Noncommutative differential geometry, quantization, and smooth symmetries of the \(C^*\)-algebras associated to topological dynamics
type: Item
instance of: scholarly article
title: Noncommutative differential geometry, quantization, and smooth symmetries of the \(C^*\)-algebras associated to topological dynamics
zbMATH Open document ID: 0695.46030
DOI: 10.1007/BF01194558
published in: Integral Equations and Operator Theory
publication date: 1989-01-01T00:00:00Z
review text: A given Lie algebra \(\{{\mathfrak A}^{\infty}\{.,.\})\) is said to be of \(C^*\)-type (Definition 2.1) if: (i) there exists a \(C^*\)-algebra completion \({\mathfrak A}\) of \({\mathfrak A}^{\infty},\) (ii) a completely positive, unital linear mapping \(\sigma: {\mathfrak A}\to {\mathfrak A},\) (iii) a linear mapping \(\pi: {\mathfrak A}^{\infty}\to Der({\mathfrak A}^{\infty})\) such that \(\{a_ 1,a_ 2\}=K_{\pi}(a_ 1,a_ 2)- K_{\sigma}(a_ 1,a_ 2)\) \((a_ 1,a_ 2\in {\mathfrak A}^{\infty})\) and \((\pi_{a^*}(b))^*=-\pi_ a(b^*)\) (a,b\(\in {\mathfrak A}^{\infty})\), where by definition \(K_{\pi}(a_ 1,a_ 2)=\pi_{a_ 1}(a_ 2)-\pi_{a_ 2}(a_ 1)-[a_ 1,a_ 2],\) \(K_{\sigma}(a_ 1,a_ 2)=\sigma ([a_ 1,a_ 2])-[\sigma (a_ 1),\sigma (a_ 2)].\) In the case where \({\mathfrak A}^{\infty}={\mathbb{C}}^{\infty}(M,\omega)\), * is the complex conjugation, \(\sigma\) a connection form, \(\pi =\nabla\), \(\{\).,.\(\}\) is the usual Poisson bracket, we have a classical Hamiltonian system. The classical quantization problem is to find out a quantum correspondence \(a\mapsto \hat a\) from the functions into operators satisfying the relations \(\{a,b\}\) \({\hat{\;}}=1/\hslash [\hat a,\hat b]\) and \({\hat 1}=Id.\) In the operator situation one says that \({\mathfrak A}^{\infty}\) has a quantization if this can be done in such a way that the quantization mapping \(\pi\) : \({\mathfrak A}^{\infty}\to Der({\mathfrak A}^{\infty})\) becomes an isomorphism in the category of Lie algebras. In the paper under reviewing the author considers three classes of \(C^*\)-algebras: (i) noncommutative tori associated to antisymmetric bicharacters, (ii) the Cuntz algebra \({\mathcal O}_ d\) on d generators \(s_ 1,...,s_ d\) with the relations \(s^*_ is_ j=\delta_{ij}1\), \(s_ is^*_ i=1\) and (iii) the Cuntz-Krieger algebras \({\mathcal O}_ A\) on the modified relations \(s^*_ is_ i=\sum a_{ij}s_ js^*_ j\), where \(A=(a_{ij})\) is a given 0-1 matrix. The main problems concerned in this paper are: (a) the decomp…
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
Mathematics Subject Classification ID: 53D50
Mathematics Subject Classification ID: 81S10
zbMATH DE Number: 4138559
zbMATH Keywords: Lie algebra
zbMATH Keywords: \(C^*\)-type
zbMATH Keywords: connection form
zbMATH Keywords: Poisson bracket
zbMATH Keywords: quantization problem
zbMATH Keywords: noncommutative tori associated to antisymmetric bicharacters
zbMATH Keywords: Cuntz algebra
zbMATH Keywords: Cuntz-Krieger algebras
zbMATH Keywords: decomposition problem for derivations in \(C^*\)-algebras
zbMATH Keywords: finding the smooth Lie action on a given \(C^*\)-algebra
zbMATH Keywords: explicit Lie bracket
author: Palle Jorgensen
reviewed by: Do Ngoc Diep
MaRDI profile type: Publication
Wikidata QID: Q115393931
cites work: Spectral resolution groups of unitary operators
cites work: Derivations Tangential to Compact Group Actions: Spectral Conditions in the Weak Closure
cites work: Unbounded derivations tangential to compact groups of automorphisms. II
cites work: A dichotomy for derivations on \({\mathcal O}_ n\)
cites work: Unbounded derivations tangential to compact groups of automorphisms
cites work: Derivations commuting with abelian gauge actions on lattice systems
cites work: Q3854185
cites work: On an automorphic action of U(n,1) on \({\mathcal O}_ n\)
cites work: The 𝐶*-algebra generated by an isometry
cites work: Simple \(C^*\)-algebras generated by isometries
cites work: A class of C*-algebras and topological Markov chains
cites work: Q4726889
cites work: On \(O_ n\).
cites work: Lie algebras of unbounded derivations
cites work: Q3897665
cites work: The integrability problem for infinite-dimensional representations of finite-dimensional Lie algebras
cites work: Q3331753
cites work: New results on unbounded derivations and ergodic groups of automorphisms
cites work: Q3905952
cites work: Q4180804
cites work: C*-algebras associated with irrational rotations
cites work: On One-Parameter Subgroups of *-Automorphisms on Operator Algebras and the Corresponding Unbounded Derivations
cites work: Diophantine approximation
cites work: Irreducible representations of operator algebras
cites work: On factor representations and the \(C^ *\)-algebra of canonical commutation relations
cites work: On a problem of Sakai in unbounded derivations
cites work: Q3803670
full work available at URL: https://doi.org/10.1007/bf01194558
OpenAlex ID: W2329230263
description: scientific article; zbMATH DE number 4138559