[Paper Review] Representations of Group Algebras in Spaces of Completely Bounded Maps

Roger R. Smith, Nico Spronk|ArXiv.org|Oct 27, 2003

Advanced Operator Algebra Research4 references3 citations

TL;DR

This paper investigates the range of the map $\Gamma_\pi: \mathrm{M}(G) \to \mathcal{CB}^\sigma(\mathcal{B}(\mathcal{H}))$, defined by $\Gamma_\pi(\mu) = \int_G \pi(s) \otimes \pi(s)^* \, d\mu(s)$, identifying it with the extended Haagerup tensor product $\mathcal{B}(\mathcal{H}) \otimes^{eh} \mathcal{B}(\mathcal{H})$. The key result is that $\Gamma_\pi(\mathrm{L}^1(G)) \subset \mathcal{C}^*_\pi \otimes^h \mathcal{C}^*_\pi$ if and only if the representation $\pi$ is norm continuous, providing a characterization of norm continuity via tensor product structure.

ABSTRACT

Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded maps on B(H). We study the range of the map Gamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\otimes pi(s)^*dmu(s) where we identify CB^sigma(B(H)) with the extended Haagerup tensor product B(H)\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated by integrating pi to L^1(G) is unital exactly when pi is norm continuous to show that Gamma_pi(L^1(G))\subset B(H)\otimes^{eh}B(H) exactly when pi is norm continuous. For the case that G is abelian, we study Gamma_pi(M(G)) as a subset of the Varopoulos algebra. We also characterise positive definite elements of the Varopoulos algebra in terms of completely positive operators.

Motivation & Objective

• To understand the range of the map $\Gamma_\pi$ from the measure algebra $\mathrm{M}(G)$ to the space of normal completely bounded maps on $\mathcal{B}(\mathcal{H})$.
• To determine when the image of $\mathrm{L}^1(G)$ under $\Gamma_\pi$ lies in the Haagerup tensor product $\mathcal{C}^*_\pi \otimes^h \mathcal{C}^*_\pi$.
• To characterize positive and completely positive elements in the Varopoulos algebra $\mathrm{V}^b(E_\pi)$ arising from the Fourier-Stieltjes transform of $\Gamma_\pi(\mathrm{M}(G))$.
• To establish a connection between the topological structure of the spectral support $E_\pi$ and the tensor product membership of $\Gamma_\pi(f)$ for $f \in \mathrm{L}^1(G)$.

Proposed method

• The authors use the identification of $\mathcal{CB}^\sigma(\mathcal{B}(\mathcal{H}))$ with the extended Haagerup tensor product $\mathcal{B}(\mathcal{H}) \otimes^{eh} \mathcal{B}(\mathcal{H})$ via duality and weak* convergence of operator series.
• They analyze the map $\Gamma_\pi$ using the Fourier-Stieltjes transform to relate $\Gamma_\pi(\mu)$ to functions on $E_\pi \times E_\pi$, leading to the Varopoulos algebra $\mathrm{V}^b(E_\pi)$.
• The paper applies results on completely positive maps to characterize when $\Gamma_\pi(\mu)$ is completely positive, linking this to positive definiteness of the function $\sigma, \tau \mapsto \hat{\mu}(\sigma \tau^{-1})$ on $E_\pi \times E_\pi$.
• A key technique is the use of the fact that the C*-algebra $\mathcal{C}^*_\pi$ is unital if and only if $\pi$ is norm continuous, which is used to characterize tensor product membership.
• The authors construct explicit examples, such as $G = \mathbb{T}$ with $\pi(z)\xi_n = z^{n^2}\xi_n$, to show that $\Gamma_\pi(f) \in \mathcal{C}^*_\pi \otimes^h \mathcal{C}^*_\pi$ even when $\pi$ is not norm continuous, under specific spectral conditions.
• They employ the Fourier transform on abelian groups to analyze the image of $\Gamma_\pi$ in terms of the support of $\hat{f}$ and its interaction with $E_\pi \cap \sigma E_\pi$.

Experimental results

Research questions

• RQ1 When is the image of $\Gamma_\pi(\mathrm{L}^1(G))$ contained in the Haagerup tensor product $\mathcal{C}^*_\pi \otimes^h \mathcal{C}^*_\pi$?
• RQ2 What characterizes the norm continuity of a strongly continuous unitary representation $\pi$ in terms of the tensor product structure of $\Gamma_\pi(\mathrm{L}^1(G))$?
• RQ3 How can complete positivity of $\Gamma_\pi(\mu)$ be characterized in terms of the Fourier-Stieltjes transform of $\mu$?
• RQ4 Under what conditions on $G$ and $\pi$ is $\Gamma_\pi$ injective or isometric on $\mathrm{L}^1(G)$?
• RQ5 When does $\Gamma_\pi(f)$ lie in $\mathcal{C}^*_\pi \otimes^h \mathcal{C}^*_\pi$ for $f \in \mathrm{L}^1(G)$, even when $\pi$ is not norm continuous?

Key findings

• The image $\Gamma_\pi(\mathrm{L}^1(G))$ lies in $\mathcal{C}^*_\pi \otimes^h \mathcal{C}^*_\pi$ if and only if the representation $\pi$ is norm continuous.
• For abelian $G$, the range of $\Gamma_\pi$ is identified with a Varopoulos algebra $\mathrm{V}^b(E_\pi)$, and $\Gamma_\pi(\mu)$ is completely positive if and only if the function $\sigma, \tau \mapsto \hat{\mu}(\sigma \tau^{-1})$ is positive definite on $E_\pi \times E_\pi$.
• The kernel of $\Gamma_\pi$ is $\{\mu \in \mathrm{M}(G) : \hat{\mu}|_{E_\pi E_\pi^{-1}} = 0\}$, showing that $\Gamma_\pi$ is injective precisely when $\hat{\mu}$ does not vanish on the difference set $E_\pi E_\pi^{-1}$.
• An explicit example shows that $\Gamma_\pi(f) \in \mathcal{C}^*_\pi \otimes^h \mathcal{C}^*_\pi$ even when $\pi$ is not norm continuous, provided $f \in \mathrm{I}_0(G)$ and $\hat{f}$ is supported on a set where $E_\pi \cap \sigma E_\pi$ is compact for some $\sigma$.
• For $G = \mathbb{R}$ with $E_\pi = [0, \infty)$, $\Gamma_\pi$ is injective but not necessarily isometric, and the question of isometry remains open.
• When $E_\pi = \widehat{G}$, such as for the left regular representation, $\Gamma_\pi(\mathrm{L}^1(G)) \cap (\mathcal{C}^*_\pi \otimes^h \mathcal{C}^*_\pi) = \{0\}$, showing that tensor product membership is rare unless $\pi$ is norm continuous.