## State space methods in operator and function theory

#### Location

Cupples I Room 113

#### Start Date

7-19-2016 4:00 PM

#### End Date

19-7-2016 4:20 PM

#### Description

It is well known that a function $K: \Omega \times \Omega \to \mathcal{L}(\mathcal{Y})$ (where $\mathcal{L}(\mathcal{Y}$) is the set of all bounded linear operators on a Hilbert space $\mathcal Y$) being (1) a positive kernel in the sense of Aronszajn (i.e. $\sum_{i,j=1}^N \langle K(\omega_i , \omega_j) y_j, y_i \rangle \geq 0$ for all $\omega_1, \dots, \omega_N \in \Omega$, $y_1, \dots, y_N \in \mathcal Y$, and $N=1,2,\dots$) is equivalent to (2) $K$ being the reproducing kernel for a reproducing kernel Hilbert space $\mathcal H (K)$, and (3) $K$ having a Kolmogorov decomposition $K(\omega, \zeta)=H(\omega)H(\zeta)^*$ for an operator-valued function $H: \Omega \to \mathcal{L}(\mathcal X, \mathcal Y)$ where $\mathcal X$ is an auxiliary Hilbert space. Recent work of the authors extended this result to the setting of free noncommutative functions (i.e. functions defined on matrices over a point set which respects direct sums and similarities) with the target set $\mathcal L ( \mathcal Y)$ of $K$ replaced by $\mathcal L (\mathcal A, \mathcal L (\mathcal Y))$ where $\mathcal A$ is a $C^*$-algebra. In this talk, we discuss the next extension where the target set of $K$ is replaced by $\mathcal L (\mathcal A, \mathcal L_a (\mathcal E))$ where $\mathcal A$ is a $W^*$-algebra and $\mathcal L_a (\mathcal E)$ is the set of adjointable operators on a Hilbert $W^*$-module over a $W^*$-algebra $\mathcal B$. Various special cases of this result correspond to results of Kasparov, Murphy, and Szafraniec in the Hilbert $C^*$-module literature.

#### Share

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Jul 19th, 4:00 PM Jul 19th, 4:20 PM

Completely positive kernels: the noncommutative correspondence setting

Cupples I Room 113

It is well known that a function $K: \Omega \times \Omega \to \mathcal{L}(\mathcal{Y})$ (where $\mathcal{L}(\mathcal{Y}$) is the set of all bounded linear operators on a Hilbert space $\mathcal Y$) being (1) a positive kernel in the sense of Aronszajn (i.e. $\sum_{i,j=1}^N \langle K(\omega_i , \omega_j) y_j, y_i \rangle \geq 0$ for all $\omega_1, \dots, \omega_N \in \Omega$, $y_1, \dots, y_N \in \mathcal Y$, and $N=1,2,\dots$) is equivalent to (2) $K$ being the reproducing kernel for a reproducing kernel Hilbert space $\mathcal H (K)$, and (3) $K$ having a Kolmogorov decomposition $K(\omega, \zeta)=H(\omega)H(\zeta)^*$ for an operator-valued function $H: \Omega \to \mathcal{L}(\mathcal X, \mathcal Y)$ where $\mathcal X$ is an auxiliary Hilbert space. Recent work of the authors extended this result to the setting of free noncommutative functions (i.e. functions defined on matrices over a point set which respects direct sums and similarities) with the target set $\mathcal L ( \mathcal Y)$ of $K$ replaced by $\mathcal L (\mathcal A, \mathcal L (\mathcal Y))$ where $\mathcal A$ is a $C^*$-algebra. In this talk, we discuss the next extension where the target set of $K$ is replaced by $\mathcal L (\mathcal A, \mathcal L_a (\mathcal E))$ where $\mathcal A$ is a $W^*$-algebra and $\mathcal L_a (\mathcal E)$ is the set of adjointable operators on a Hilbert $W^*$-module over a $W^*$-algebra $\mathcal B$. Various special cases of this result correspond to results of Kasparov, Murphy, and Szafraniec in the Hilbert $C^*$-module literature.