-case of such spaces to give a more general notion of relaxing the triangular inequality in the asymmetric case. In this paper, we first introduce the concept of C*-algebra-valued quasi-controlled K -metric spaces and prove some fixed point theorems that remain valid in this setting. To support our main results, we also furnish some examples which demonstrate the utility of our main result. Finally, as an application, we use our results to prove the existence and uniqueness of the solution to a nonlinear stochastic integral equation. [ABSTRACT FROM AUTHOR]

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]

The open shop scheduling problem (OSSP) is one of the standard scheduling problems. It consists of scheduling jobs associated with a finite set of tasks developed by different machines. In this case, each machine processes at most one operation at a time, and the job processing order on the machines does not matter. The goal is to determine the completion times of the operations processed on the machines to minimize the largest job completion time, called Cmax. This paper proves that each OSSP has associated a path algebra called Brauer configuration algebra whose representation theory (particularly its dimension and the dimension of its center) can be given using the corresponding Cmax value. It has also been proved that the dimension of the centers of Brauer configuration algebras associated with OSSPs with minimal Cmax are congruent modulo the number of machines. [ABSTRACT FROM AUTHOR]

The Lefschetz fixed point theorem provides a powerful obstruction to the existence of minimal homeomorphisms on well-behaved spaces such as finite CW-complexes. We show that these obstructions do not hold for more general spaces. Minimal homeomorphisms are constructed on compact connected metric spaces with any prescribed finitely generated K-theory or cohomology. In particular, although a non-zero Euler characteristic obstructs the existence of a minimal homeomorphism on a finite CW-complex, this is not the case on a compact metric space. We also allow for some control of the map on K-theory and cohomology induced from these minimal homeomorphisms. This allows for the construction of many minimal homeomorphisms that are not homotopic to the identity. Applications to C*-algebras will be discussed in another paper. [ABSTRACT FROM AUTHOR]

In this paper we study free actions of groups on separated graphs and their C^*-algebras, generalizing previous results involving ordinary (directed) graphs. We prove a version of the Gross-Tucker Theorem for separated graphs yielding a characterization of free actions on separated graphs via a skew product of the (orbit) separated graph by a group labeling function. Moreover, we describe the C^*-algebras associated to these skew products as crossed products by certain coactions coming from the labeling function on the graph. Our results deal with both the full and the reduced C^*-algebras of separated graphs. To prove our main results we use several techniques that involve certain canonical conditional expectations defined on the C^*-algebras of separated graphs and their structure as amalgamated free products of ordinary graph C^*-algebras. Moreover, we describe Fell bundles associated with the coactions of the appearing labeling functions. As a byproduct of our results, we deduce that the C^*-algebras of separated graphs always have a canonical Fell bundle structure over the free group on their edges. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend Boyd and Wong's fixed point theorem in complete generalized metric spaces and to apply this result under the so-called G-weak topology in generalized Banach algebras. Some tools will be introduced and involved in our work as the generalized measures of weak non-compactness and the sequential condition (P G). Our results will extend many known theorems in the literature. Also, we will give an application for an infinite system of nonlinear integral equations defined on the generalized Banach algebra (C ([ 0 , 1 ] , c 0) × C ([ 0 , 1 ] , c 0)) , where c0 is the space of all real sequences converging to zero. [ABSTRACT FROM AUTHOR]

In this study of the relative Dixmier property for inclusions of von Neumann algebras and of C ∗ -algebras, Popa considered a certain property of automorphisms on C ∗ -algebras, that we here call the strong averaging property. In this paper, we characterize when an automorphism on a C ∗ -algebra has the strong averaging property. In particular, automorphisms on commutative C ∗ -algebras possess this property precisely when they are free. An automorphism on a unital separable simple C ∗ -algebra with at least one tracial state has the strong averaging property precisely when its extension to the finite part of the bi-dual of the C ∗ -algebra is properly outer, and in the simple, non-tracial case the strong averaging property is equivalent to being outer. To illustrate the usefulness of the strong averaging property we give three examples where we can provide simpler proofs of existing results on crossed product C ∗ -algebras, and we are also able to extend these results in different directions. [ABSTRACT FROM AUTHOR]