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{\Large Department of Mathematics, BGU}

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{\Huge Operator Algebras and Operator Theory}\\[0.2\baselineskip]

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\textbf{On} \emph{Monday, December  4, 2017}
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\textbf{At} \emph{16:00 -- 17:00}
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\textbf{In} \emph{-101}

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{\large\scshape Gregory Marx 
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  (BGU)
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will talk about
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{\Large\bfseries Completely Positive Noncommutative Kernels, part 2\par}
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\textsc{Abstract:}
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.

Last time, I introduced free noncommutative function theory and wrote down the analogue of the result above for noncommutative kernels. In part two, I will give a sketch of our proof and discuss some well-known results (e.g. Stinespring's dilation theorem for completely positive maps) which follow as corollaries. With any remaining time, I will talk about applications and more recent related results.








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