Representations of invariant multilinear maps on HilbertC*-modules

A Radon–Nikodým theorem for completely positive invariant multilinear maps and its applications

“…We only sketch the construction, since the proof is similar to that of theorem 2·1 in [7]. We only sketch the construction, since the proof is similar to that of theorem 2·1 in [7].…”

A Radon-Nikodým theorem for local completely positive invariant multilinear maps

“…Note that if k = 1, then Definitions 2.4, 2.5 coincide with the notion of local completely contractive and local completely positive linear maps defined between local C * -algebras [6]. Furthermore, if , are C * -algebras, then Definitions 2.4, 2.5 coincide with the notion of completely contractive and completely positive k-linear maps studied in [15,16]. Now we introduce the notion of invariance from [16] to k-linear maps on locally C * -algebras.…”

“…] for details). Motivated by this work, Heo [15] came up with an additional property (called as invariant) on k-linear CP-maps as an analogue of invariant sesquilinear forms [1]. With this additional invariance property, Heo obtained a dilation for invariant k-linear CP-map to a multilinear representation which can be factorized as a product of m-number of commuting representations, where m = [ k+1 2 ].…”

A Russo-Dye type Theorem and Stinespring representation for invariant block multilinear completely positive maps

“…In 1987, E. Christensen Further, in 1999, J. Heo studied multilinear CP maps using invariant property [12]. Imposing invariant property, the author obtained an excellent form of the Stinespring dilation theorem for invariant multilinear completely positive maps which are completely bounded and symmetric as well (cf.…”

“…For instance, we show that every invariant multilinear CP map is automatically symmetric and completely bounded. Surprisingly these results are unknown in the literature (see [12,13,15]). As a negative result, we provide a concrete example of positive multilinear map ϕ : C(X ) 3 → which is not CP.…”