Analytic functionals for the non-commutative disc algebra
| Autor: | Raphaël Clouâtre, Edward J. Timko, Robert T. W. Martin |
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| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Function space
010102 general mathematics Mathematics - Operator Algebras Absolute continuity 01 natural sciences Measure (mathematics) Projection (linear algebra) Functional Analysis (math.FA) Algebra Mathematics - Functional Analysis Cuntz algebra 0103 physical sciences FOS: Mathematics 010307 mathematical physics 0101 mathematics Representation (mathematics) Operator Algebras (math.OA) Commutative property Analysis Mathematics Operator system |
| Popis: | The main objects of study in this paper are those functionals that are analytic in the sense that they annihilate the non-commutative disc algebra. In the classical univariate case, a theorem of F. and M. Riesz implies that such functionals must be given as integration against an absolutely continuous measure on the circle. We develop generalizations of this result to the multivariate non-commutative setting, upon reinterpreting the classical result. In one direction, we show that the GNS representation naturally associated to an analytic functional on the Cuntz algebra cannot have any singular summand. Following a different interpretation, we seek weak-$*$ continuous extensions of analytic functionals on the free disc operator system. In contrast with the classical setting, such extensions do not always exist, and we identify the obstruction precisely in terms of the so-called universal structure projection. We also apply our ideas to commutative algebras of multipliers on some complete Nevanlinna--Pick function spaces. 25 pages |
| Databáze: | OpenAIRE |
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