Journal article
Tensorial Function Theory: From Berezin Transforms to Taylor’s Taylor Series and Back
Integral equations and operator theory, Vol.76(4), pp.463-508
08/2013
DOI: 10.1007/s00020-013-2062-4
Abstract
Let H
∞(E) be the Hardy algebra of a W*-correspondence E over a W*-algebra M. Then the ultraweakly continuous completely contractive representations of H
∞(E) are parametrized by certain sets
$${{\mathcal{AC}}(\sigma)}$$
indexed by NRep(M)—the normal *-representations σ of M. Each set
$${{\mathcal{AC}}(\sigma)}$$
has analytic structure, and each element
$${F \in H^{\infty}(E)}$$
gives rise to an analytic operator-valued function
$${\widehat{F}_{\sigma}}$$
on
$${{\mathcal{AC}}(\sigma)}$$
that we call the σ-Berezin transform of F. The sets
$${\{{\mathcal{AC}}(\sigma)\}_{\sigma\in\Sigma}}$$
and the family of functions
$${\{\widehat{F}_{\sigma}\}_{\sigma\in\Sigma}}$$
exhibit “matricial structure” that was introduced by Joeseph Taylor in his work on noncommutative spectral theory in the early 1970s. Such structure has been exploited more recently in other areas of free analysis and in the theory of linear matrix inequalities. Our objective here is to determine the extent to which the matricial structure characterizes the Berezin transforms.
Details
- Title: Subtitle
- Tensorial Function Theory: From Berezin Transforms to Taylor’s Taylor Series and Back
- Creators
- Paul Muhly - Department of Mathematics University of Iowa Iowa IA 52242 USABaruch Solel - Department of Mathematics Technion 32000 Haifa Israel
- Resource Type
- Journal article
- Publication Details
- Integral equations and operator theory, Vol.76(4), pp.463-508
- Publisher
- Springer Basel
- DOI
- 10.1007/s00020-013-2062-4
- ISSN
- 0378-620X
- eISSN
- 1420-8989
- Language
- English
- Date published
- 08/2013
- Academic Unit
- Mathematics; Statistics and Actuarial Science
- Record Identifier
- 9984083860902771
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