Normalizers of Operator Algebras and Reflexivity

The set of normalizers between von Neumann (or, more generally, reflexive) algebras A and B, (that is, the set of all operators x such that xAx* is a subset of B and x*Bx is a subset of A) possesses `local linear structure': it is a union of reflexive linear spaces. These spaces belong to the interesting class of normalizing linear spaces, namely, those linear spaces U for which UU*U is a subset of U. Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form U={x:xp=h(p)x, for all p in P}, where P is a set of projections and h a certain map defined on P. A normalizing space consists of normalizers between appropriate von Neumann algebras A and B. Necessary and sufficient conditions are found for a normalizing space to consist of normalizers between two reflexive algebras. Normalizing spaces which are bimodules over maximal abelian selfadjoint algebras consist of operators `supported' on sets of the form [f=g] where f and g are appropriate Borel functions. They also satisfy spectral synthesis in the sense of Arveson.Comment: 20 pages; to appear in the Proceedings of the London Mathematical Societ

Operator algebras from the discrete Heisenberg semigroup

We study reflexivity and structure properties of operator algebras generated by representations of the discrete Heisenberg semi-group. We show that the left regular representation of this semi-group gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation which gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of H^{\infty}(\bb{T})\otimes\cl B(\cl H)

On the ranges of bimodule projections

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 $\|P\| < 2/\sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps.Comment: Please refer to the journal for the final versio

The Jacobson radical for analytic crossed products

We characterise the (Jacobson) radical of the analytic crossed product of C_0(X) by the non-negative integers (Z_+), answering a question first raised by Arveson and Josephson in 1969. In fact, we characterise the radical of analytic crossed products of C_0(X) by (Z_+)^d. The radical 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.Comment: 17 pages; AMS-LaTeX; minor correction

Topics in non-commutative probability theory with applications to statistical mechanics

Chapter I contains a presentation of Non-Commutative Integration theory. The relation Between Segal's and Nelson's definition of measurability is investigated, and a new proof of duality for non-commutative probability Lp spaces is given. In chapter II, known results on isometries between Banach spaces of functions and operators are presented, and a new proof of the fact that unit-preserving isometries of abelian C* algebras are *-isomorphisms is given. It is shown that unit-preserving *-isometries between non-commutative probability L p spaces come from Jordan *-homomorphisms and several conclusions are drawn. Chapter III is a presentation of Tomita-Takesaki theory. Possible generalizations are pointed out, and the Radon-Nikodym theorem is discussed. In chapter IV the characterization of equilibrium in Quantum Statistical Mechanics by the KMS condition is investigated. In chapter V, a class of Gibbs states wbeta is defined on the algebra M of the canonical commutation relations in infinitely many degrees of freedom. This is done by showing that for any beta > 0 the second quantization H of a hamiltonian with positive polynomially bounded discrete spectrum defines a nuclear operator exp(-betaH) from Fock space into g, a generalization of Schwartz space for infinitely many variables. This allows the construction of an "almost modular" Hiblert subalgebra on which the modular automorphisms may he defined, and satisfy the KMS condition. The final chapter contains a proof of a commutation theorem, namely that the commutant of in the GNS representation induced by wg is invariant under the modular automorphisms, and is isomorphic to its own commutant via an antiunitary involution of the GNS Hilbert space. This is done by showing that pi beta is unitarily equivalent to left multiplication on Hilbert-Schmidt operators on Pock space, acting on a suitable tensor product of g with Fock space.<p

Ideals of the Fourier algebra, supports and harmonic operators

We examine the common null spaces of families of HerzSchurmultipliers and apply our results to study jointly harmonic operatorsand their relation with jointly harmonic functionals. We show howan annihilation formula obtained in [1] can be used to give a short proofas well as a generalisation of a result of Neufang and Runde concerningharmonic operators with respect to a normalised positive definite function.We compare the two notions of support of an operator that havebeen studied in the literature and show how one can be expressed interms of the other

Reflexivity of the translation-dilation algebras on L^2(R)

The hyperbolic algebra A_h, studied recently by Katavolos and Power, is the weak star closed operator algebra on L^2(R) generated by H^\infty(R), as multiplication operators, and by the dilation operators V_t, t \geq 0, given by V_t f(x) = e^{t/2} f(e^t x). We show that A_h is a reflexive operator algebra and that the four dimensional manifold Lat A_h (with the natural topology) is the reflexive hull of a natural two dimensional subspace.Comment: 10 pages, no figures To appear in the International Journal of Mathematic

Tensor products of subspace lattices and rank one density

We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. As a consequence, we show that the Lattice Tensor Product Formula holds for \Alg M and any other reflexive operator algebra and give several further corollaries of these results.Comment: 15 page

Compactness properties of operator multipliers

We continue the study of multidimensional operator multipliers initiated in [arXiv:math/0701645]. We introduce the notion of the symbol of an operator multiplier. We characterise completely compact operator multipliers in terms of their symbol as well as in terms of approximation by finite rank multipliers. We give sufficient conditions for the sets of compact and completely compact multipliers to coincide and characterise the cases where an operator multiplier in the minimal tensor product of two C*-algebras is automatically compact. We give a description of multilinear modular completely compact completely bounded maps defined on the direct product of finitely many copies of the C*-algebra of compact operators in terms of tensor products, generalising results of Saar