2017/03/08 by Raphaël Clouâtre, Christopher Bronk Ramsey, Clouâtre, Raphaël +1
Mathematics · #Advanced Operator Algebra Research #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1703.02924
We develop a completely bounded counterpart to the non-commutative Choquet\nboundary of an operator space. We show how the class of completely bounded\nlinear maps is too large to accommodate our purposes. To overcome this\nobstacle, we isolate the subset of completely bounded linear maps on an\noperator space admitting a dilation of the same norm which is multiplicative on\nthe generated C^*-algebra. We view such maps as analogues of the familiar\nunital completely contractive maps, and we exhibit many of their structural\nproperties. Of particular interest to us are those maps which are extremal with\nrespect to a natural dilation order. We establish the existence of extremals\nand show that they have a certain unique extension property. In particular,\nthey give rise to *-homomorphisms which we use to associate to any\nrepresentation of an operator space an entire scale of C^*-envelopes. We\nconjecture that these C^*-envelopes are all *-isomorphic, and verify this\nin some important cases.\n