31/10/2023, 15:00 — 16:00 — Room P3.31, Mathematics Building
Tomás Pacheco, Instituto Superior Técnico, Universidade de Lisboa
On the Crossed Product $C^*$-algebras by Endomorphisms
In this presentatiom, we discuss two theories of the crossed product by endomorphisms. First being when the action is given by a single one, whose definition relies heavily on the concept of a transfer operator $Li$. We also define the crossed product for the case where the action is given by a plural number of endomorphisms, either via a group $G$ in the form of an interaction group $(A,G,V)$ or given by a semigroup homomorphism $\alpha:P \xrightarrow{}\text{End}(A)$. We give sufficient conditions for these theories of the crossed product to be isomorphic between themselves. Let $A=C(S^1)$ be regarded as multiplication operators on $B(L^2(S^1))$ and $\alpha: A \xrightarrow{} A$ an endomorphism given by $f(z) \mapsto f(z^2)$, throughout this presentation, we will use this example as guidance.
