526 research outputs found
Normalizers of Operator Algebras and Reflexivity
The set of normalizers between von Neumann (or, more generally, reflexive)
algebras A and B, (that is, the set of all operators x such that xAx* is a
subset of B and x*Bx is a subset of A) possesses `local linear structure': it
is a union of reflexive linear spaces. These spaces belong to the interesting
class of normalizing linear spaces, namely, those linear spaces U for which
UU*U is a subset of U. Such a space is reflexive whenever it is ultraweakly
closed, and then it is of the form U={x:xp=h(p)x, for all p in P}, where P is a
set of projections and h a certain map defined on P. A normalizing space
consists of normalizers between appropriate von Neumann algebras A and B.
Necessary and sufficient conditions are found for a normalizing space to
consist of normalizers between two reflexive algebras. Normalizing spaces which
are bimodules over maximal abelian selfadjoint algebras consist of operators
`supported' on sets of the form [f=g] where f and g are appropriate Borel
functions. They also satisfy spectral synthesis in the sense of Arveson.Comment: 20 pages; to appear in the Proceedings of the London Mathematical
Societ
Lie Groupoids and Lie algebroids in physics and noncommutative geometry
The aim of this review paper is to explain the relevance of Lie groupoids and
Lie algebroids to both physicists and noncommutative geometers. Groupoids
generalize groups, spaces, group actions, and equivalence relations. This last
aspect dominates in noncommutative geometry, where groupoids provide the basic
tool to desingularize pathological quotient spaces. In physics, however, the
main role of groupoids is to provide a unified description of internal and
external symmetries. What is shared by noncommutative geometry and physics is
the importance of Connes's idea of associating a C*-algebra C*(G) to a Lie
groupoid G: in noncommutative geometry C*(G) replaces a given singular quotient
space by an appropriate noncommutative space, whereas in physics it gives the
algebra of observables of a quantum system whose symmetries are encoded by G.
Moreover, Connes's map G -> C*(G) has a classical analogue G -> A*(G) in
symplectic geometry due to Weinstein, which defines the Poisson manifold of the
corresponding classical system as the dual of the so-called Lie algebroid A(G)
of the Lie groupoid G, an object generalizing both Lie algebras and tangent
bundles. This will also lead into symplectic groupoids and the conjectural
functoriality of quantization.Comment: 39 pages; to appear in special issue of J. Geom. Phy
On differential graded categories
Differential graded categories enhance our understanding of triangulated
categories appearing in algebra and geometry. In this survey, we review their
foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen
and Toen-Vaquie.Comment: 30 pages, correction at the end of 3.9, corrections and added
references in 5.
Applications of operator space theory to nest algebra bimodules
Recently Blecher and Kashyap have generalized the notion of W* modules over
von Neumann algebras to the setting where the operator algebras are \sigma-
weakly closed algebras of operators on a Hilbert space. They call these modules
weak* rigged modules. We characterize the weak* rigged modules over nest
algebras . We prove that Y is a right weak* rigged module over a nest algebra
Alg(M) if and only if there exists a completely isometric normal representation
\phi of Y and a nest algebra Alg(N) such that Alg(N)\phi(Y)Alg(M) \subset
\phi(Y) while \phi(Y) is implemented by a continuous nest homomorphism from M
onto N. We describe some properties which are preserved by continuous CSL
homomorphisms
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