##
**Coactions and Yang-Baxter equations for ergodic actions and subfactors.**
*(English)*
Zbl 0809.46079

Operator algebras and applications. Vol 2: Mathematical physics and subfactors, Pap. UK-US Jt. Semin., Warwick/UK 1987, Lond. Math. Soc. Lect. Note Ser. 136, 203-236 (1988).

[For the entire collection see Zbl 0668.00015.]

The aim of this paper is to bring into evidence the usefulness of considering not only actions of compact groups on operator algebras, but also the dual notion of coaction. A lot of what we say is contained in much greater detail in three series of papers, due to be published in the near future: four papers on ergodic actions, two on product type actions and two on equivariant \(K\)-theory. These all have their rather primitive origins in the three chapters of my thesis. The other main proponent of coactions of compact groups is A. Ocneanu, and we have occasion to make frequent reference to his still unpublished work.

We briefly summarize the contents of this paper. In Section II we recall the basic definitions of coactions of compact groups on von Neumann and \(C^*\)-algebras. We present two examples of \(C^*\)-algebras which arise perhaps unexpectedly as crossed products by coactions, and show how this observation can be used to explore their structure. The basic idea here is an old one: to use symmetry properties to simplify and elucidate computations. In Section III we exhibit two general principles in equivariant \(KK\)-theory, namely Frobenius reciprocity and Dirac induction. When combined with the equivariant Thom isomorphism, these lead to a generalization of a spectral theorem of Hodgkin (for the \(K\)- theory of spaces) which in principle provides a homological machine whereby ordinary \(KK\)-theory (of a pair of algebras) can be deduced from equivariant \(KK\)-theory. In particular, if \(G\) is a compact Lie group which is both connected and simply connected, this leads one to suspect that if \(A\) is a \(G\)-algebra for which \(K^ G_ *(A)\) is just \(\mathbb{Z}\) (with the augmentation action of \(R(G)\)), then \(A\) is \(KK\)-equivalent to \(C(G)\). Examples of such actions arise naturally in the theory of ergodic actions.

In Section IV we outline the general theory of ergodic actions and show how they can be understood better by exploiting a link between equivariant \(K\)-theory and spectral multiplicities. This leads to the concept of the ‘multiplicity map’ and its associated diagrams. We explain how these ideas can be used to classify the ergodic actions of SU(2). To achieve this we need two additional tools: the first exploits the theory of coactions, while the second hinges on the presence of gaps in the spectra of certain homogeneous spaces of SU(2). We reach the perhaps disappointing conclusion that SU(2) has no exotic ergodic actions on operator algebras, i.e. all its ergodic actions are necessarily on type I von Neumann algebras. On the other hand, looking on the bright side, we recapture the classification of the closed subgroups of SU(2) up to conjugacy by a new if somewhat long-winded method. In Section V we explain how it is possible to develop a theory which parallels in every respect the easy classification of ergodic actions of compact Abelian groups, but this time for noncommutative groups. The natural restriction here is that the crossed product by the ergodic action should be a factor. One finds that such actions are classified by cocycles and bicharacters of the group dual, plus analogues of the usual nondegeneracy criteria for the action to be on a factor. The noncommutativity, however, introduces a quite novel feature, the symmetric quantum Yang-Baxter equation. This equation (with parameters) has been much studied in Russia in connection with completely integrable Hamiltonian systems and exactly solved models in quantum field theory. It also turns out to be the key to understanding the second cohomology of the dual of a simple compact group. In the remainder of this section we illustrate how these cocycles can be classified for certain classical groups of low rank.

In Section VI we study actions of compact groups on von Neumann algebras for which the crossed product is a factor. We consider in particular actions where there is an invariant subalgebra satisfying the same hypothesis. We then examine the position of the fixed point algebra of the subalgebra in the fixed point algebra of the whole algebra using the framework provided by the theory of subfactors. We find that there is a particularly simple ‘invariance principle’ operating here. Finally in Section VII, we bring together the ideas of the two preceding sections. Every solution of the symmetric quantum Yang-Baxter equation gives rise to a factor representation of the infinite symmetric group in the hyperfinite \(\text{II}_ 1\) factor, and hence we naturally obtain a subfactor. We conjecture that the position of this subfactor is more or less equivalent to the information provided by the bicharacter and go some way to lending credibility to this idea, by showing the analogy with the theory developed in Section VI.

The aim of this paper is to bring into evidence the usefulness of considering not only actions of compact groups on operator algebras, but also the dual notion of coaction. A lot of what we say is contained in much greater detail in three series of papers, due to be published in the near future: four papers on ergodic actions, two on product type actions and two on equivariant \(K\)-theory. These all have their rather primitive origins in the three chapters of my thesis. The other main proponent of coactions of compact groups is A. Ocneanu, and we have occasion to make frequent reference to his still unpublished work.

We briefly summarize the contents of this paper. In Section II we recall the basic definitions of coactions of compact groups on von Neumann and \(C^*\)-algebras. We present two examples of \(C^*\)-algebras which arise perhaps unexpectedly as crossed products by coactions, and show how this observation can be used to explore their structure. The basic idea here is an old one: to use symmetry properties to simplify and elucidate computations. In Section III we exhibit two general principles in equivariant \(KK\)-theory, namely Frobenius reciprocity and Dirac induction. When combined with the equivariant Thom isomorphism, these lead to a generalization of a spectral theorem of Hodgkin (for the \(K\)- theory of spaces) which in principle provides a homological machine whereby ordinary \(KK\)-theory (of a pair of algebras) can be deduced from equivariant \(KK\)-theory. In particular, if \(G\) is a compact Lie group which is both connected and simply connected, this leads one to suspect that if \(A\) is a \(G\)-algebra for which \(K^ G_ *(A)\) is just \(\mathbb{Z}\) (with the augmentation action of \(R(G)\)), then \(A\) is \(KK\)-equivalent to \(C(G)\). Examples of such actions arise naturally in the theory of ergodic actions.

In Section IV we outline the general theory of ergodic actions and show how they can be understood better by exploiting a link between equivariant \(K\)-theory and spectral multiplicities. This leads to the concept of the ‘multiplicity map’ and its associated diagrams. We explain how these ideas can be used to classify the ergodic actions of SU(2). To achieve this we need two additional tools: the first exploits the theory of coactions, while the second hinges on the presence of gaps in the spectra of certain homogeneous spaces of SU(2). We reach the perhaps disappointing conclusion that SU(2) has no exotic ergodic actions on operator algebras, i.e. all its ergodic actions are necessarily on type I von Neumann algebras. On the other hand, looking on the bright side, we recapture the classification of the closed subgroups of SU(2) up to conjugacy by a new if somewhat long-winded method. In Section V we explain how it is possible to develop a theory which parallels in every respect the easy classification of ergodic actions of compact Abelian groups, but this time for noncommutative groups. The natural restriction here is that the crossed product by the ergodic action should be a factor. One finds that such actions are classified by cocycles and bicharacters of the group dual, plus analogues of the usual nondegeneracy criteria for the action to be on a factor. The noncommutativity, however, introduces a quite novel feature, the symmetric quantum Yang-Baxter equation. This equation (with parameters) has been much studied in Russia in connection with completely integrable Hamiltonian systems and exactly solved models in quantum field theory. It also turns out to be the key to understanding the second cohomology of the dual of a simple compact group. In the remainder of this section we illustrate how these cocycles can be classified for certain classical groups of low rank.

In Section VI we study actions of compact groups on von Neumann algebras for which the crossed product is a factor. We consider in particular actions where there is an invariant subalgebra satisfying the same hypothesis. We then examine the position of the fixed point algebra of the subalgebra in the fixed point algebra of the whole algebra using the framework provided by the theory of subfactors. We find that there is a particularly simple ‘invariance principle’ operating here. Finally in Section VII, we bring together the ideas of the two preceding sections. Every solution of the symmetric quantum Yang-Baxter equation gives rise to a factor representation of the infinite symmetric group in the hyperfinite \(\text{II}_ 1\) factor, and hence we naturally obtain a subfactor. We conjecture that the position of this subfactor is more or less equivalent to the information provided by the bicharacter and go some way to lending credibility to this idea, by showing the analogy with the theory developed in Section VI.

### MSC:

46L55 | Noncommutative dynamical systems |

46L37 | Subfactors and their classification |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

19K35 | Kasparov theory (\(KK\)-theory) |

19L47 | Equivariant \(K\)-theory |

58J22 | Exotic index theories on manifolds |