Operator Algebras

We provide an exposition and review of the theory of Hilbert algebras. We show that every right Hilbert algebra of a left Hilbert algebra is also a left Hilbert algebra. We prove Tomita's Fundamental Theorem which posits that for every generalized Hilbert algebra A, there exists a modular Hilbert algebra B which is equivalent to A. We thence prove the existence and properties of a conjugate linear unitary involutive operator J on H, which is the completion of A. We, in particular, show that the left von Neumann algebra, L(A) of A, is anti-isomorphic to its commutant. We also establish that the tensor product of two modular Hilbert algebras A is a modular Hilbert algebra, and hence, we prove that the left von Neumann algebra, generated by the tensor product of two modular Hilbert algebras is the tensor product of the von Neumann algebras generated by each of the modular Hilbert algebras. Furthermore, if A
is a family of modular Hilbert algebras, then the algebraic direct sum of A is also a modular Hilbert algebra with the left von Neumann algebra

We show that every von Neumann algebra endowed with a seminite trace has a canonical
completed or full Hilbert algebra associated with it; and conversely a completed Hilbert
algebra of exactly this form can be canonically associated with every von Neumann algebra. It is shown that if M is a von Neumann algebra acting on a Hilbert space H and if M admits a
separating and generating vector in H, then there exists a modular Hilbert algebra B such
that M is spatially isomorphic to the left von Neumann algebra L(B) of B. Therefore, there
exists an isometric involution J in H such that JMJ = M.

Questo libro fornisce una introduzione completa ai metodi matematici che sono a fondamento della meccanica quantistica e che possono servire come prodromi per la teoria quantistica dei campi.

Nella prima parte si riassumono alcune nozioni di base di algebra, topologia e teoria della misura.

Nella seconda parte si discute la teoria degli operatori continui e non, con particolare enfasi sulla teoria spettrale e delle rappresentazioni, e sulle proprietà delle algebre di von Neumann e C*.

Nella terza parte si introducono i fondamenti della teoria quantistica in finiti gradi di libertà e la seconda quantizzazione, illustrando il materiale precedente.

Il testo origina da note: non vi figurano esercizi ma molti esempi.

"Il lavoro che qui si presenta racchiude le ricerche da me svolte col sostegno della borsa di studio di Dottorato di Ricerca presso il  Dipartimento di Matematica U. Dini dell'Universitµa di Firenze.
Si tratta di una indagine che è possibile riassumere secondo tre linee principali: l'applicazione dei concetti classici della Geometria Differenziale alle varietà di Poisson; l'introduzione di nuovi invarianti analitici per queste varietà e le loro generalizzazioni algebriche; l'estensione del calcolo integrale alle varietà di Poisson."

The purpose of this thesis is to analyse the Hilbert Space requirement for Quantum Mechanics. In particular, we justify sharp observables but question the requirement of completeness of the inner product space and the underlying Öeld. We view our mathematical framework as a dynamical theory but with a mysterious probabilistic interpretation instead of the otherway round. Whenever we speak of "Quantum Mechanics", we mean Non-relativistic Quantum Mechanics. To make things less messy, we assume associativity through-out. No attempt has been made to refer to QFT and statistical quantum mechanics and we use conventional mathematical symbols instead of Diracís formalism.

After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectral triples), we present a "panoramic view" of the status of our current research program on the use of categorical methods in the setting of A.Connes' non-commutative geometry: morphisms/categories of spectral triples, categorification of Gel'fand duality. We conclude with a summary of the expected applications of "categorical non-commutative geometry" to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.

Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows to expand the range of applications of cyclic cohomology. It is the goal of the present paper to illustrate these recent developments, with special emphasis on the application to transverse index theory, and point towards future directions. In particular, we highlight the remarkable accord between our framework for cyclic cohomology of Hopf algebras on one hand and both the algebraic as well as the analytic theory of quantum groups on the other, manifest in the construction of the modular square.

The Gelfand - Na\u{i}mark theorem supplies the one to one correspondence between commutative $C^*$-algebras and locally compact Hausdorff spaces. So any noncommutative $C^*$-algebra can be regarded as a generalization of a topological space.  Generalizations of several topological invariants can be defined by algebraical methods. This article contains a pure algebraical construction of inverse limits in the category of (noncommutative) covering projections. It is proven that Moyal planes are  inverse limits of covering projections of noncommutative tori.

A satisfactory marriage between higher categories and operator algebras has never been achieved: although (monoidal) C*-categories have been systematically used since the development of the theory of superselection sectors, higher category theory has more recently evolved along lines closer to classical higher homotopy.
We present axioms for strict involutive n-categories (a vertical
categorification of dagger categories) and a definition for strict
higher C*-categories and Fell bundles (possibly equipped with
involutions of arbitrary depth), that were developed in
collaboration with Roberto Conti, Wicharn Lewkeeratiyutkul and Noppakhun Suthichitranont.

In this paper, we investigate some properties of Toeplitz matrices with respect to different matrix products. We also give some results regarding circulant matrices, skew-circulant matrices and approximation by Toeplitz matrices over the field of complex numbers.

To each quantum system, described by a von Neumann algebra of physical quantities, we associate a complete bi-Heyting algebra. The elements of this algebra represent contextualised propositions about the values of the physical quantities of the quantum system.

Let $A$ be a dense Fr\'echet *-subalgebra of a C*-algebra $B$. (We do not require Fr\'echet algebras to be $m$-convex.)  Let $G$ be a Lie group, not necessarily connected, which acts on both $A$ and $B$ by *-automorphisms, and let $\s$ be a submultiplicative function from $G$ to the nonnegative real numbers.  If $\s$ and the action of $G$ on $A$ satisfy certain simple properties,  we define a
dense Fr\'echet *-subalgebra $G\rtimes^{\s} A$ of the crossed product $L^{1}(G, B)$.  Our algebra consists of differentiable $A$-valued functions on $G$, rapidly vanishing in $\s$.

We give conditions on the action of $G$ on $A$ which imply
the $m$-convexity of the dense subalgebra $G\rtimes^{\s}A$.  A locally convex algebra is said to be $m$-convex if there is a family of submultiplicative seminorms for the topology of the algebra.  The property of $m$-convexity is  important  for a Fr\'echet algebra, and is useful in modern operator theory.

If $G$ acts as a transformation group on a manifold $M$, we develop a class of dense subalgebras for the crossed product
$L^{1}(G, C_{0}(M))$, where $C_{0}(M)$ denotes the continuous functions on $M$ vanishing at infinity with the sup norm topology.  We define Schwartz functions $\Cal S(M)$ on $M$, which are differentiable with respect to some group action on $M$, and are rapidly vanishing with respect to some scale on $M$.  We then form a dense $m$-convex Fr\'echet *-subalgebra $G\rtimes^{\s}\Cal S (M)$ of rapidly vanishing, $G$-differentiable functions from $G$ to $\Cal S(M)$.

If the reciprocal of $\s$ is in $L^{p}(G)$ for some $p$, we prove that our group algebras $\Cal S^{\s}(G)$ are nuclear Fr\'echet spaces, and that $G\rtimes^{\s}A$ is the projective completion $\Cal S^{\s}(G) {\widehat \otimes} A$.

We give a short and very general proof of the fact that the property of a dense Fr\'echet subalgebra of a Banach algebra being local, or closed under the holomorphic functional calculus in the Banach algebra, is preserved by tensoring with the $n\times n$ matrix algebra of the complex numbers.

In "Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order" [arXiv:1208.4724], it was shown that self-adjoint operators affiliated with a von Neumann algebra N can equivalently be described as certain real-valued functions on the projection lattice P(N) of the algebra, which we call q-observable functions. Here, we show that q-observable functions can be interpreted as generalised quantile functions for quantum observables interpreted as random variables. More generally, when L is a complete meet-semilattice, we show that L-valued cumulative distribution functions and the corresponding L-quantile functions form a Galois connection. An ordinary CDF can be written as an L-CDF composed with a state. For classical probability, one picks L=B(\Omega), the complete Boolean algebra of measurable subsets modulo null sets of a measurable space \Omega. For quantum probability, one uses L=P(N), the projection lattice of a nonabelian von Neumann algebra N. Moreover, using some constructions from the topos approach to quantum theory, we show that there is a joint sample space for all quantum observables, despite no-go results such as the Kochen-Specker theorem. Specifically, the spectral presheaf \Sigma\ of a von Neumann algebra N, which is not a mere set, but a presheaf (i.e., a 'varying set'), plays the role of the sample space. The relevant meet-semilattice L in this case is the complete bi-Heyting algebra of clopen subobjects of \Sigma. We show that using the spectral presheaf \Sigma\ and associated structures, quantum probability can be formulated in a way that is structurally very similar to classical probability.

We provide a conceptual discussion and physical interpretation of some of the quite abstract constructions in the topos approach to physics. In particular, the daseinisation process for projection operators and for self-adjoint operators is motivated and explained from a physical point of view. Daseinisation provides the bridge between the standard Hilbert space formalism of quantum theory and the new topos-based approach to quantum theory. As an illustration, we will show all constructions explicitly for a three-dimensional Hilbert space and the spin-z operator of a spin-1 particle. This article is a companion to the article by Isham in the same volume.

The topos approach to the formulation of physical theories includes a new form of quantum logic. We present this topos quantum logic, including some new results, and compare it to standard quantum logic, all with an eye to conceptual issues. In particular, we show that topos quantum logic is distributive, multi-valued, contextual and intuitionistic. It incorporates superposition without being based on linear structures, has a built-in form of coarse-graining which automatically avoids interpretational problems usually associated with the conjunction of propositions about incompatible physical quantities, and provides a material implication that is lacking from standard quantum logic. Importantly, topos quantum logic comes with a clear geometrical underpinning. The representation of pure states and truth-value assignments are discussed. It is briefly shown how mixed states fit into this approach.

The Kochen-Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type I_n factor as algebra of observables, including I_infinity. Afterwards, we give a proof of the Kochen-Specker theorem for an arbitrary von Neumann algebra R without summands of types I_1 and I_2, using a known result on two-valued measures on the projection lattice P(R). Some connections with presheaf formulations as proposed by Isham and Butterfield are made.

The Novikov-Shubin numbers are defined for open manifolds with bounded geometry, the -trace of Atiyah being replaced by a s emicontinuous semifinite trace on the C � -algebra of almost local operators. It is proved that they are invariant under quasi-isometries and, making use of the theory of singular traces for C � -algebras developed in [29], they are interpreted as asymptotic dimensions since, in analogy with what happens in Connes’ noncommutative geometry, they indicate which power of the Laplacian gives rise to a singular trace. Therefore, as in geometric measure theory, these numbers furnish the order of infinitesimal giving rise to a non trivial measure. The dimensional interpretation is strenghtened in the case of the 0-th Novikov-Shubin invariant, which is shown to coincide, under suitable geometric conditions, with the asymptotic counterpart of the box dimension of a metric space. Since this asymptotic dimension coincides with the polynomial growth of a discrete group, th...

These two mathematical operators (D and 1/D) define the differential and integration-term in turn. This module covers the solution of homogeneous and linear ODE using conventional and parameter variation (I & II) methods. Beyond thah, the partial differential equations (PDE), both linear-homogeneous and-non homogeneous are also studies being ended-up with some applications related to the mechanical, electrical and block diagrams systems.

Abstract: General semifinite factor representations of the diffeomorphism group of euclidean space are constructed by means of a canonical correspondence with the finite factor representations of the inductive limit unitary group. This construction includes the quasi-free representations of the canonical commutation and anti-commutation relations. To establish this correspondence requires a non-linear form of complete positivity as developed by Arveson. We also compare the asymptotic character formula for the unitary group with the ...

We present a duality between the category of compact Riemannian spin manifolds (equipped with a given spin bundle and charge conjugation) with isometries as mor-phisms and a suitable “metric” category of spectral triples over commutative pre-C*-algebras. We ...

Fields Institute Communications Volume 39. 2nn3 Category Theory for Conformal Boundary Conditions Jiirgen Puchs Institutionen for fysik Karlstad University Universitetsgatan 1. S-65188 Karlstad ifuchs&fuchs. tekn. kau. se Christoph Schweigert Fachbereich Mathematik. ...

We define the notion of strong spectral invariance for a dense Frechet subalgebra A of a Banach algebra B. We show that if A is strongly spectral invariant in a C*-algebra B, and G is a compactly generated polynomial growth Type R Lie group, not necessarily connected, then the smooth crossed product G\rtimes A is spectral invariant in the C*-crossed product G\rtimes B. Examples of such groups are given by finitely generated polynomial growth discrete groups, compact or connected nilpotent Lie groups, the group of Euclidean motions on the plane, the Mautner group, or any closed subgroup of one of these. Our theorem gives the spectral invariance of G\rtimes A if A is the set of C^{\infty}-vectors for the action of G on B, or if B= C_{0}(M), and A is a set of G-differentiable Schwartz functions S(M) on M. This gives many examples of spectral invariant dense subalgebras for the C*-algebras associated with dynamical systems. We also obtain relevant results about exact sequences, subalgebras, tensoring by smooth compact operators, and strong spectral invariance in L_{1}(G, B).

For von Neumann algebras M, N not isomorphic to C^2 and without type I_2 summands, we show that for an order-isomorphism f:AbSub(M)->AbSub(N) between the posets of abelian von Neumann subalgebras of M and N, there is a unique Jordan *-isomorphism g:M->N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of a von Neumann algebra not isomorphic to C^2 and without type I_2 summands is determined by the poset of its abelian subalgebras, and has implications in recent approaches to foundational issues in quantum mechanics.

After a brief introduction to the spectral presheaf, which serves as an analogue of state space in the topos approach to quantum theory, we show that every state of the von Neumann algebra of physical quantities of a quantum system determines a certain measure on the spectral presheaf of the system. The so-called clopen subobjects of the spectral presheaf play the role of measurable sets. Measures on the spectral presheaf can be characterised abstractly, and the main result is that every abstract measure induces a unique state of the von Neumann algebra. Finally, we show how quantum-theoretical expectation values can be calculated from measures associated to quantum states.

We define smooth generalized crossed products and prove six-term exact sequences of Pimsner-Voiculescu type.This sequence may, in particular, be applied to smooth subalgebras of the Quantum Heisenberg Manifolds in order to compute the generators of their cyclic cohomology. Our proof is based on a combination of arguments from the setting of (Cuntz-)Pimsner algebras and the Toeplitz proof of Bott-periodicity.

We show that a $C^\star$-algebra $B$ contains a dense left or right Fr\'echet ideal $A$, with $A$ a nuclear locally convex space, if and only if the primitive ideal space Prim$(B)$ of $B$ is discrete and countable, and $B/I$ is finite dimensional for each $I \in $ Prim$(B)$.  We show the forward implication holds for a general Banach algebra $B$, if the ideal is assumed two-sided.  For $C^\star$-algebras, we construct all two-sided dense nuclear ideals by defining a set of matrix-valued Schwartz functions on the countable discrete space Prim$(B)$.