nLab Connes fusion



The operation called Connes fusion (after Alain Connes) is a tensor product operation of Hilbert space bimodules over von Neumann algebras that adjusts the naive algebraic tensor product.


Connes fusion is used to define fusion of positive energy representations of the loop group SU(N)\mathcal{L}S U(N) in

  • Antony Wassermann, Operator algebras and conformal field theory III (arXiv) .

It is used in a proposal for a construction of string group-principal bundles in the context of finding geometric models for elliptic cohomology in


The fusion operation on Hilbert modules of von Neumann algebras has been described in

  • Alain Connes, On the spatial theory of von Neumann algebras , J. of functional analysis 35 (1980) 153–164;

and in 1983 studied further by V. Jones in context of subfactors.

A useful review of Connes fusion observing some simplifications is

An attempt to summarize that in a blog entry is here

R. M Brower and N. P. Landsman described a bicategory of von Neumann algebras, Connes’ correspondences as morphisms and (bounded) intertwiners as 2-cells. They proved that the equivalence in that bicategory is the same as usual Morita equivalence of von Neumann algebras as studied by Mark Rieffel and others.

Last revised on May 25, 2012 at 13:38:52. See the history of this page for a list of all contributions to it.