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Universal Operator Algebras Associated to Contractive Sequences of Non‐Commuting Operators
Author(s) -
Popescu Gelu
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006656
Subject(s) - mathematics , subalgebra , pure mathematics , operator algebra , nest algebra , compact operator , commutative property , discrete mathematics , algebra over a field , universal enveloping algebra , non associative algebra , extension (predicate logic) , computer science , programming language
We characterize the minimal isometric dilation of a non‐commutative contractive sequence of operators as a universal object for certain diagrams of completely positive maps. A non‐spatial construction of the minimal isometric dilation is also given, using Hilbert modules over C *‐algebras. It is shown that the non‐commutative disc algebras A n ( n ⩾2) are the universal algebras generated by contractive sequences of operators and the identity, and C *( S 1 , …, S n ) ( n ⩾2), the extension through compact operators of the Cuntz algebra O n , is the universal C *‐algebra generated by a contractive sequence of isometries. It is also shown that the algebras A n and C *( S 1 , …, S n ) are completely isometrically isomorphic to some free operator algebras considered by D. Blecher. In particular, the universal operator algebra of a row (respectively column) contraction is identified with a subalgebra of C *( S 1 , …, S n ). The internal characterization of the matrix norm on a universal algebra leads to some factorization theorems.