algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
(geometry Isbell duality algebra)
The generalization of the process of one-point compactification of possibly non-compact topological spaces from topology to non-commutative topology is the process of adding units to possibly non-unital C*-algebras, thought of as formal duals of non-commutative spaces in noncommutative topology.
For a non-unital C*-algebra, its unitisation is the -algebra whose underlying vector space is the direct sum
of with the field of complex numbers, equipped with the multiplication law
and the involution
(complex conjugation is taking place on the right). This really is a -algebra. The norm can be characterised as
if does not have a multiplicative unit. (Note that the operatornorm vanishes on if does have a multiplicative unit). For -algebras with a multiplicative unit one can show that the norm equals
For a locally compact Hausdorff topological space and its possibly non-unital C*-algebra of functions, then
this is equivalently the unital -algebra of functions on the one-point compactification of .
There is hence a canonical projection
The topological K-theory of is the kernel of the induced map in operator K-theory
The unitisation of comes with a canonical projection homomorphism of C*-algebras
given by
Dually this corresponds to the inclusion of the “point at infinity”.
The map of remark induces a homomorphism of operator K-theory groups of the form
The kernel of this map is the operator K-theory of the original possibly non-unital -algebra :
Heuristically it is clear that this is the “compactly suppported” K-theory of the possibly non-compact non-commutative space given by the algebra . This statement has been made precise for instance in (Emerson 07, theorem 3.8)):
For a locally compact Hausdorff topological space, there is a natural isomorphism
where on the right we homotopy classes of maps of compact support into the classifying space for K-theory (space of Fredholm operators).
Last revised on November 25, 2017 at 18:50:55. See the history of this page for a list of all contributions to it.