Narrow Results

The notion of quasi-product actions of a compact group on a C${}^{\ast }$-algebra was introduced by Bratteli et al. in their attempt to seek an equivariant analogue of Glimm’s characterization of non-type I C${}^{\ast }$-algebras. We show that a faithful minimal action of a second countable compact group on a separable C${}^{\ast }$-algebra is quasi-product whenever its fixed point algebra is simple. This was previously known only for compact abelian groups and for profinite groups. Our proof relies on a subfactor technique applied to finite index inclusions of simple C${}^{\ast }$-algebras in the purely infinite case, and also uses ergodic actions of compact groups in the general case. As an application, we show that if moreover the fixed point algebra is a Kirchberg algebra, such an action is always isometrically shift-absorbing, and hence is classifiable by the equivariant KK-theory due to a recent result of Gabe-Szabó.

Almost flat finitely generated projective Hilbert $C^{\ast }$-module bundles were successfully used by Hanke and Schick to prove special cases of the Strong Novikov Conjecture. Dadarlat later showed that it is possible to calculate the index of a $K$-homology class $\eta \in K_{\ast }(M)$ twisted with an almost flat bundle in terms of the image of $\eta$ under Lafforgue’s assembly map and the almost representation associated with the bundle. Mishchenko used flat infinite-dimensional bundles equipped with a Fredholm operator in order to prove special cases of the Novikov higher signature conjecture. We show how to generalize Dadarlat’s theorem to the case of an infinite-dimensional bundle equipped with a continuous family of Fredholm operators on the fibers. Along the way, we show that special cases of the Strong Novikov Conjecture can be proven if there exist sufficiently many almost flat bundles with Fredholm operator. To this end, we introduce the concept of an asymptotically flat Fredholm bundle and its associated asymptotic Fredholm representation, and prove an index theorem which relates the index of the asymptotic Fredholm bundle with the so-called asymptotic index of the associated asymptotic Fredholm representation.

We consider how the outputs of the Kadison transitivity theorem and Gelfand–Naimark–Segal (GNS) construction may be obtained in families when the initial data are varied. More precisely, for the Kadison transitivity theorem, we prove that for any nonzero irreducible representation $({\mathscr{H}},\pi )$ of a $C^{\ast }$-algebra $𝔄$ and $n\in \mathbb{N}$, there exists a continuous function $A:X\to 𝔄$ such that $\pi (A(x,y))x_i=y_i$ for all $i\in \{1,\dots ,n\}$, where $X$ is the set of pairs of $n$-tuples $(x,y)\in {{\mathscr{H}}}^n\times {{\mathscr{H}}}^n$ such that the components of $x$ are linearly independent. Versions of this result where $A$ maps into the self-adjoint or unitary elements of $𝔄$ are also presented. Regarding the GNS construction, we prove that given a topological $C^{\ast }$-algebra fiber bundle $p:𝔄\to Y$, one may construct a topological fiber bundle $\mathcal{𝒫}(𝔄)\to Y$ whose fiber over $y\in Y$ is the space of pure states of ${𝔄}_y$ (with the norm topology), as well as bundles $\mathcal{\mathscr{H}}\to \mathcal{𝒫}(𝔄)$ and $\mathcal{𝒩}\to \mathcal{𝒫}(𝔄)$ whose fibers ${\mathcal{\mathscr{H}}}_{\omega }$ and ${\mathcal{𝒩}}_{\omega }$ over $\omega \in \mathcal{𝒫}(𝔄)$ are the GNS Hilbert space and closed left ideal, respectively, corresponding to $\omega$. When $p:𝔄\to Y$ is a smooth fiber bundle, we show that $\mathcal{𝒫}(𝔄)\to Y$ and $\mathcal{\mathscr{H}}\to \mathcal{𝒫}(𝔄)$ are also smooth fiber bundles; this involves proving that the group of ∗-automorphisms of a $C^{\ast }$-algebra is a Banach Lie group. In service of these results, we review the topology and geometry of the pure state space. A simple non-interacting quantum spin system is provided as an example illustrating the physical meaning of some of these results.

We study the radiation of photons from a classical charged particle. We particularly consider a situation where the particle has a constant velocity in the distant past, then is accelerated, and then has a constant velocity in the distant future. Starting with no photons in the distant past we seek to characterize the quantum state of the photon field in the distant future. Working in the Coulomb gauge and in a $C^{\ast }$-algebra formulation, we give sharp conditions on whether this state is or is not in Fock space.

We prove that discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of $C^{\ast }$-algebras are again free via a Fell-topology characterization for $C^{\ast }$-field continuity, recovering a result of Blanchard’s in a somewhat more general setting.

Generalizing the well-known correspondence between two-sided adjunctions and Frobenius algebras, we establish a one-to-one correspondence between local adjunctions of $C^{\ast }$-correspondences, as defined and studied in prior work with Clare and Higson; and Frobenius $C^{\ast }$-algebras, a natural $C^{\ast }$-algebraic adaptation of the standard notion of Frobenius algebras that we introduce here.

We formulate a notion of the quantum automorphism group of a $2$-graph. After some preliminary computations, we define quantum isomorphism between a pair of $2$-graphs. We produce a “nontrivial” example of a pair of $2$-graphs that are not quantum isomorphic to each other.

In this paper, we establish a strong connection between groups and gyrogroups, which provides the machinery for studying gyrogroups via group theory. Specifically, we prove that there is a correspondence between the class of gyrogroups and a class of triples with components being groups and twisted subgroups. This in particular provides a construction of a gyrogroup from a group with an automorphism of order two that satisfies the uniquely 2-divisible property. We then present various examples of such groups, including the general linear groups over $ℝ$ and $ℂ$, the Clifford group of a Clifford algebra, the Heisenberg group on a module, and the group of units in a unital C${}^{\ast }$-algebra. As a consequence, we derive polar decompositions for the groups mentioned previously.

Let ${𝒜}$ and $𝔄$ be two $C^{\ast }$-algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible action of $𝔄$ on ${𝒜}$. We define ${𝒜}⋊𝔄$ as a $C^{\ast }$-algebra, where it is a strongly splitting $C^{\ast }$-algebra extension of $𝔄$ by ${𝒜}$. Normal, self-adjoint, unitary, invertible and projection elements of ${𝒜}⋊𝔄$ are characterized; sufficient and necessary conditions for existing unit and bounded approximate identity of ${𝒜}⋊𝔄$ as a Banach algebra and as a $C^{\ast }$-algebra are given. We characterize ∗-automorphisms on ${𝒜}⋊𝔄$ and give some results related to ∗-homomorphisms, ∗-representations and completely bounded maps on this $C^{\ast }$-algebra. Also, we have constructed a new Hilbert $C^{\ast }$-module $X⋊Y$ over ${𝒜}⋊𝔄$, where $X$ is a Hilbert $C^{\ast }$-module over ${𝒜}$ and $Y$ is a Hilbert $C^{\ast }$-module over $𝔄$.

For Hilbert spaces $H$ and $K_i(i\in 𝕁)$, we use the notations $B_H^g({\{K_j\}}_{j\in 𝕁})$, $F_H^g({\{K_j\}}_{j\in 𝕁})$ and $R_H^g({\{K_j\}}_{j\in 𝕁})$ to denote the sets of all $g$-Bessel sequences, $g$-frames and Riesz bases in $H$ with respect to ${\{K_j\}}_{j\in 𝕁}$, respectively. By defining a linear operation and a norm, we prove that $B_H^g({\{K_j\}}_{j\in 𝕁})$ becomes a Banach space and is isometrically isomorphic to the operator space $B(H,K)$, where $K={\oplus }_{j\in 𝕁}{K}_j$. In light of operator theory, it is proved that $F_H^g({\{K_j\}}_{j\in 𝕁})$ and $R_H^g({\{K_j\}}_{j\in 𝕁})$ are open sets in $B_H^g({\{K_j\}}_{j\in 𝕁})$. This implies that both $g$-frames and Riesz bases are stable under a small perturbation. By introducing a linear bijection $\pi$ from $B_H^g({\{K_j\}}_{j\in 𝕁})$ onto the $C^{\ast }$-algebra $B(H)$, a multiplication and an involution on the Banach space $B_H^g({\{K_j\}}_{j\in 𝕁})$ are defined so that $B_H^g({\{K_j\}}_{j\in 𝕁})$ becomes a unital $C^{\ast }$-algebra that is isometrically isomorphic to the $C^{\ast }$-algebra $B(H)$, provided that $H$ and $K$ are isomorphic.

This paper collects and extends the lectures I gave at the “XXIV International Fall Workshop on Geometry and Physics” held in Zaragoza (Spain) during September 2015. Within these lectures I review the formulation of Quantum Mechanics, and quantum theories in general, from a mathematically advanced viewpoint, essentially based on the orthomodular lattice of elementary propositions, discussing some fundamental ideas, mathematical tools and theorems also related to the representation of physical symmetries. The final step consists of an elementary introduction the so-called ($C^{\ast }$-) algebraic formulation of quantum theories.

The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, To appear in IJGMMP (2019)] and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analyzed. For that, the algebra generated by the observables, as well as the notion of state, are discussed, and the structure of the transition functions, that plays an instrumental role in Schwinger’s picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures is discussed.

Schwinger’s algebra of selective measurements has a natural interpretation in the formalism of groupoids. Its kinematical foundations, as well as the structure of the algebra of observables of the theory, were presented in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, Int. J. Geom. Meth. Mod. Phys. (2019), arXiv: 1905.12274 [math-ph]. https://doi.org/10.1142/S0219887819501196. F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics II: Algebras and observables, Int. J. Geom. Meth. Mod. Phys. (2019). https://doi.org/10.1142/S0219887819501366]. In this paper, a closer look to the statistical interpretation of the theory is taken and it is found that an interpretation in terms of Sorkin’s quantum measure emerges naturally. It is proven that a suitable class of states of the algebra of virtual transitions of the theory allows to define quantum measures by means of the corresponding decoherence functionals. Quantum measures satisfying a reproducing property are described and a class of states, called factorizable states, possessing the Dirac–Feynman “exponential of the action” form are characterized. Finally, Schwinger’s transformation functions are interpreted similarly as transition amplitudes defined by suitable states. The simple examples of the qubit and the double slit experiment are described in detail, illustrating the main aspects of the theory.

In this paper, we will present the main features of what can be called Schwinger’s foundational approach to Quantum Mechanics. The basic ingredients of this formulation are the selective measurements, whose algebraic composition rules define a mathematical structure called groupoid, which is associated with any physical system. After the introduction of the basic axioms of a groupoid, the concepts of observables and states, statistical interpretation and evolution are derived. An example is finally introduced to support the theoretical description of this approach.

The groupoid description of Schwinger’s picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger’s groupoid of the system. The groupoid picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, as well as the notion of free independence, find a natural realm in the groupoid formalism. The ideas described in this paper will be illustrated by using the EPRB experiment. It will be observed that, in addition to the notion of the non-separability provided by the entangled state of the system, there is an intrinsic “non-separability” associated to the impossibility of identifying the entangled particles as subsystems of the total system.

We investigate to what extent a nilpotent Lie group is determined by its $C^{\ast }$-algebra. We prove that, within the class of exponential Lie groups, direct products of Heisenberg groups with abelian Lie groups are uniquely determined even by their unitary dual, while nilpotent Lie groups of dimension $\le 5$ are uniquely determined by the Morita equivalence class of their $C^{\ast }$-algebras. We also find that this last property is shared by the filiform Lie groups and by the $6$-dimensional free two-step nilpotent Lie group.

Let ${𝒜}$ and ${ℬ}$ be two unital $C^{\ast }$-algebras with unit $I$. It is shown that the mapping $\varphi :{{𝒜}}_s\to {{ℬ}}_s$ which preserves arithmetic mean and Jordan triple product is a difference of two Jordan homomorphisms provided that $0\in \mathrm{\text{Ran}}\varphi$. The structure of $\varphi$ is more refined when $\varphi (I)\ge 0$ or $\varphi (I)\le 0$. Furthermore, if ${𝒜}$ is a $C^{\ast }$-algebra of real rank zero and $\varphi :{𝒜}\to {ℬ}$ is additive and preserves absolute value of product, then $\varphi ={\varphi }_1\oplus {\varphi }_2$ such that ${\varphi }_1$ (respectively, ${\varphi }_2$) is a complex linear (respectively, antilinear) ∗-homomorphism.

In this paper, we introduce the notion of ∗-$K$-operator frame as a generalization of the notion of $K$-operator frame and we study the corresponding frame operator. It is completed by a result on the frame operator of the tensor product of two frame operators.

We introduce the notion of 2-Hilbert $\mathcal{𝒜}$ modules and give some characterizations. The key idea is to consider a 2-Hilbert space instead of a Hilbert space. We employ our results to obtain reverse Cauchy–Schwarz and triangle inequality in this setting.

The aim of this paper is to introduce a new class of generalized centralizers on $C^{\ast }$-algebras which is termed as extended $(k_1,k_2)$-centralizer. We study different properties of this class of centralizers. Moreover, we define a class of $C^{\ast }$-algebras which contains the prime $C^{\ast }$-algebras as a subclass. Different properties of extended centralizers with respect to this class of $C^{\ast }$-algebras are discussed with suitable examples.