Rocky Mountain Journal of Mathematics

$\mathfrak K$-families and CPD-H-extendable families

Santanu Dey and Harsh Trivedi

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We introduce, for any set $S$, the concept of a $\mathfrak {K}$-family between two Hilbert $C^*$-modules over two $C^*$-algebras, for a given completely positive definite (CPD-) kernel $\mathfrak {K}$ over $S$ between those $C^*$-algebras, and we obtain a factorization theorem for such $\mathfrak {K}$-families. If $\mathfrak {K}$ is a CPD-kernel and $E$ is a full Hilbert $C^*$-module, then any $\mathfrak {K}$-family which is covariant with respect to a dynamical system $(G,\eta ,E)$ on $E$, extends to a $\widetilde {\mathfrak {K}}$-family on the crossed product $E \times _\eta G$, where $\widetilde {\mathfrak {K}}$ is a CPD-kernel. Several characterizations of $\mathfrak {K}$-families, under the assumption that ${E}$ is full, are obtained, and covariant versions of these results are also given. One of these characterizations says that such $\mathfrak {K}$-families extend as CPD-kernels, between associated (extended) linking algebras, whose $(2,2)$-corner is a homomorphism and vice versa. We discuss a dilation theory of CPD-kernels in relation to $\mathfrak {K}$-families.

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Rocky Mountain J. Math., Volume 47, Number 3 (2017), 789-816.

First available in Project Euclid: 24 June 2017

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Zentralblatt MATH identifier

Primary: 46L07: Operator spaces and completely bounded maps [See also 47L25] 46L08: $C^*$-modules 46L53: Noncommutative probability and statistics 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]

Completely positive definite kernels dilations crossed product Hilbert $C^*$-modules Kolmogorov decomposition linking algebras


Dey, Santanu; Trivedi, Harsh. $\mathfrak K$-families and CPD-H-extendable families. Rocky Mountain J. Math. 47 (2017), no. 3, 789--816. doi:10.1216/RMJ-2017-47-3-789.

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