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FOCK SPACE TECHNIQUES IN TENSOR ALGEBRAS OF DIRECTED ALVARO ARIAS
 

Summary: FOCK SPACE TECHNIQUES IN TENSOR ALGEBRAS OF DIRECTED
GRAPHS
ALVARO ARIAS
Abstract. In [MS1], Muhly and Solel developed a theory of tensor algebras over C -
correspondences that extends the model theory of contractions in B (H) : The main exam-
ples are generated by Fock spaces, directed graphs, and analytic cross products. In this
paper we show that many results of tensor algebras of directed graphs, including dilations
and commutant lifting theorems for C 0 completely contractive representations, can be
deduced from results on Fock spaces. One of the main tools we use is the Poisson kernels,
which we de...ne for arbitrary C -correspondences. The Fock space approach allows us
to consider "weighted" graphs, where the dilation and commutant lifting theorems hold.
Additionally we prove a rigidity result for submodules of the induced representations of
directed graphs and we obtain projective resolutions of graph deformations.
1. Introduction
In the last thirty years there have been many attempts to generalize the model theory
for contractions in B (H), particularly the Nagy-Foias dilation theory and the Commutant
Lifting Theorem. For example, Douglas and Paulsen proposed the Hilbert module lan-
guage to extend these results to multivariate function theory. Popescu extended them to
a noncommutative multivariate setting. And Muhly and Solel extended them to tensor al-
gebras over C -Correspondences (Hilbert bimodules over a C -algebra A): The language of

  

Source: Arias, Alvaro - Department of Mathematics, University of Denver

 

Collections: Mathematics