Math-Fi seminar on 4 Jun. (Co-organized as a Quantum Walk Seminar)

Local completely contractive maps are a particular class of continuous linear maps on

locally C∗-algebras, which are inverse limit of inverse system of C∗-algebras in the category of

topological ∗-algebras. In this talk, we present a structure theorem for unbounded operator valued

local completely contractive map ψ on a locally C∗-algebra A. There is a unique commutant

operator T in the structure of ψ with norm at most 2. We show that the operator T is a contraction

if and only if the block map

Φ= φ ψ

ψ∗ φ

defined on M2(A) is local completely positive, for some local completely positive and local

completely contractive map φ on A. In general, such a map φ may not exist for a given ψ, we

illustrate this situation with an example. However, we prove a block representation of ψ in the sense

that there always exist a local completely positive and local completely bounded map φ such that Φ

is a local completely positive map. This is a joint work with R. Siddique

立命館大学数理科学科