Non-tracial Free Graph von Neumann Algebras
- PDF / 519,397 Bytes
- 40 Pages / 439.37 x 666.142 pts Page_size
- 30 Downloads / 186 Views
Communications in
Mathematical Physics
Non-tracial Free Graph von Neumann Algebras Michael Hartglass1 , Brent Nelson2 1 Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, California, USA.
E-mail: [email protected]
2 Department of Mathematics, Michigan State University, East Lansing, Michigan, USA
E-mail: [email protected] Received: 15 November 2018 / Accepted: 15 June 2020 Published online: 14 September 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: Given a finite, directed, connected graph equipped with a weighting μ on its edges, we provide a construction of a von Neumann algebra equipped with a faithful, normal, positive linear functional (M(, μ), ϕ). When the weighting μ is instead on the vertices of , the first author showed the isomorphism class of (M(, μ), ϕ) depends only on the data (, μ) and is an interpolated free group factor equipped with a scaling of its unique trace (possibly direct sum copies of C). Moreover, the free dimension of the interpolated free group factor is easily computed from μ. In this paper, we show for a weighting μ on the edges of that the isomorphism class of (M(, μ), ϕ) depends only on the data (, μ), and is either as in the vertex weighting case or is a free Araki–Woods factor equipped with a scaling of its free quasi-free state (possibly direct sum copies of C). The latter occurs when the subgroup of R+ generated by μ(e1 ) · · · μ(en ) for loops e1 · · · en in is non-trivial, and in this case the point spectrum of the free quasi-free state will be precisely this subgroup. As an application, we give the isomorphism type of some infinite index subfactors considered previously by Jones and Penneys. List of symbols p1 ,..., pn
p
t1 ,...,tn
t
Mn (C), (A, φ)
Implicit states or weighting on states
ψλ φp (Tλ , ϕλ )
A state on B(H) or Mn (C) determined by a set of matrix units Compression of a positive linear functional The free Araki–Woods factor generated by two variables and its free quasi-free state A generalized circular element A free product of free Araki–Woods factors The central support of a projection p in a von Neumann algebra M A graph with vertices V and edges E The source, target, and opposite of an edge e An edge weighting
yλ (TH , ϕ H ) z( p : M) = (V, E) s(e), t (e), eop μ
2
Ye , u e S(, μ) M(, μ) H (, μ) Tr ϕ
M. Hartglass, B. Nelson
An edge operator and its polar part The C ∗ -algebra associated to a graph and an edge weighting μ The von Nemann algebra associated to a graph and an edge weighting μ The space of paths in a graph The space of loops in a graph The subgroup of R+ generated by μ(e1 ) · · · μ(en ) for loops e1 · · · en in a graph with edge weighting μ A maximal subgraph of (, μ) subject to the condition H (Tr , μ) is trivial A faithful normal positive linear functional on M(, μ) induced by Tr
Introduction Given a finite, directed, connected graph = (V, E), there has been for some time an interest in von Neumann algebras associated to this
Data Loading...
