Browsing by Author "Clouâtre, Raphaël"

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    A completely bounded noncommutative Choquet boundary for operator spaces
    (2019) Clouâtre, Raphaël; Ramsey, Christopher
    We develop a completely bounded counterpart to the noncommutative Choquet boundary of an operator space. We show how the class of completely bounded linear maps is too large to accommodate our purposes. To overcome this obstacle, we isolate the subset of completely bounded linear maps admitting a dilation of the same norm that is multiplicative on the associated C*-algebra. We view such maps as analogs of the familiar unital completely contractive maps, and we exhibit many of their structural properties. Of particular interest to us are those maps that are extremal with respect to a natural dilation order. We establish the existence of extremals and show that they have a certain unique extension property. In particular, they give rise to *-homomorphisms that we use to associate to any representation of an operator space an entire scale of C*-envelopes. We conjecture that these C*-envelopes are all *-isomorphic and verify this in some important cases.
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    Residually finite-dimensional operator algebras
    (2019) Clouâtre, Raphaël; Ramsey, Christopher
    We study non-selfadjoint operator algebras that can be entirely understood via their finite-dimensional representations. In contrast with the elementary matricial description of finite-dimensional C∗-algebras, in the non-selfadjoint setting we show that an additional level of flexibility must be allowed. Motivated by this peculiarity, we consider a natural non-selfadjoint notion of residual finite-dimensionality. We identify sufficient conditions for the tensor algebra of a C∗-correspondence to enjoy this property. To clarify the connection with the usual self-adjoint notion, we investigate the residual finite-dimensionality of the minimal and maximal C∗-covers associated to an operator algebra.