Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Tree algebras, semidiscreteness and dilation theory.
AU - Davidson, K. R.
AU - Power, S. C.
AU - Paulsen, V. I.
PY - 1994
Y1 - 1994
N2 - We introduce a class of finite-dimensional algebras built from a partial order generated as a transitive relation from a finite tree. These algebras, known as tree algebras, have the property that every locally contractive representation has a *-dilation. Furthermore, they satisfy an appropriate analogue of the Sz. Nagy–Foia Commutant Lifting Theorem. Then we define the infinite-dimensional analogue of these algebras in the class of completely distributive CSL algebras. These algebras are shown to have the semidiscreteness and complete compact approximation properties with respect to the class of finite-dimensional tree algebras. Consequently, they also have the property that contractive weak-* continuous representations have *-dilations, and satisfy the Sz. Nagy–Foia Commutant Lifting Theorem.
AB - We introduce a class of finite-dimensional algebras built from a partial order generated as a transitive relation from a finite tree. These algebras, known as tree algebras, have the property that every locally contractive representation has a *-dilation. Furthermore, they satisfy an appropriate analogue of the Sz. Nagy–Foia Commutant Lifting Theorem. Then we define the infinite-dimensional analogue of these algebras in the class of completely distributive CSL algebras. These algebras are shown to have the semidiscreteness and complete compact approximation properties with respect to the class of finite-dimensional tree algebras. Consequently, they also have the property that contractive weak-* continuous representations have *-dilations, and satisfy the Sz. Nagy–Foia Commutant Lifting Theorem.
U2 - 10.1112/plms/s3-68.1.178
DO - 10.1112/plms/s3-68.1.178
M3 - Journal article
VL - 68
SP - 178
EP - 202
JO - Proceedings of the London Mathematical Society
JF - Proceedings of the London Mathematical Society
SN - 1460-244X
IS - 1
ER -