Department of

February 2019March 2019 April 2019 Su MoTuWe Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 1 2 1 2 3 4 5 6 3 4 5 6 7 8 9 3 4 5 6 7 8 9 7 8 9 10 11 12 13 10 11 12 13 14 15 16 10 11 12 13 14 15 16 14 15 16 17 18 19 20 17 181920 21 22 23 17 18 19 20 21 22 23 21 22 23 24 25 26 27 24 25 26 27 28 24 25 26 27 28 29 30 28 29 30 31

Friday, March 8, 2019

**Abstract:** Given a unital operator algebra, it is natural to seek the smallest $C^*$-algebra generated by a completely isometric image of it, by analogy with the classical Shilov boundary of a uniform algebra. In keeping with this analogy, one method for constructing the so-called $C^*$-envelope is through a non-commutative version of the Choquet boundary. It is known that such a procedure can be also be applied to operator spaces, although in this case the envelope has less structure. In this talk, I will present a certain completely bounded version of the non-commutative Choquet boundary of an operator space that yields the structure of a $C^*$-algebra for the associated Shilov boundary. I will explain how the resulting $C^*$-algebras enjoy some of the properties expected of an envelope, but I will also highlight their shortcomings along with some outstanding questions about them. This is joint work with Christopher Ramsey.