8 research outputs found
Continuous extension of a densely parameterized semigroup
Let S be a dense sub-semigroup of the positive real numbers, and let X be a
separable, reflexive Banach space. This note contains a proof that every weakly
continuous contractive semigroup of operators on X over S can be extended to a
weakly continuous semigroup over the positive real numbers. We obtain similar
results for non-linear, non-expansive semigroups as well. As a corollary we
characterize all densely parametrized semigroups which are extendable to
semigroups over the positive real numbers.Comment: 8 pages, minor modification
Coherent States of the q--Canonical Commutation Relations
For the -deformed canonical commutation relations for in some Hilbert
space we consider representations generated from a vector
satisfying , where .
We show that such a representation exists if and only if .
Moreover, for these representations are unitarily equivalent
to the Fock representation (obtained for ). On the other hand
representations obtained for different unit vectors are disjoint. We
show that the universal C*-algebra for the relations has a largest proper,
closed, two-sided ideal. The quotient by this ideal is a natural -analogue
of the Cuntz algebra (obtained for ). We discuss the Conjecture that, for
, this analogue should, in fact, be equal to the Cuntz algebra
itself. In the limiting cases we determine all irreducible
representations of the relations, and characterize those which can be obtained
via coherent states.Comment: 19 pages, Plain Te