algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In noncommutative topology the standard notion of homomorphism of C*-algebras is too restrictive for some applications, related to the fact that some noncommutative $C^\ast$-algebras correspond to “locally badly behaved” noncommutative topological spaces. The notion of asymptotic $C^\ast$-homomorphism is more flexible than that of plain $C^\ast$-homomorphisms and designed to correct this problem. Homotopy classes of asymptotic $C^\ast$-homomorphisms are the hom-sets in a category called E-theory. See there for more details.
For $A,B$ two C*-algebras, an asymptotic homomorphism between them is a $[0,\infty)$-parameterized collection of continuous functions $\{\phi_t \colon A \to B\}_{t \in [0, \infty)}$, such that
for each $a \in A$, the function $t \mapsto \phi_t(a)$ is a continuous function;
in the limit $t \to \infty$, $\phi_t$ becomes a star-algebra homomorphism.
As for ordinary $C^\ast$-algebra homomorphisms one puts:
For $f_t, g_t \colon A \to B$ to asymptotic $C^\ast$-homomorphisms, def. , a (right) homotopy between them is an asyptotic homomorphism $\eta_t \colon A \to C([0,1],B)$ which restricts to $f$ at 0 and to $g$ at $1$, hence such that it fits into a commuting diagram of the form
Homotopy of asymptotic $C^\ast$-homomorphisms is clearly an equivalence relation. Write $[A,B]$ for the set of homotopy-equivalence classes of asymptotic homomorphisms $A \to B$.
For $A,B \in$ C*Alg, the set $[A, C_0((0,1), B)]$ is naturally an abelian group under the composition operation which sends the homotopy classes presented by $f,g \colon A \times (0,1) \to B$ to the homotopy class of
The $t$-wise composition of two asymptotic $C^\ast$-homomorphisms is not in general itself an asymptotic $C^\ast$-homomorphims. However, every asympotic homomorphism is homotopic to one which is an equicontinuous function, and $t$-wise composition of equicontinuous asymptotic $C^\ast$-homomorphisms is again an asymptotic homomorphism.
Two asymptotic $C^\ast$-homomorphisms which differe just by a reparameterization of $[0,\infty)$ while having the same limit can be related by a homotopy, def. .
(self-adjointification homotopy)
For $f_t \colon A \to B$ an asymptotic $C^\ast$-homomorphism, there is a homotopy to the asymptotic morphism
The notion was introduced in
A review is for instance around p. 23 of
Last revised on April 10, 2013 at 20:33:13. See the history of this page for a list of all contributions to it.