"Matrix locally
convex topologies"

By

**Anar Dosiev **

**(Atılım University)**

**Abstract**The known
representation theorem for operator spaces asserts that each abstract
operator space V can be realized as a subspace of the space B( H) of all
bounded linear operators on a Hilbert space H. This result plays one of the
central role in the operator space theory and allows us to have an abstract
characterization of a linear space of bounded linear operators on a Hilbert
space. Physically well motivated, operator spaces can be thought as quantized
normed spaces, where we have replaced functions with operators regarding
classical normed spaces as abstract function spaces. To have more solid justification
of the quantum physics it is necessary to consider an operator analogues of
locally convex spaces too, that is, a quantization of locally convex spaces.
This amounts considering linear spaces of unbounded Hilbert space operators or
more generally, projective limits of operator spaces. In recent years, this
theory started to develop by Effros and Webster under the title of "local
operator spaces", or quantizations of locally convex spaces.

In this talk we propose an elementary introduction to
these topics and as a final aim we will have an intrinsic description of local operator
spaces as above mentioned characterization for operator spaces. More precisely,
we prove that each local operator space can be realized as a subspace of
unbounded operators on a Hilbert space. Furthermore, if the given local
operator space has a bounded locally convex topology then it can be realized as
bounded operators on a Hilbert space.

**Date: ****Monday, November 28, 2005**

**Time: ****15.40**

**Place: ****Mathematics Seminar Room, SA-141**