Two dimensional unital Riesz algebras, their representations and norms

We describe all two dimensional unital Riesz algebras and study representations of them in Riesz algebras of regular operators. Although our results are not complete, we do demonstrate that very varied behaviour can occur even though all these algebras can be given a Banach lattice algebra norm.

STOPPING THE CCR FLOW AND ITS ISOMETRIC COCYCLES

It is shown how to use non-commutative stopping times in order to stop the CCR flow of arbitrary index and also its isometric cocycles, i.e. left operator Markovian cocycles on Boson Fock space. Stopping the CCR flow yields a homomorphism from the semigroup of stopping times, equipped with the convolution product, into the semigroup of unital endomorphisms of the von Neumann algebra of bounded operators on the ambient Fock space. The operators produced by stopping cocycles themselves satisfy a cocycle relation.

A probabilistic index for completely positive maps and an application

Gohm, Rolf, (2003) 'A probabilistic index for completely positive maps and an application', Journal of Operator Theory 54(2) pp.339-361 RAE2008

Spectrally bounded operators from von Neumann algebras

We prove that every unital spectrally bounded operator from a properly infinite von Neumann algebra onto a semisimple Banach algebra is a Jordan homomorphism.

The maximal C*-algebra of quotients as an operator bimodule

We establish a description of the maximal C*-algebra of quotients of a unital C*-algebra A as a direct limit of spaces of completely bounded bimodule homomorphisms from certain operator submodules of the Haagerup tensor product of A with itself labelled by the essential closed right ideals of A into A. In addition the invariance of the construction of the maximal C*-algebra of quotients under strong Morita equivalence is proved.

Jordan isomorphism of purely infinite C*-algebras

We prove that every unital bounded linear mapping from a unital purely infinite C*-algebra of real rank zero into a unital Banach algebra which preserves elements of square zero is a Jordan homomorphism. This entails a description of unital surjective spectral isometries as the Jordan isomorphisms in this setting.

Spectral isometries on non-simple C*-algebras

We prove that unital surjective spectral isometries on certain non-simple unital C*-algebras are Jordan isomorphisms. Along the way, we establish several general facts in the setting of semisimple Banach algebras.

Chaotic Banach algebras

We construct an infinite dimensional non-unital Banach algebra $A$ and $a\in A$ such that the sets $\{za^n:z\in\C,\ n\in\N\}$ and $\{({\bf 1}+a)^na:n\in\N\}$ are both dense in $A$, where $\bf 1$ is the unity in the unitalization $A^{\#}=A\oplus \spann\{{\bf 1}\}$ of $A$. As a byproduct, we get a hypercyclic operator $T$ on a Banach space such that $T\oplus T$ is non-cyclic and $\sigma(T)=\{1\}$.

Crossed products by twisted partial actions and graded algebras

For a twisted partial action e of a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A x(Theta) G is proved to be associative. Given a G-graded k-algebra B = circle plus(g is an element of G) B-g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B-1 x(Theta) G for some twisted partial action of G on B-1. The equality BgBg-1 B-g = B-g (for all g is an element of G) is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G. (c) 2008 Elsevier Inc. All rights reserved.

CLASSIFICATION OF SUBALGEBRAS OF THE CAYLEY ALGEBRA OVER A FINITE FIELD

We classify all unital subalgebras of the Cayley algebra O(q) over the finite field F(q), q = p(n). We obtain the number of subalgebras of each type and prove that all isomorphic subalgebras are conjugate with respect to the automorphism group of O(q). We also determine the structure of the Moufang loops associated with each subalgebra of O(q).

Structurable superalgebras of Cartan type

We describe the simple Lie superalgebras arising from the unital structurable superalgebras of characteristic 0 and construct four series of the unital simple structurable superalgebras of Cartan type. We give a classification of simple structurable superalgebras of Cartan type over an algebraically closed field F of characteristic 0. Together with the Faulkner theorem on the classification of classical such superalgebras, it gives a classification of the simple structurable superalgebras over F. Crown Copyright (C) 2010 Published by Elsevier Inc. All rights reserved.

Lie algebras and triple systems

This thesis is dedicated to the Tits-Kantor-Koecher (TKK) construction which establishes a bijective correspondence between unital Jordan algebras and shortly graded Lie algebras with Z-grading induced by an sl_2-triple. It is based on the observation that if g is a Lie algebra with a short Z-grading and f lies in g_1, then the formula ab=[[a,f],b] defines a structure of a Jordan algebra on g_{-1}. The TKK construction has been extended to Jordan triple systems and, more recently, to the so-called Kantor triple systems. These generalizations are studied in the thesis.

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.

Lorentz invariant intrinsic decoherence

We present a Lorentz invariant extension of a previous model for intrinsic decoherence (Milburn 1991 Phys. Rev. A 44 5401). The extension uses unital semigroup representations of space and time translations rather than the more usual unitary representation, and does the least violence to physically important invariance principles. Physical consequences include a modification of the uncertainty principle and a modification of field dispersion relations, similar to modifications suggested by quantum gravity and string theory, but without sacrificing Lorentz invariance. Some observational signatures are discussed.