Papers by Aristides Katavolos
Computers & Mathematics with Applications, 1998
3 Spectral Theory 6 3.1 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... more 3 Spectral Theory 6 3.1 The spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 Gelfand theory for commutative C*-algebras . . . . . . . . . . . . . . . . . . 9 3.3 Functional calculus and spectral theorem . . . . . . . . . . . . . . . . . . . . 13 3.3.1 The Continuous Functional Calculus for selfadjoint operators . . . . . 13 3.3.2 Connection with Gelfand Theory . . . . . . . . . . . . . . . . . . . . 14 3.3.3 The Spectral Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Canadian Journal of Mathematics, 1982
Let M be a w*-algebra (Von Neumann algebra), τ a semifinite, faithful, normal trace on M. There e... more Let M be a w*-algebra (Von Neumann algebra), τ a semifinite, faithful, normal trace on M. There exists a w*-dense (i.e., dense in the σ(M, M *)-topology, where M * is the predual of M) *-ideal J of M such that τ is a linear functional on J, and (where |x| = (x*x)1/2) is a norm on J. The completion of J in this norm is Lp (M, τ) (see [2], [8], [7], and [4]). If M is abelian, in which case there exists a measure space (X, μ) such that M = L ∞(X, μ), then Lp (X, τ) is isometric, in a natural way, to Lp (X, μ). A natural question to ask is whether this situation persists if M is non-abelian. In a previous paper [5] it was shown that it is not possible to have a linear mapping
Canadian Journal of Mathematics, 1976
If the Lp spaces of two measure spaces are “the same”, to what extent can we identify the underly... more If the Lp spaces of two measure spaces are “the same”, to what extent can we identify the underlying measure spaces? This question has been partially answered by Schneider [7] (extending results of Forelli [2]). He proves that a linear isometry between the Lv spaces of two finite measure spaces is in fact an (isometric) homomorphism between the corresponding L ∞ spaces, if it preserves the identity.
Canadian Journal of Mathematics, 1981
1. The central objects in integration theory can be considered to be an abelian Von Neumann algeb... more 1. The central objects in integration theory can be considered to be an abelian Von Neumann algebra, L ∞, of the measure space, together with a (not necessarily finite-valued) positive linear functional on it, the integral (see [10]). It is natural, therefore, to attempt to construct a “non-commutative” integration theory starting with a non-abelian Von Neumann algebra. Segal [9] and Dixmier [2] have developed such a theory, and constructed the Non-Commutative Lp spaces associated with a Von Neumann algebra M and a normal, faithful, semifinite trace (i.e. a unitarily invariant weight) t on M. They show that there exists a unique ultra-weakly dense *-ideal J of M such that t (extends to) a positive linear form on J . A generalisation of the Hölder inequality then shows that, for 1 ≦ p < ∞, the function is a norm on J, denoted by || • ||p.
Bulletin of the London Mathematical Society, 1995
Starting with a left ideal J of L^1(G) we consider its annihilator J^ in L^∞(G) and the generated... more Starting with a left ideal J of L^1(G) we consider its annihilator J^ in L^∞(G) and the generated VN(G)-bimodule in B(L^2(G)), Bim(J^). We prove that Bim(J^)=( Ran J)^ when G is weakly amenable discrete, compact or abelian, where Ran J is a suitable saturation of J in the trace class. We define jointly harmonic functions and jointly harmonic operators and show that, for these classes of groups, the space of jointly harmonic operators is the VN(G)-bimodule generated by the space of jointly harmonic functions. Using this, we give a proof of the following result of Izumi and Jaworski - Neufang: the non-commutative Poisson boundary is isomorphic to the crossed product of the space of harmonic functions by G.
We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of... more We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are able to easily give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if P is a normal bimodule idempotent and ‖P ‖ < 2 / √ 3 then P is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.
If K,L and M are (closed) subspaces of a Banach space X sat- isfying K \ M = (0), K _ L = X and L... more If K,L and M are (closed) subspaces of a Banach space X sat- isfying K \ M = (0), K _ L = X and L M, then P = {(0),K,L,M,X} is a pentagon subspace lattice on X. If P1 and P2 are pentagons, every (al- gebraic) isomorphism ' : AlgP1 ! AlgP2 is quasi-spatial. The SOT-closure of the fin-
![Research paper thumbnail of O A ] 1 6 A ug 2 00 1 THE JACOBSON RADICAL FOR ANALYTIC CROSSED PRODUCTS](https://attachments.academia-assets.com/77676417/thumbnails/1.jpg)
We characterise the Jacobson radical of an analytic crossed product C0(X) ×φ Z+, answering a ques... more We characterise the Jacobson radical of an analytic crossed product C0(X) ×φ Z+, answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the Jacobson radical of analytic crossed products C0(X)×φ Z+. This consists of all elements whose ‘Fourier coefficients’ vanish on the recurrent points of the dynamical system (and the first one is zero). The multi-dimensional version requires a variation of the notion of recurrence, taking into account the various degrees of freedom. There is a rich interplay between operator algebras and dynamical systems, going back to the founding work of Murray and von Neumann in the 1930’s. Crossed product constructions continue to provide fundamental examples of von Neumann algebras and C-algebras. Comparatively recently, Arveson [1] in 1967 introduced a nonselfadjoint crossed product construction, called the analytic crossed product or the semicrossed product, which has the remarkable property of capturing all of the inf...
Studia Mathematica
Starting with a left ideal J of L 1 (G) we consider its annihilator J ⊥ in L ∞ (G) and the genera... more Starting with a left ideal J of L 1 (G) we consider its annihilator J ⊥ in L ∞ (G) and the generated VN(G)-bimodule in B(L 2 (G)), Bim(J ⊥). We prove that Bim(J ⊥) = (Ran J) ⊥ when G is weakly amenable discrete, compact or abelian, where Ran J is a suitable saturation of J in the trace class. We define jointly harmonic functions and jointly harmonic operators and show that, for these classes of groups, the space of jointly harmonic operators is the VN(G)-bimodule generated by the space of jointly harmonic functions. Using this, we give a proof of the following result of Izumi and Jaworski-Neufang: the noncommutative Poisson boundary is isomorphic to the crossed product of the space of harmonic functions by G.
Operator Theory: Advances and Applications, 2013
Mathematical Proceedings of the Cambridge Philosophical Society
We examine the common null spaces of families of Herz–Schur multipliers and apply our results to ... more We examine the common null spaces of families of Herz–Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in [1] can be used to give a short proof as well as a generalisation of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other.
The Fourier binest algebra is de ned as the intersection of the Volterra nest algebra on L 2 (IR)... more The Fourier binest algebra is de ned as the intersection of the Volterra nest algebra on L 2 (IR) with its conjugate by the Fourier transform. Despite the absence of nonzero nite rank operators this algebra is equal to the closure in the weak operator topology of the Hilbert-Schmidt bianalytic pseudo-di erential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp(?isx 2 =2) for s > 0. This multinest is the re exive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identi cation the unitary automorphism group of the algebra is determined as the semi-direct product IR 2 IR for the action t (;) =
Contemporary Mathematics, 1991
ABSTRACT Let M and N be nonzero subspaces of a Hilbert space H satisfying M∩N={0} and M∨N=H and l... more ABSTRACT Let M and N be nonzero subspaces of a Hilbert space H satisfying M∩N={0} and M∨N=H and let T∈ℬ(H). Consider the question: If T leaves each of M and N invariant, respectively, intertwines M and N, does T decompose as a sum of two operators with the same property and each of which, in addition, annihilates one of the subspaces? If the angle between M and N is positive the answer is affirmative. If the angle is zero, the answer is still affirmative for finite rank operators but there are even trace class operators for which it is negative. An application gives an alternative proof that no distance estimate holds for the algebra of operators leaving M and N invariant if the angle is zero, and an analogous result is obtained for the set of operators intertwining M and N.
Operator Algebras and Applications, 1997
The purpose of this talk is twofold. In the first part (sections 1-4) I will briefly describe the... more The purpose of this talk is twofold. In the first part (sections 1-4) I will briefly describe the notions of generalised reflexivity and strong reflexivity for linear space of operators, as well as the problem of the density of the rank one subspace. The second part is devoted to a presentation of recent joint work with John Erdos and Victor Shulman [9] concerning reflexive subspaces admitting actions of masas. The perhaps surprising solution of the rank one density problem will be given, and a new "simultaneous coordinatisation" of such subspaces will be presented. This will be given in measure-theoretic terms, and so the blanket assumption of separability of all Hilbert spaces will be made (although many results, particularly in the first part, are valid generally). The results in the first part are mostly known, apart from a few exceptions (Theorem 2.2, for example); the treatment is somewhat new 1. Let me emphasise that the objects in this talk are linear spaces of operators acting on a Hilbert space. The basic notions we are interested in (reflexivity, rank one operators etc.) are not invariant under isomorphism, but only under spatial isomorphism. Thus the notion of equivalence will be unitary equivalence, not some more general notion of isomorphism. Why linear spaces and not (unital) algebras? As we will see, many results in operator algebra theory do not depend essentially on multiplication of operators; furthermore, in some problems concerning operator algebras,
Uploads
Papers by Aristides Katavolos