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Showing papers on "Operator algebra published in 2016"


Journal ArticleDOI
TL;DR: Gauge-invariant observables for quantum gravity are described in this article, with explicit constructions given primarily to leading order in Newton's constant, analogous to and extending constructions first given by Dirac in quantum electrodynamics.
Abstract: Gauge-invariant observables for quantum gravity are described, with explicit constructions given primarily to leading order in Newton's constant, analogous to and extending constructions first given by Dirac in quantum electrodynamics. These can be thought of as operators that create a particle, together with its inseparable gravitational field, and reduce to usual field operators of quantum field theory in the weak-gravity limit; they include both Wilson-line operators and those creating a Coulombic field configuration. We also describe operators creating the field of a particle in motion; as in the electromagnetic case, these are expected to help address infrared problems. An important characteristic of the quantum theory of gravity is the algebra of its observables. We show that the commutators of the simple observables of this paper are nonlocal, with nonlocality becoming significant in strong field regions, as predicted previously on general grounds.

195 citations


Posted Content
TL;DR: In this paper, it was shown that the fixed-point vertex operator subalgebra (T^\sigma) is regular under the action of any finite solvable group, and that the twisted twining characters for commuting finite order automorphisms of holomorphic vertex operator algebras are also regular.
Abstract: We show that if $T$ is a simple non-negatively graded regular vertex operator algebra with a nonsingular invariant bilinear form and $\sigma$ is a finite order automorphism of $T$, then the fixed-point vertex operator subalgebra $T^\sigma$ is also regular. This yields regularity for fixed point vertex operator subalgebras under the action of any finite solvable group. As an application, we obtain an $SL_2(\mathbb{Z})$-compatibility between twisted twining characters for commuting finite order automorphisms of holomorphic vertex operator algebras. This resolves one of the principal claims in the Generalized Moonshine conjecture.

93 citations


Posted Content
TL;DR: In this article, it was shown that if the fixed point vertex operator subalgebra is regular, then the regular vertex operator algebra of CFT type with a nonsingular invariant bilinear form is also regular.
Abstract: We show that if $T$ is a simple regular vertex operator algebra of CFT type with a nonsingular invariant bilinear form and $\sigma$ is a finite automorphism of $T$, then the fixed point vertex operator subalgebra $T^\sigma$ is also regular.

85 citations


Journal ArticleDOI
TL;DR: In this paper, the index problems of topological insulators and superconductors were formulated using Kasparov theory, both complex and real, and a spectral triple encoding the geometry of the sample's (possibly non-commutative) Brillouin zone was derived.
Abstract: We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of non-commutative index theory of operator algebras. In particular, we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realized as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample’s (possibly non-commutative) Brillouin zone.

76 citations


Journal ArticleDOI
TL;DR: It is shown in this Letter that this Liouville conformal field theory can be interpreted in terms of microscopic loop models and in particular a family of geometrical operators are introduced, and it is shown that their operator algebra corresponds exactly to that of vertex operators V_{α[over ^]} in c≤1Liouville theory.
Abstract: The possibility of extending the Liouville Conformal Field Theory from values of the central charge $c \geq 25$ to $c \leq 1$ has been debated for many years in condensed matter physics as well as in string theory. It was only recently proven that such an extension -- involving a real spectrum of critical exponents as well as an analytic continuation of the DOZZ formula for three-point couplings -- does give rise to a consistent theory. We show in this Letter that this theory can be interpreted in terms of microscopic loop models. We introduce in particular a family of geometrical operators, and, using an efficient algorithm to compute three-point functions from the lattice, we show that their operator algebra corresponds exactly to that of vertex operators $V_{\hat{\alpha}}$ in $c \leq 1$ Liouville. We interpret geometrically the limit $\hat{\alpha} \to 0$ of $V_{\hat{\alpha}}$ and explain why it is not the identity operator (despite having conformal weight $\Delta=0$).

71 citations


Journal ArticleDOI
TL;DR: In this article, the authors give an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators and present a mixture of classic results, presented from the point of view of tensor networks, and of new results.
Abstract: Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.

62 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider the general properties of bounded approximate units in non-self-adjoint operator algebras and show that these properties arise naturally from the differential structure of spectral triples and unbounded KK modules.

59 citations


Journal ArticleDOI
TL;DR: In this paper, the authors define Euclidean quantum space as a choice of polarization for the Heisenberg algebra of quantum theory and show that such polarizations contain a fundamental length scale and, contrary to what is implied by the Schrodinger polarization, they possess topologically distinct spectra.
Abstract: At present, our notion of space is a classical concept. Taking the point of view that quantum theory is more fundamental than classical physics, and that space should be given a purely quantum definition, we revisit the notion of Euclidean space from the point of view of quantum mechanics. Since space appears in physics in the form of labels on relativistic fields or Schrodinger wave functionals, we propose to define Euclidean quantum space as a choice of polarization for the Heisenberg algebra of quantum theory. We show, following Mackey, that generically, such polarizations contain a fundamental length scale and that contrary to what is implied by the Schrodinger polarization, they possess topologically distinct spectra. These are the modular spaces. We show that they naturally come equipped with additional geometrical structures usually encountered in the context of string theory or generalized geometry. Moreover, we show how modular space reconciles the presence of a fundamental scale with translation and rotation invariance. We also discuss how the usual classical notion of space comes out as a form of thermodynamical limit of modular space while the Schrodinger space is a singular limit.

56 citations


Journal ArticleDOI
TL;DR: In this article, the numerical conformal bootstrap was used to search for finite, closed sub-algebras of the OPE without assuming unitarity, and the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism.
Abstract: We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the “Gliozzi” bootstrap method, and provides a simpler setting in which to study technical challenges with the method. In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.

50 citations


Journal ArticleDOI
TL;DR: In this article, the symmetry algebra A n of the associated Dirac-Dunkl equation on S n − 1 is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra.

50 citations


Journal ArticleDOI
TL;DR: In this paper, the authors consider the generalizations of the free U(N) and O(n) scalar conformal field theories to actions with higher powers of the Laplacian box^k, in general dimension d. They study the spectra, Verma modules, anomalies and OPE of these theories.
Abstract: We consider the generalizations of the free U(N) and O(N) scalar conformal field theories to actions with higher powers of the Laplacian box^k, in general dimension d. We study the spectra, Verma modules, anomalies and OPE of these theories. We argue that in certain d and k, the spectrum contains zero norm operators which are both primary and descendant, as well as extension operators which are neither primary nor descendant. In addition, we argue that in even dimensions d <= 2k, there are well-defined operator algebras which are related to the box^k theories and are novel in that they have a finite number of single-trace states.

Journal ArticleDOI
TL;DR: In this article, the authors consider the algebra of simple operators defined in a time band in a CFT with a holographic dual and show that this algebra obeys a version of the Reeh-Schlieder theorem: the action of the algebra can approximate any low energy state in the CFT arbitrarily well, but no operator within this algebra can exactly annihilate the vacuum.
Abstract: We consider the algebra of simple operators defined in a time band in a CFT with a holographic dual. When the band is smaller than the light crossing time of AdS, an entire causal diamond in the center of AdS is separated from the band by a horizon. We show that this algebra obeys a version of the Reeh-Schlieder theorem: the action of the algebra on the CFT vacuum can approximate any low energy state in the CFT arbitrarily well, but no operator within the algebra can exactly annihilate the vacuum. We show how to relate local excitations in the complement of the central diamond to simple operators in the band. Local excitations within the diamond are invisible to the algebra of simple operators in the band by causality, but can be related to complicated operators called “precursors”. We use the Reeh-Schlieder theorem to write down a simple and explicit formula for these precursors on the boundary. We comment on the implications of our results for black hole complementarity and the emergence of bulk locality from the boundary.

Journal ArticleDOI
TL;DR: In this paper, the authors studied a family of non-C2-cofinite vertex operator algebras, called the singlet vertex operator algebra, and connected several important concepts in the theory of vertex operators, quantum modular forms, and modular tensor categories.
Abstract: We study a family of non-C2-cofinite vertex operator algebras, called the singlet vertex operator algebras, and connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet vertex operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter ϵϵ. We then apply modular transformation properties of regularisedpartial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of ϵϵ. The map from the space of characters equipped with the Verlinde product to the space of regularised quantum dimensions turns out to be a genuine ring isomorphism for positive real part of ϵϵ, while for sufficiently negative real part of ϵϵ its surjective image gives the fusion ring of a rational vertex operator algebra. The category of modules of this rational vertex operator algebra should be viewed as obtained through the process of a semi-simplification procedure widely used in the theory of quantum groups. Interestingly, the modular tensor category structure constants of this vertex operator algebra can be also detected from vector-valued quantum modular forms formed by distinguished atypical characters.

Journal ArticleDOI
TL;DR: In this article, an elementary relativistic system within Wigner's approach is defined as a locally-faithful irreducible continuous unitary representation of the Poincare group in a real Hilbert space.
Abstract: As established by Soler, Quantum Theories may be formulated in real, complex or quaternionic Hilbert spaces only. Stuckelberg provided physical reasons for ruling out real Hilbert spaces relying on Heisenberg principle. Focusing on this issue from another viewpoint, we argue that there is a fundamental reason why elementary quantum systems are not described in real Hilbert spaces: their symmetry group. We consider an elementary relativistic system within Wigner's approach defined as a locally-faithful irreducible continuous unitary representation of the Poincare group in a real Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincare invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation. All that leads to a physically equivalent formulation in a complex Hilbert space. Differently from what happens in the real picture, here all selfadjoint operators are observables in accordance with Soler's thesis, and the standard quantum version of Noether theorem holds. We next focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them and making our model physically more general. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions and we adopt a notion of continuity referred to the states. Also in this case, the final result proves that there exist a unique (up to sign) Poincare invariant complex structure making the theory complex and completely fitting into Soler's picture. This complex structure reveals a nice interplay of Poincare symmetry and the classification of the commutant of irreducible real von Neumann algebras.

Journal ArticleDOI
TL;DR: In this paper, the authors developed an orbifold theory for a finite, cyclic group G acting on a suitably regular, holomorphic vertex operator algebra V. The theory was extended to vertex operator algebras with group-like fusion.
Abstract: In this thesis we develop an orbifold theory for a finite, cyclic group G acting on a suitably regular, holomorphic vertex operator algebra V. To this end we describe the fusion algebra of the fixed-point vertex operator subalgebra V^G and show that V^G has group-like fusion. Then we solve the extension problem for vertex operator algebras with group-like fusion. We use these results to construct five new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds, contributing to the classification of the V_1-structures of suitably regular, holomorphic vertex operator algebras of central charge 24. As another application we present the BRST construction of ten Borcherds-Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight.

Journal ArticleDOI
TL;DR: In this paper, the mirror extensions of vertex operator algebras are considered via tensor categories, and it is proved that the mirror extension conjecture can be proven in the tensor category.
Abstract: In this paper, mirror extensions of vertex operator algebras is considered via tensor categories. The mirror extension conjecture is proved.

Journal ArticleDOI
TL;DR: In this paper, the authors use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity.
Abstract: We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the "Gliozzi" bootstrap method, and provides a simpler setting in which to study technical challenges with the method. In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.

Journal ArticleDOI
TL;DR: In this paper, the authors constructed three holomorphic vertex operator algebras of central charge 24 using the \({\mathbb{Z}_{2}}\)-orbifold construction associated to inner automorphisms.
Abstract: In this article, we construct three new holomorphic vertex operator algebras of central charge 24 using the \({\mathbb{Z}_{2}}\)-orbifold construction associated to inner automorphisms. Their weight one subspaces have the Lie algebra structures D7,3A3,1G2,1, E7,3A5,1, and \({A_{8,3}A_{2,1}^2}\). In addition, we discuss the constructions of holomorphic vertex operator algebras with Lie algebras A5,6C2,3A1,2 and \({D_{6,5}A_{1,1}^2}\) from holomorphic vertex operator algebras with Lie algebras C5,3G2,2A1,1 and \({A_{4,5}^2}\), respectively.

Journal ArticleDOI
TL;DR: In this paper, the fusion rules of the free wreath product quantum groups G≀⁎SN+ for all compact matrix quantum groups of Kac type G and N≥4.

Posted Content
TL;DR: In this article, a conformal field theory description of the spaces of ground states for the T[SU(N)] theories was proposed, which can play a role of S-duality kernel in maximally supersymmetric SU(N) gauge theory and provide a kernel for the Geometric Langlands duality for special unitary groups.
Abstract: Three-dimensional N = 4 supersymmetric quantum field theories admit two topological twists, the Rozansky-Witten twist and its mirror. Either twist can be used to define a supersymmetric compactification on a Riemann surface and a corre- sponding space of supersymmetric ground states. These spaces of ground states can play an interesting role in the Geometric Langlands program. We propose a description of these spaces as conformal blocks for certain non-unitary Vertex Operator Algebras and test our conjecture in some important examples. The two VOAs can be constructed respectively from a UV Lagrangian description of the N = 4 theory or of its mirror. We further conjecture that the VOAs associated to an N = 4 SQFT inherit properties of the theory which only emerge in the IR, such as enhanced global symmetries. Thus knowledge of the VOAs should allow one to compute the spaces of supersymmetric ground states for a theory coupled to supersymmetric background connections for the full symmetry group of the IR SCFT. In particular, we propose a conformal field theory description of the spaces of ground states for the T[SU(N)] theories. These theories play a role of S-duality kernel in maximally supersymmetric SU(N) gauge theory and thus the corresponding spaces of supersymmetric ground states should provide a kernel for the Geometric Langlands duality for special unitary groups.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the structure of certain protected operator algebras that arise in three-dimensional N=4 superconformal field theories, and they found that these algebra can be understood as a quantization of (either of) the half-BPS chiral ring(s).
Abstract: We investigate the structure of certain protected operator algebras that arise in three-dimensional N=4 superconformal field theories. We find that these algebras can be understood as a quantization of (either of) the half-BPS chiral ring(s). An important feature of this quantization is that it has a preferred basis in which the structure constants of the quantum algebra are equal to the OPE coefficients of the underlying superconformal theory. We identify several nontrivial conditions that the quantum algebra must satisfy in this basis. We consider examples of theories for which the moduli space of vacua is either the minimal nilpotent orbit of a simple Lie algebra or a Kleinian singularity. For minimal nilpotent orbits, the quantum algebras (and their preferred bases) can be uniquely determined. These algebras are related to higher spin algebras. For Kleinian singularities the algebras can be characterized abstractly - they are spherical subalgebras of symplectic reflection algebras - but the preferred basis is not easily determined. We find evidence in these examples that for a given choice of quantum algebra (defined up to a certain gauge equivalence), there is at most one choice of canonical basis. We conjecture that this is the case for general N=4 SCFTs.

Journal ArticleDOI
TL;DR: For a nontrivial locally compact group, this paper showed that the Banach algebras of a topological dynamical system can be represented on an operator algebra if and only if p = 2.
Abstract: For a nontrivial locally compact group $G$, and $p\in [1,\infty)$, consider the Banach algebras of $p$-pseudofunctions, $p$-pseudomeasures, $p$-convolvers, and the full group $L^p$-operator algebra. We show that these Banach algebras are operator algebras if and only if $p=2$. More generally, we show that for $q\in [1,\infty)$, these Banach algebras can be represented on an $L^q$-space if and only if one of the following holds: (a) $p=2$ and $G$ is abelian; or (b) $|\frac 1p - \frac 12|=|\frac 1q - \frac 12|$. This result can be interpreted as follows: for $p,q\in [1,\infty)$, the $L^p$- and $L^q$-representation theories of a group are incomparable, except in the trivial cases when they are equivalent. As an application, we show that, for distinct $p,q\in [1,\infty)$, if the $L^p$ and $L^q$ crossed products of a topological dynamical system are isomorphic, then $\frac 1p + \frac 1q=1$. In order to prove this, we study the following relevant aspects of $L^p$-crossed products: existence of approximate identities, duality with respect to $p$, and existence of canonical isometric maps from group algebras into their multiplier algebras.

Journal ArticleDOI
TL;DR: In this article, a new class of infinite-dimensional Lie algebras, called Lax operator algesas, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface.
Abstract: A new class of infinite-dimensional Lie algebras, called Lax operator algebras, is presented, along with a related unifying approach to finite-dimensional integrable systems with a spectral parameter on a Riemann surface such as the Calogero-Moser and Hitchin systems. In particular, the approach includes (non-twisted) Kac-Moody algebras and integrable systems with a rational spectral parameter. The presentation is based on quite simple ideas about the use of gradings of semisimple Lie algebras and their interaction with the Riemann-Roch theorem. The basic properties of Lax operator algebras and the basic facts about the theory of the integrable systems in question are treated (and proved) from this general point of view. In particular, the existence of commutative hierarchies and their Hamiltonian properties are considered. The paper concludes with an application of Lax operator algebras to prequantization of finite-dimensional integrable systems. Bibliography: 51 titles.

Journal ArticleDOI
TL;DR: In this article, the Hao-Ng isomorphism for the reduced crossed product and all discrete groups was established by calculating the C*-envelope of an operator algebra by a discrete group.
Abstract: Using non-selfadjoint techniques, we establish the Hao-Ng isomorphism for the reduced crossed product and all discrete groups. For the full crossed product an analogous result holds for all discrete groups but the C*-correspondences involved have to be hyperrigid. These results are obtained by calculating the C*-envelope of the reduced crossed product of an operator algebra by a discrete group.

Posted Content
TL;DR: In this article, the authors use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d = 4$ superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets.
Abstract: We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d ${\cal N} = 4$ superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2- and 3-point functions of Higgs branch operators in the full 3d ${\cal N}=4$ theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d ${\cal N} = 2$ subalgebra of the ${\cal N}=4$ algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on $S^3$ that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.

Journal ArticleDOI
TL;DR: In this article, the authors define operator algebras internal to a rigid C*-tensor category and prove the analog of the Stinespring dilation theorem, which unifies the definitions of analytic properties for discrete quantum groups and rigid C *tensor categories.
Abstract: In this article, we define operator algebras internal to a rigid C*-tensor category $\mathcal{C}$. A C*/W*-algebra object in $\mathcal{C}$ is an algebra object $\mathbf{A}$ in $\operatorname{ind}$-$\mathcal{C}$ whose category of free modules ${\sf FreeMod}_{\mathcal{C}}(\mathbf{A})$ is a $\mathcal{C}$-module C*/W*-category respectively. When $\mathcal{C}={\sf Hilb_{f.d.}}$, the category of finite dimensional Hilbert spaces, we recover the usual notions of operator algebras. We generalize basic representation theoretic results, such as the Gelfand-Naimark and von Neumann bicommutant theorems, along with the GNS construction. We define the notion of completely positive maps between C*-algebra objects in $\mathcal{C}$ and prove the analog of the Stinespring dilation theorem. As an application, we discuss approximation and rigidity properties, including amenability, the Haagerup property, and property (T) for a connected W*-algebra $\mathbf{M}$ in $\mathcal{C}$. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories.

Posted Content
TL;DR: In this article, a quantitative algorithm for computing K-theory for the class of filtered C *-algebras with asymptotic finite nuclear decomposition is presented.
Abstract: In this paper, we apply quantitative operator K-theory to develop an algorithm for computing K-theory for the class of filtered C *-algebras with asymptotic finite nuclear decomposition. As a consequence, we prove the K{u}nneth formula for C *-algebras in this class. Our main technical tool is a quantitative Mayer-Vietoris sequence for K-theory of filtered C *-algebras.

Journal ArticleDOI
TL;DR: The main goal of this paper is to find operator algebra variants of certain deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and Stormer as discussed by the authors concerning projections on C*-algebras and their ranges.
Abstract: The main goal of this paper is to find operator algebra variants of certain deep results of Stormer, Friedman and Russo, Choi and Effros, Effros and Stormer, Robertson and Youngson, Youngson, and others, concerning projections on C*-algebras and their ranges. (See papers of these authors referenced in the bibliography.) In particular we investigate the `bicontractive projection problem' and related questions in the category of operator algebras. To do this, we will add the ingredient of `real positivity' from recent papers of the first author with Read.

Journal ArticleDOI
TL;DR: In this paper, the connection between algebra homomorphisms defined on subalgebras of the Banach algebra l 1(N 0) and fractional versions of Cesaro sums of a linear operator T ∈ B(X) is established.
Abstract: Let X be a complex Banach space. The connection between algebra homomorphisms defined on subalgebras of the Banach algebra l 1(N0) and fractional versions of Cesaro sums of a linear operator T ∈ B(X) is established. In particular, we show that every (C, α)-bounded operator T induces an algebra homomorphism — and it is in fact characterized by such an algebra homomorphism. Our method is based on some sequence kernels, Weyl fractional difference calculus and convolution Banach algebras that are introduced and deeply examined. To illustrate our results, improvements to bounds for Abel means, new insights on the (C, α)-boundedness of the resolvent operator for temperated a-times integrated semigroups, and examples of bounded homomorphisms are given in the last section.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every nonlinear *-Lie derivation δ of 𝒜 is automatically linear and δ is an inner *-derivation.
Abstract: Let ℋ be an infinite dimensional complex Hilbert space and 𝒜 be a standard operator algebra on ℋ which is closed under the adjoint operation. We prove that every nonlinear *-Lie derivation δ of 𝒜 is automatically linear. Moreover, δ is an inner *-derivation.