Abstract
Let (G; P) be a quasi-lattice ordered group. In this paper we present a modied proof of Laca and Raeburn's theorem about the covariant isometric representations of amenable quasi-lattice ordered groups [7, Theorem 3.7], by following a two stage strategy. First, we construct a universal covariant representation for a given quasi-lattice ordered group (G; P) and show that it is unique. The construction of this object is new; we have not followed either Nica's approach in [10] or Laca and Raeburn's approach in [7], although all three objects are essentially the same. Our approach is a very natural one and avoids some of the intricacies of the other approaches. Then we show if (G; P) is amenable, true representations of (G; P) generate C-algebras which are canonically isomorphic to the universal object.
Highlights
Which says that all C∗-algebras generated by non-unitary isometries are the same and in some sense, universal
There is a unique covariant representation with the universal property. Nica used this universal object to define amenability, which is an interesting property of some quasi-lattice ordered groups
Laca and Raeburn introduced a subclass of covariant representations for a given quasi-lattice ordered group, which are called here true representations
Summary
C∗-algebras Generated by Isometries implies zx ≤ zy for any x, y, z ∈ G It is the natural partial order determined by P. Let (G, P ) be a quasi-lattice ordered group. A covariant isomeric representation of the quasi-lattice ordered group (G, P ) may be defined as a pair (A, V ) consisting of a unital C∗-algebra A and a map V from P to A such that. Vp−1(p∨q)Vq∗−1(p∨q), when p, q have a common upper bound in P ; To see that the first definition implies the second, notice first that if p, q ∈ P have no common upper bound in P the covariance condition gives.
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