"Matrix locally convex topologies"
AbstractThe 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.
Place: Mathematics Seminar Room, SA-141