Crossed Product Algebra Research Articles

Noncommutative differential geometry on crossed product algebras

We provide a differential structure on arbitrary cleft extensions B:=AcoH⊆A for an H-comodule algebra A. This is achieved by constructing a covariant calculus on the corresponding crossed product algebra B#σH from the data of a bicovariant calculus on the structure Hopf algebra H and a calculus on the base algebra B, which is compatible with the 2-cocycle and measure of the crossed product. The result is a quantum principal bundle with canonical strong connection and we describe the induced bimodule covariant derivatives on associated bundles of the crossed product. All results specialize to trivial extensions and smash product algebras B#H and we give a characterization of the smash product calculus in terms of the differentials of the cleaving map j:H→A and the inclusion B↪A. The construction is exemplified for pointed Hopf algebras. In particular, the case of Radford Hopf algebras H(r,n,q) is spelled out in detail.

Quantum reference frames from top-down crossed products

All physical observations are made relative to a reference frame, which is a system in its own right. If the system of interest admits a group symmetry, the reference frame observing it must transform commensurately under the group to ensure the covariance of the combined system. We point out that the crossed product is a way to realize quantum reference frames from the bottom-up; adjoining a quantum reference frame and imposing constraints generates a crossed product algebra. We provide a top-down specification of crossed product algebras and show that one cannot obtain inequivalent quantum reference frames using this approach. As a remedy, we define an abstract algebra associated to the system and symmetry group built out of relational crossed product algebras associated with different choices of quantum reference frames. We term this object the G-framed algebra, and show how potentially inequivalent frames are realized within this object. We comment on this algebra’s analog of the classical Gribov problem in gauge theory, its importance in gravity where we show that it is relevant for semiclassical de Sitter and potentially beyond the semiclassical limit, and its utility for understanding the frame dependence of physical notions like observables, density states, and entropies. Published by the American Physical Society 2024

Crossed products, conditional expectations and constraint quantization

Recent work has highlighted the importance of crossed products in correctly elucidating the operator algebraic approach to quantum field theories. In the gravitational context, the crossed product simultaneously promotes von Neumann algebras associated with subregions in diffeomorphism covariant quantum field theories from type III to type II, and provides the necessary ingredients to gravitationally dress operators, thereby enforcing the constraints of the theory. In this note we enhance the crossed product construction to the context of general gauge theories with arbitrary combinations of internal and spacetime local symmetries. This is done by leveraging the correspondence between the crossed product and the extended phase space. We then undertake a detailed study of constraint quantization from the perspective of a generic crossed product algebra. We study and compare four distinct approaches to constraint quantization from this point of view: refined algebraic quantization, BRST quantization, path integral quantization, and the commutation theorem for crossed products. Far from simply reproducing existing analyses, the operator algebraic viewpoint sheds new light on old problems by reformulating the dressing of operators in terms of conditional expectations and other closely related projection maps. We conclude by applying our approach to the constraint quantization of three distinct gauge theories including a discussion of gravity on null hypersurfaces.

Algebras and their covariant representations in quantum gravity

We study a physically motivated representation of an algebra of operators in gravitational and non gravitational theories called the covariant representation of an algebra. This is a representation where the symmetries of the operator algebra are implemented unitarily on the Hilbert space. We emphasize the very close similarity of this representation to the crossed product of an algebra. In fact, as an example of (and sometimes identified with) a covariance algebra, the crossed product of an algebra is in one to one correspondence with the covariant representation of the algebra. This will in turn illuminate physically what the crossed product algebra is in the context of quantum gravity.

Crossed product algebras and generalized entropy for subregions

An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III_{1}1, in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II_∞∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.

Hochschild homology of reductive $p$-adic groups

Consider a reductive p -adic group G , its (complex-valued) Hecke algebra \mathcal H (G) , and the Harish-Chandra–Schwartz algebra \mathcal S (G) . We compute the Hochschild homology groups of \mathcal H (G) and of \mathcal S (G) , and we describe the outcomes in several ways. Our main tools are algebraic families of smooth G -representations. With those we construct maps from HH_{n} (\mathcal H (G)) and HH_{n} (\mathcal S(G)) to modules of differential n -forms on affine varieties. For n = 0 , this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) G -representations. It is known from [J. Algebra 606 (2022), 371–470] that every Bernstein ideal \mathcal H (G)^{\mathfrak s} of \mathcal H (G) is closely related to a crossed product algebra of the form \mathcal O (T)\rtimes W . Here \mathcal O (T) denotes the regular functions on the variety T of unramified characters of a Levi subgroup L of G , and W is a finite group acting on T . We make this relation even stronger by establishing an isomorphism between HH_{*} (\mathcal H (G)^{\mathfrak s}) and HH_{*} (\mathcal O (T)\rtimes W) , although we have to say that in some cases it is necessary to twist \mathbb{C} [W] by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal \mathcal S (G)^{\mathfrak s} of \mathcal S (G) is isomorphic to HH_{*} (C^{\infty} (T_{u})\rtimes W) , where T_{u} denotes the Lie group of unitary unramified characters of L . In these pictures of HH_{*} (\mathcal H (G)) and HH_{*} (\mathcal S (G)) , we also show how the Bernstein centre of \mathcal H (G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of \mathcal H (G) and of \mathcal S (G) and we relate that to topological K-theory.

On the hereditarity and maximality of weak crossed product orders over discrete valuation rings

Consider a weak crossed product order [Formula: see text] in the crossed product algebra [Formula: see text], where [Formula: see text] is the integral closure of a discrete valuation ring [Formula: see text] in a tamely ramified Galois extension [Formula: see text] of the field of fractions of [Formula: see text]. Assume that [Formula: see text] is local. In this paper, we develop a method that can help to determine whether [Formula: see text] is a maximal order by looking for certain ideals in [Formula: see text] where [Formula: see text] is the inertia group. We demonstrate this method by classifying specific examples of hereditary weak crossed product orders as being either maximal or nonmaximal among the [Formula: see text]-orders of [Formula: see text] and we give a criterion for determining maximality of [Formula: see text] when [Formula: see text] is cyclic.

Decomposition of idempotent 2-cocycles

Let $L$ be a finite Galois field extension of $K$ with Galois group $G$. We decompose any idempotent 2-cocycle $f$ using finite sequences of descending two-sided ideals of the corresponding weak crossed product algebra $A_f$. We specialize the results in

Double crossed biproducts and related structures

Let H be a bialgebra. Let be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action Let be a linear map, where B is a right H-comodule coalgebra, and an algebra with a right H-weak action In this paper, we improve the necessary conditions for the two-sided crossed product algebra and the two-sided smash coproduct coalgebra to form a bialgebra (called double crossed biproduct) such that the condition in Majid’s double biproduct (or double-bosonization) is one of the necessary conditions. On the other hand, we provide a more general two-sided crossed product algebra structure via Brzezński’s crossed product and give some applications.

A thermodynamically consistent time integration scheme for non-linear thermo-electro-mechanics

The aim of this paper is the design of a new one-step implicit and thermodynamically consistent Energy–Momentum (EM) preserving time integration scheme for the simulation of thermo-electro-elastic processes undergoing large deformations. The time integration scheme takes advantage of the notion of polyconvexity and of a new tensor cross product algebra. These two ingredients are shown to be crucial for the design of so-called discrete derivatives fundamental for the calculation of the second Piola–Kirchhoff stress tensor, the entropy and the electric field. In particular, the exploitation of polyconvexity and the tensor cross product, enable the derivation of comparatively simple formulas for the discrete derivatives. This is in sharp contrast to much more elaborate discrete derivatives which are one of the main downsides of classical EM time integration schemes. The newly proposed scheme inherits the advantages of EM schemes recently published in the context of thermo-elasticity and electro-mechanics, whilst extending to the more generic case of nonlinear thermo-electro-mechanics. Furthermore, the manuscript delves into suitable convexity/concavity restrictions that thermo-electro-mechanical strain energy functions must comply with in order to yield physically and mathematically admissible solutions. Finally, a series of numerical examples will be presented in order to demonstrate robustness and numerical stability properties of the new EM scheme.

A Cuntz-Pimsner model for the C⁎-algebra of a graph of groups

We provide a Cuntz-Pimsner model for graph of groups C⁎-algebras. This allows us to compute the K-theory of a range of examples and show that graph of groups C⁎-algebras can be realised as Exel-Pardo algebras. We also make a preliminary investigation of whether the crossed product algebra of Baumslag-Solitar groups acting on the boundary of certain trees satisfies Poincaré duality in KK-theory. By constructing a K-theory duality class we compute the K-homology of these crossed products.

The Rokhlin property for inclusions of $C^*$-algebras

Let P⊂A be an inclusion of σ-unital C∗-algebras with a finite index in the sense of Pimsner–Popa. Then we introduce the Rokhlin property for a conditional expectation E from A onto P and show that if A is simple and satisfies any of the property (1)–(12) listed in the below, and E has the Rokhlin property, then so does P: simplicity; nuclearity; C∗-algebras that absorb a given strongly self-absorbing C∗-algebra 𝒟; C∗-algebras of stable rank one; C∗-algebras of real rank zero; C∗-algebras of nuclear dimension at most n, where n∈ℤ+; C∗-algebras of decomposition rank at most n, where n∈ℤ+; separable simple C∗-algebras that are stably isomorphic to AF algebras; separable simple C∗-algebras that are stably isomorphic to AI algebras; separable simple C∗-algebras that are stably isomorphic to AT algebras; separable simple C∗-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes; separable C∗-algebras with strict comparison of positive elements. In particular, when α:G→ Aut(A) is an action of a finite group G on A with the Rokhlin property in the sense of Nawata, the properties (1)–(12) are inherited to the fixed point algebra Aα and the crossed product algebra A⋊αG from A. In the case of a finite index inclusion of unital C∗-algebras P⊂A if the conditional expectation E:A→P has the Rokhlin property in the sense of Izumi, the previous results except (11) are observed in previous works.

A new energy–momentum time integration scheme for non-linear thermo-mechanics

The aim of this paper is the design a new one-step implicit and thermodynamically consistent Energy–Momentum (EM) preserving time integration scheme for the simulation of thermo-elastic processes undergoing large deformations and temperature fields. Following Bonet et al. (2020), we consider well-posed constitutive models for the entire range of deformations and temperature. In that regard, the consideration of polyconvexity inspired constitutive models and a new tensor cross product algebra are shown to be crucial in order to derive the so-called discrete derivatives, fundamental for the construction of the algorithmic derived variables, namely the second Piola–Kirchoff stress tensor and the entropy (or the absolute temperature). The proposed scheme inherits the advantages of the EM scheme recently published by Franke et al. (2018), whilst resulting in a simpler scheme from the implementation standpoint. A series of numerical examples will be presented in order to demonstrate the robustness and applicability of the new EM scheme. Although the examples presented will make use of a temperature-based version of the EM scheme (using the Helmholtz free energy as the thermodynamical potential and the temperature as the thermodynamical state variable), we also include in an Appendix an entropy-based analogue EM scheme (using the internal energy as the thermodynamical potential and the entropy as the thermodynamical state variable).

The block Schur product is a Hadamard product

Given two $n \times n $ matrices $A = (a_{ij})$ and $B=(b_{ij}) $ with entries in $B(H)$ for some Hilbert space $H$, their block Schur product is the $n \times n$ matrix $ A\square B := (a_{ij}b_{ij})$. Given two continuous functions $f$ and $g$ on the torus with Fourier coefficients $(f_n)$ and $(g_n)$ their convolution product $f \star g$ has Fourier coefficients $(f_n g_n)$. Based on this, the Schur product on scalar matrices is also known as the Hadamard product.
 We show that for a C*-algebra $\mathcal{A} $, and a discrete group $G$ with an action $\alpha _g$ of $G$ on $\mathcal{A} $ by *-automorphisms, the reduced crossed product C*-algebra $\mathrm {C}^*_r(\mathcal{A} , \alpha , G)$ possesses a natural generalization of the convolution product, which we suggest should be named the Hadamard product.
 We show that this product has a natural Stinespring representation and we lift some known results on block Schur products to this setting, but we also show that the block Schur product is a special case of the Hadamard product in a crossed product algebra.

Hecke operators in $KK$-theory and the $K$-homology of Bianchi groups

Let \Gamma be a torsion-free arithmetic group acting on its associated global symmetric space X . Assume that X is of non-compact type and let \Gamma act on the geodesic boundary \partial X of X . Via general constructions in KK -theory, we endow the K -groups of the arithmetic manifold X / \Gamma , of the reduced group C^* -algebra C^*_r(\Gamma) and of the boundary crossed product algebra C(\partial X) \rtimes\Gamma with Hecke operators. In the case when \Gamma is a group of real hyperbolic isometries, the K -theory and K -homology groups of these C^{*} -algebras are related by a Gysin six-term exact sequence and we prove that this Gysin sequence is Hecke equivariant. Finally, when \Gamma is a Bianchi group, we assign explicit unbounded Fredholm modules (i.e. spectral triples) to (co)homology classes, inducing Hecke-equivariant isomorphisms between the integral cohomology of \Gamma and each of these K -groups. Our methods apply to case \Gamma \subset \mathbf {PSL}(\mathbf Z) as well. In particular we employ the unbounded Kasparov product to push the Dirac operator an embedded surface in the Borel–Serre compactification of \mathbf H/\Gamma to a spectral triple on the purely infinite geodesic boundary crossed product algebra C(\partial \mathbf H) \rtimes\Gamma .

Dynamics of compact quantum metric spaces

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. Moreover, we provide a flexible set of assumptions ensuring that a continuous family of$\ast$-automorphisms of a compact quantum metric space yields a field of crossed product algebras which varies continuously in Rieffel’s quantum Gromov–Hausdorff distance. Finally, we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney$C^{1}$-topology.

Smooth crossed product of minimal unique ergodic diffeomorphisms of a manifold and cyclic cohomology

Different diffeomorphisms can give the same [Formula: see text] crossed product algebra. Our main purpose is to show that we can still classify dynamical systems with some appropriate smooth crossed product algebras when their corresponding [Formula: see text] crossed product algebras are isomorphic. For this purpose, we construct two minimal unique ergodic diffeomorphisms [Formula: see text] and [Formula: see text] of [Formula: see text]. The [Formula: see text] algebras classification theory, smooth crossed product algebras considered by R. Nest and cyclic cohomology are used to show that [Formula: see text] and [Formula: see text] give the same [Formula: see text] algebra and induce different smooth crossed product algebras.

K-automorphisms of a weak-crossed product F-algebra over a Galois extension K/F

We compute the K-automorphism group of AK=K⊗FAf of a weak crossed product algebra Af for a weak 2-cocycle f over a Galois extension K/F with Galois group G. The K-automorphism group of AK decomposes into its unipotent part and the reductive part Hˆ. Automorphisms in Hˆ are computed via their restriction to semi-simple part of AK. There is a strong relationship between Hˆ and the lower-subtractive relation (≤) induced by f on G. We introduce a subgroup of Hˆ, namely Λ, which contains interesting combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lower-subtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph.

The ideal structure of Steinberg algebras

Given an ample action of an inverse semigroup on a locally compact and Hausdorff topological space, we study the ideal structure of the crossed product algebra associated with it. By developing a theory of induced ideals, we manage to prove that every ideal in the crossed product algebra may be obtained as the intersection of ideals induced from isotropy group algebras. This can be interpreted as an algebraic version of the Effros-Hahn conjecture. Finally, as an application of our result, we study the ideal structure of a Steinberg algebra associated with an ample groupoid by interpreting it as an inverse semigroup crossed product algebra.

SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

AbstractFor any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.