[Paper Review] Morita Transforms of Tensor Algebras

Paul S. Muhly, Baruch Solel|arXiv (Cornell University)|Jul 20, 2010

Advanced Operator Algebra Research16 references3 citations

TL;DR

This paper establishes that a Morita equivalence between C*-correspondence pairs (E,M) and (F,N) induces an isometric functor between the categories of Hilbert modules over their respective tensor algebras T₊(E) and T₊(F). The functor preserves absolute continuity and provides a new interpretation of Popescu’s reconstruction operator via stabilization using Cuntz families of isometries.

ABSTRACT

We show that if $M$ and $N$ are $C^{*}$-algebras and if $E$ (resp. $F$) is a $C^{*}$-correspondence over $M$ (resp. $N$), then a Morita equivalence between $(E,M)$ and $(F,N)$ implements a isometric functor between the categories of Hilbert modules over the tensor algebras of $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$. We show that this functor maps absolutely continuous Hilbert modules to absolutely continuous Hilbert modules and provides a new interpretation of Popescu's reconstruction operator.

Motivation & Objective

• To clarify the explicit isometric isomorphism between the representation theories of tensor algebras T₊(E) and T₊(F) when (E,M) and (F,N) are Morita equivalent C*-correspondence pairs.
• To show that tensoring with the equivalence bimodule X implements a functor that preserves key structural properties of Hilbert modules, particularly absolute continuity.
• To provide a new, concrete interpretation of Popescu’s reconstruction operator through stabilization using Cuntz families of isometries in the context of Morita equivalence.
• To demonstrate that the induced representation theory of T₊(E) and T₊(F) is isometrically isomorphic via a construction rooted in Rieffel induction and intertwiners.
• To explore the role of different stabilizations of C*-algebras in the context of Morita equivalence, particularly in the case of (C^d, C) and its stabilization via compact operators.

Proposed method

• Utilizes the notion of Morita equivalence for C*-correspondence pairs via a C*-equivalence bimodule X satisfying X⊗ₙF ≅ E⊗ₘX as C*-correspondences.
• Applies Rieffel’s induced representation theory to construct Hilbert modules over T₊(E) and T₊(F) via the induced action on E⊗σH and F⊗σH.
• Establishes a bijective correspondence between completely contractive bimodule maps T:E→B(H) and contractive intertwiners T̃:E⊗σH→H, enabling explicit realization of representations.
• Constructs the isometric functor between Hilbert module categories by tensoring with the equivalence bimodule X, showing it preserves the structure of absolutely continuous modules.
• Uses Cuntz families {Sᵢ} of isometries on ℓ²(ℕ) to define endomorphisms α of 𝒦(ℓ²(ℕ)), enabling the identification of stabilized correspondences like C_d(𝒦) ≅ α𝒦.
• Reinterprets Popescu’s reconstruction operator as the adjoint of the induced intertwiner (T̃^X)* = ∑ᵢ Sᵢ ⊗ Tᵢ*, linking it explicitly to the Morita equivalence framework.

Experimental results

Research questions

• RQ1 How does a Morita equivalence between C*-correspondence pairs (E,M) and (F,N) induce an isometric isomorphism between the categories of Hilbert modules over T₊(E) and T₊(F)?
• RQ2 What is the precise role of the equivalence bimodule X in implementing this isometric functor, and how does it preserve the property of absolute continuity?
• RQ3 Can Popescu’s reconstruction operator be reinterpreted as a natural consequence of Morita equivalence and stabilization in the context of C*-correspondences?
• RQ4 How do different stabilizations of C*-algebras—such as 𝒦(ℓ²(ℕ))—interact with Morita equivalence and influence the structure of tensor algebras?
• RQ5 To what extent does the choice of Cuntz family {Sᵢ} affect the resulting reconstruction operator in the stabilized setting?

Key findings

• A Morita equivalence between (E,M) and (F,N) via a C*-equivalence bimodule X induces an isometric functor between the categories of Hilbert modules over T₊(E) and T₊(F).
• This functor maps absolutely continuous Hilbert modules over T₊(E) to absolutely continuous Hilbert modules over T₊(F), preserving their structural and analytic properties.
• The induced representation of T₊(F) on H is realized as the composition of the induced representation on E⊗σH and the intertwiner T̃, with the action of X providing the isometric isomorphism.
• Popescu’s reconstruction operator is reinterpreted as the adjoint of the induced intertwiner (T̃^X)* = ∑ᵢ Sᵢ ⊗ Tᵢ*, where {Sᵢ} is a Cuntz family of isometries on ℓ²(ℕ).
• The stabilization of (C^d, C) via the equivalence bimodule C_∞(𝒦) yields a correspondence isomorphic to α𝒦, where α is the endomorphism defined by ∑ᵢ Sᵢ a Sᵢ*, showing that α𝒦 is a valid stabilization of C^d.
• The construction demonstrates that the dependence of the reconstruction operator on the choice of Cuntz family {Sᵢ} is intrinsic and potentially significant, suggesting deeper structural implications in noncommutative function theory.