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Local Operator Multipliers and Positivity
We establish an unbounded version of Stinespring's Theorem and a lifting
result for Stinespring representations of completely positive modular maps
defined on the space of all compact operators. We apply these results to study
positivity for Schur multipliers. We characterise positive local Schur
multipliers, and provide a description of positive local Schur multipliers of
Toeplitz type. We introduce local operator multipliers as a non-commutative
analogue of local Schur multipliers, and obtain a characterisation that extends
earlier results concerning operator multipliers and local Schur multipliers. We
provide a description of the positive local operator multipliers in terms of
approximation by elements of canonical positive cones.Comment: 31 page
From vertex operator algebras to conformal nets and back
We consider unitary simple vertex operator algebras whose vertex operators
satisfy certain energy bounds and a strong form of locality and call them
strongly local. We present a general procedure which associates to every
strongly local vertex operator algebra V a conformal net A_V acting on the
Hilbert space completion of V and prove that the isomorphism class of A_V does
not depend on the choice of the scalar product on V. We show that the class of
strongly local vertex operator algebras is closed under taking tensor products
and unitary subalgebras and that, for every strongly local vertex operator
algebra V, the map W\mapsto A_W gives a one-to-one correspondence between the
unitary subalgebras W of V and the covariant subnets of A_V. Many known
examples of vertex operator algebras such as the unitary Virasoro vertex
operator algebras, the unitary affine Lie algebras vertex operator algebras,
the known c=1 unitary vertex operator algebras, the moonshine vertex operator
algebra, together with their coset and orbifold subalgebras, turn out to be
strongly local. We give various applications of our results. In particular we
show that the even shorter Moonshine vertex operator algebra is strongly local
and that the automorphism group of the corresponding conformal net is the Baby
Monster group. We prove that a construction of Fredenhagen and J\"{o}rss gives
back the strongly local vertex operator algebra V from the conformal net A_V
and give conditions on a conformal net A implying that A= A_V for some strongly
local vertex operator algebra V.Comment: Minor correction
Local and Semilocal Vertex Operator Algebras
We investigate a general structure theory for a vertex operator algebra. We
discuss the center and blocks, the Jacobson radical and solvable radical and
local vertex operator algebras. The main consequence of our structure theory is
that if the vertex operator algebra satisfies some mild conditions then it is
necessarily semilocal, i.e. a direct sum of local vertex operator algebras.Comment: 17 page
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