Multipliers and Dual Operator Algebras

Abstract

In a previous paper we showed how the main theorems characterizing operator algebras and operator modules, fit neatly into the framework of the``noncommutative Shilov boundary,'' and more particularly via the left multiplier operator algebra of an operator space. As well as giving new characterization theorems, the approach of that paper allowed many of the hypotheses of the earlier theorems to be eliminated. Recent progress of the author with Effros and Zarikian now enables weak*-versions of these characterization theorems. For example, we prove a result analogous to Sakai's famous characterization of von Neumann algebras as the C*-algebras with predual, namely, that the _-weakly closed unital (not-necessarilyselfadjoint) subalgebras of B(H ) for a Hilbert space H, are exactly the unital operator algebras which possess an operator space predual. This removes one of the hypotheses from an earlier characterization due to Le Merdy. We also show that the multiplier operator algebras of dual operator spaces are dual operator algebras. Using this we refine several known results characterizing dual operator modules. Academic Press

Duality and operator algebras: automatic weak* continuity and applications

“…Theorem 4.1 will also be an useful tool for future work on operator modules. For example, in [4] the first author was able to refine, in several ways, a theorem of Effros and Ruan characterizing certain operator modules over von Neumann algebras [12]. Theorem 4.1 allows precisely the same improvements for 'normal dual operator modules' over unital dual operator algebras.…”

“…Theorem 4.1 allows precisely the same improvements for 'normal dual operator modules' over unital dual operator algebras. In particular, Theorem 4.1 shows that the left normal hypothesis used in [4] is automatic, and may therefore be removed. We state a sample of other consequences:…”

“…These are the left multipliers as defined in the second paragraph of our paper in terms of a complete isometry , but also satisfying a * (X) ⊂ (X) in the language of that definition. It is shown in [4,6] that if X is a dual operator space then M l (X) is a dual operator algebra, A l (X) is a W * -algebra, and every T ∈ A l (X) is w * -continuous. These results have been key to work on one-sided multipliers and M-ideals following [6] (see e.g.…”

Amenability, Completely Bounded projections, Dynamical Systems and Smooth Orbits

“…Moreover note that condition (b) in Definition 2.2 is satisfied. In case (i), it follows by Theorem 2.1 in [Bl01] that the multiplication in A is separately weak * -continuous, hence A is actually a dual operator G-space. Now we see that in either of the cases (i) and (ii) it follows by Theorem 3.1 that there exists a completely bounded idempotent mapping P : A → A with Ran P = A G = ρ(G) ′ , and we are done.…”

Logmodularity and isometries of operator algebras

“…This hypothesis was removed by the first author in [11]. However, in fact, Proposition 5.1 gives a much simpler way to remove the hypothesis.…”