[Paper Review] Simple reduced $L^p$ operator crossed products with unique trace

Sanaz Pooya, Shirin Hejazian|arXiv (Cornell University)|Feb 13, 2014

Advanced Operator Algebra Research8 references3 citations

TL;DR

This paper establishes that for $ p \in (1,\infty) $, the reduced $ L^p $ operator crossed product $ F^p_{\mathrm{r}}(G,A,\alpha) $ is simple and admits a unique normalized trace when $ G $ is a countable Powers group, $ A $ is a unital, $ G $-simple $ L^p $ operator algebra, and the action is isometric and nondegenerate. The result extends Powers' $ C^* $-algebra simplicity and trace uniqueness to the $ L^p $ setting using duality between $ L^p $ and $ L^q $ spaces and conditional expectations.

ABSTRACT

In this article we study simplicity and traces of reduced $L^p$ operator crossed products $F^p_{\mathrm{r}}(G, A, α)$. Given $p \in (1, \infty)$, let $G$ be a Powers group, and let $α\colon G o Aut(A)$ be an isometric action of $G$ on a unital $L^p$ operator algebra $A$ such that $A$ is $G$-simple. We prove that the reduced $L^p$ operator crossed product of $A$ by $G$, $F^p_{\mathrm{r}}(G, A, α)$, is simple. Moreover, we show that traces on $F^p_{\mathrm{r}}(G, A, α)$ are in correspondence with $G$-invariant traces on A. Our results generalize the results obtained by de la Harpe for reduced $C^*$crossed products in 1985. By letting $G$ be a countable nonabelian free group as a special case, we recover an analogue of a result of Powers from 1975. For the case $p = 1$, it turns out that (reduced) $L^p$ operator group algebras are not simple.

Motivation & Objective

• To generalize de la Harpe and Skandalis' results on $ C^* $-algebra crossed products to the $ L^p $ operator algebra setting for $ p \in (1,\infty) $.
• To establish the simplicity of the reduced $ L^p $ operator crossed product $ F^p_{\mathrm{r}}(G,A,\alpha) $ under conditions of $ G $-simplicity, unitality of $ A $, and isometric $ G $-action.
• To characterize traces on $ F^p_{\mathrm{r}}(G,A,\alpha) $ as pullbacks of $ G $-invariant traces on $ A $ via the standard conditional expectation $ E $.
• To show that the reduced $ L^p $ group algebra $ F^p_{\mathrm{r}}(G) $ of a countable Powers group is simple with a unique normalized trace.
• To demonstrate that the result fails for $ p = 1 $, highlighting the necessity of $ p > 1 $ in the framework.

Proposed method

• Use of duality between $ L^p $ and $ L^q $ spaces for conjugate exponents $ p, q \in (1,\infty) $ to overcome the lack of orthogonal projections in $ L^p $-spaces.
• Application of the standard conditional expectation $ E: F^p_{\mathrm{r}}(G,A,\alpha) \to A $, which maps elements to their $ e $-component in the crossed product.
• Construction of an element $ a \in I $ with $ E(a) = 1_A $ for any nonzero ideal $ I $, using $ G $-simplicity of $ A $ and the density of compactly supported functions.
• Employment of the mean ergodic theorem-type argument: averaging conjugates $ u_{h_j} a u_{h_j}^{-1} $ to approximate the identity, showing invertibility in the ideal.
• Leveraging the fact that $ G $-invariant traces on $ A $ induce traces on $ F^p_{\mathrm{r}}(G,A,\alpha) $ via $ \sigma \mapsto \sigma \circ E $, and proving the converse via trace invariance under conjugation.
• Use of the conditional expectation to relate traces on the crossed product to traces on the base algebra $ A $, establishing a natural bijection.

Experimental results

Research questions

• RQ1 Under what conditions is the reduced $ L^p $ operator crossed product $ F^p_{\mathrm{r}}(G,A,\alpha) $ simple for $ p \in (1,\infty) $?
• RQ2 How are traces on $ F^p_{\mathrm{r}}(G,A,\alpha) $ related to $ G $-invariant traces on $ A $?
• RQ3 Does the uniqueness of the normalized trace on $ F^p_{\mathrm{r}}(G,A,\alpha) $ hold when $ A $ has a unique normalized trace?
• RQ4 Why does the simplicity result fail for $ p = 1 $, even for countable Powers groups?
• RQ5 To what extent can Powers' $ C^* $-algebra results on simplicity and trace uniqueness be generalized to the $ L^p $ operator algebra framework?

Key findings

• For $ p \in (1,\infty) $, the reduced $ L^p $ operator crossed product $ F^p_{\mathrm{r}}(G,A,\alpha) $ is simple when $ G $ is a countable Powers group, $ A $ is unital and $ G $-simple, and $ (G,A,\alpha) $ is a separable, nondegenerately representable isometric $ G $-$ L^p $ operator algebra.
• Traces on $ F^p_{\mathrm{r}}(G,A,\alpha) $ are in natural bijection with $ G $-invariant traces on $ A $, given by $ \tau = \sigma \circ E $, where $ E $ is the standard conditional expectation.
• If $ A $ has a unique normalized trace, then $ F^p_{\mathrm{r}}(G,A,\alpha) $ also has a unique normalized trace.
• The reduced $ L^p $ group algebra $ F^p_{\mathrm{r}}(G) $ of a countable Powers group $ G $ is simple and admits a unique normalized trace for $ p \in (1,\infty) $.
• The result does not hold for $ p = 1 $: $ F^1_{\mathrm{r}}(G) $ and $ F^1(G) $ are not simple for any countable discrete group $ G $, as shown by the existence of nontrivial ideals via the trivial homomorphism to $ \mathbb{C} $.
• The unitality of $ A $ is essential; the result fails for nonunital $ L^p $ operator algebras, even in the $ p=2 $ case.