Seminarium:
Dyskretna analiza harmoniczna i niekomutatywna probabilistyka
Osoba referująca:
Biswarup Das (University of Oulu, Finland)
Data:
czwartek, 23. Marzec 2017 - 10:15
Sala:
604
Opis:
Let S be a topological space, which is also a semigroup with identity,
such that the multiplication is separately continuous. Such semigroups
are called semitopological semigroups. These type of objects occur
naturally, if onestudies weakly almost periodic compactification of a
topological group.
Now if we assume the following:
(a) The topology of S is locally compact.
(b) Abstract algebraically speaking, S is a group (i.e. every element has
an inverse).
(c) The multiplication is separately continuous as above (no other
assumption. This is the only assumption concerning the interaction of the
topology with the group structure).
Then it follows that S becomes a topological group i.e. :
(a) The multiplication becomes jointly continuous.
(b) The inverse is also continuous.
This extremely beautiful fact was proven by R. Ellis in 1957 and is known
in the literature as Ellis joint continuity theorem.
In this talk, we will prove a non-commutative version of this result. Upon
briefly reviewing the notion of semitopological semigroup, we will
introduce ''compact semitopological quantum semigroup'' which were before
introduced by M. Daws in 2014 as a tool to study almost periodicity of
Hopf von Neumann algebras. Then we will give a necessary and sufficient
condition on these objects, so that they become a compact quantum group.
As a corollary, we will give a new proof of the Ellis joint continuity
theorem as well.
This is the joint work with Colin Mrozinski.