Noncommutative differential geometry, quantization, and smooth symmetries of the \(C^*\)-algebras associated to topological dynamics
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 decomposition problem for derivations in \(C^*\)-algebras, and (b) the problem of finding the smooth Lie action on a given \(C^*\)- algebra. The main results are: ``to give an explicit Lie bracket on each of the \(C^*\)-algebras (i)-(iii) which set up a Lie isomorphism to the commutator Lie algebra \({\mathcal D}\) of all smooth derivations in \({\mathfrak A}.\) For (ii) we (the author) intoduce a noncommutative symbol mapping on \({\mathcal D}\) and a Hilbert space theoretic notion of approximately innerness for derivations where the approximation is given with respect to a new inner product on \({\mathcal D}\). This inner product, in turn is defined in terms of the quantization isomorphism. We then show that a given derivation \(\delta\) is approximately inner (in the Hilbert norm) iff the symbol of \(\delta\) vanihes identically. The quantization mapping is used in the study of Lie actions on \({\mathcal O}_ A\) (case (iii)) and we show that such actions are determined by two curvature obstructions.''
noncommutative tori associated to antisymmetric bicharacters
decomposition problem for derivations in \(C^*\)-algebras
finding the smooth Lie action on a given \(C^*\)-algebra
scientific article; zbMATH DE number 4138559
scientific article