Geodesic
Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument
Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 2, pp. 25-41

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We extend the classical construction of operator colligations and characteristic functions. Consider the group G of finitary block unitary matrices of order $\alpha+\infty+\dots+\infty$ ($m$ times) and its subgroup $K \cong U(\infty)$, which consists of block diagonal unitary matrices with the identity block of order $\alpha$ and a matrix $u \in U(\infty)$ repeated $m$ times. It turns out that there is a natural multiplication on the space $G$//$K$ of conjugacy classes. We construct “spectral data” of conjugacy classes, which visualize the multiplication and are sufficient for reconstructing a conjugacy class.
Keywords: characteristic function, colligation, spectral data, infinite-dimensional group, inner function, Grassmannian, Hermitian symmetric space, invariant theory. 
@article{FAA_2017_51_2_a3,
     author = {Yu. A. Neretin},
     title = {Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {25--41},
     publisher = {mathdoc},
     volume = {51},
     number = {2},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2017_51_2_a3/}
}
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Yu. A. Neretin. Multiplication of conjugacy classes, colligations, and characteristic functions of matrix argument. Funkcionalʹnyj analiz i ego priloženiâ, Tome 51 (2017) no. 2, pp. 25-41. http://geodesic.mathdoc.fr/item/FAA_2017_51_2_a3/