2015/07/24 by Serdar Ay, Ay, Serdar, Aurelian Gheondea +1
Mathematics · #46E22 #46L89 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 47A20 #Secondary 43A35
paper · pdf · doi:10.48550/arxiv.1507.06840
openalex publication_date 2015/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
We consider positive semidefinite kernels valued in the *-algebra of continuous and continuously adjointable operators on a VH-space (Vector Hilbert space in the sense of Loynes) and that are invariant under actions of *-semigroups. For such a kernel we obtain two necessary and sufficient boundedness conditions in order for there to exist *-representations of the underlying *-semigroup on a VH-space linearisation, equivalently, on a reproducing kernel VH-space. We exhibit several situations when the latter boundedness condition is automatically fulfilled. For example, when specialising to the case of Hilbert modules over locally C^*-algebras, we show that both boundedness conditions are automatically fulfilled and, consequently, this general approach provides a rather direct proof of the general Stinespring-Kasparov type dilation theorem for completely positive maps on locally C^*-algebras and with values adjointable operators on Hilbert modules over locally C^*-algebras.