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arXiv · 2401.01775

Isometric Dilations for Representations of Product Systems

Sibaprasad Barik·M. Bhattacharjee·B. Solel
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Abstract

We discuss representations of product systems (of $W^*$-correspondences) over the semigroup $\mathbb{Z}^n_+$ and show that, under certain pureness and Szego positivity conditions, a completely contractive representation can be dilated to an isometric representation. For $n=1,2$ this is known to hold in general (without assuming the conditions) but, for $n\geq 3$, it does not hold in general (as is known for the special case of isometric dilations of a tuple of commuting contractions). Restricting to the case of tuples of commuting contractions, our result reduces to a result of Barik, Das, Haria and Sarkar. Our dilation is explicitly constructed and we present some applications.

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Sibaprasad Barik, M. Bhattacharjee, B. Solel. 2024-01-03. Isometric Dilations for Representations of Product Systems. https://arxiv.org/abs/2401.01775

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