[Paper Review] A reflexivity criterion for Hilbert C*-modules over commutative C*-algebras

Michael Frank, Vladimir Manuilov+1 more|arXiv (Cornell University)|Aug 10, 2009

Advanced Operator Algebra Research3 references3 citations

TL;DR

This paper establishes a topological criterion for C*-reflexivity of Hilbert C*-modules over commutative C*-algebras, showing that a commutative C*-algebra $ C(X) $ is C*-reflexive if and only if the canonical inclusion $ \bigoplus_k I_k \subset C(X) $ does not extend to $ \prod_k I_k \subset C(X) $ for any sequence of pairwise orthogonal, non-zero C*-subalgebras $ I_k $. The result provides a complete characterization using properties of the underlying topological space $ X $, particularly its Baire category and Stone-Čech compactification structure.

ABSTRACT

A C*-algebra $A$ is C*-reflexive if any countably generated Hilbert C*-module $M$ over $A$ is C*-reflexive, i.e. the second dual module $M''$ coincides with $M$. We show that a commutative C*-algebra $A$ is C*-reflexive if and only if for any sequence $I_k$ of disjoint non-zero C*-subalgebras, the canonical inclusion $\oplus_k I_k\subset A$ doesn't extend to an inclusion of $\prod_k I_k$.

Motivation & Objective

• To characterize C*-reflexive Hilbert C*-modules over commutative C*-algebras using topological and algebraic conditions.
• To establish a necessary and sufficient condition for $ C(X) $ to be C*-reflexive in terms of the non-embedding of infinite products of ideals into $ C(X) $.
• To generalize and refine prior results on C*-reflexivity by connecting it to Baire space properties and the Stone-Čech compactification.
• To provide a criterion applicable to countably generated Hilbert C*-modules over commutative C*-algebras via the stabilization theorem.

Proposed method

• Use of the dual and second dual module constructions for Hilbert C*-modules over $ C^* $-algebras, with canonical inclusions $ M \subset M'' \subset M' $.
• Application of the Kasparov stabilization theorem to reduce the problem to the standard module $ H_A = l_2(A) $.
• Construction of a functional $ F \in H_A'' $ via $ F(f) = \sum_k a_k^* f_k $ for sequences $ (a_k) \in \prod_k I_k $, with $ \|a_k\| = 1 $.
• Topological analysis of the spectrum $ X $ of $ C(X) $, focusing on disjoint open sets $ U_k $ and their associated $ C_0(U_k) $ ideals.
• Use of the Baire category theorem and properties of the Stone-Čech compactification $ \beta Y $ to analyze the extension of bounded continuous functions.
• Proof by contradiction: assuming $ \prod_k C_0(U_k) \subset C(X) $ leads to sequences violating the $ \lim |a_{n+1} - a_n| = 0 $ condition in the Higson compactification $ \nu\mathbb{N} $.

Experimental results

Research questions

• RQ1 When is a commutative $ C^* $-algebra $ C(X) $ C*-reflexive for all countably generated Hilbert $ C^* $-modules over it?
• RQ2 What topological conditions on the spectrum $ X $ ensure that $ \prod_k I_k \not\subset C(X) $ for orthogonal ideals $ I_k $?
• RQ3 How does the Baire property of $ X $ relate to the reflexivity of $ H_{C(X)} = l_2(C(X)) $?
• RQ4 Can the condition $ \prod_k I_k \not\subset C(X) $ serve as a necessary and sufficient criterion for C*-reflexivity in the non-commutative case?
• RQ5 What role does the Stone-Čech compactification play in determining whether $ C(X) $ is C*-reflexive?

Key findings

• A commutative $ C^* $-algebra $ A $ is C*-reflexive if and only if for any sequence $ \{I_k\} $ of pairwise orthogonal, non-zero $ C^* $-subalgebras, the inclusion $ \bigoplus_k I_k \subset A $ does not extend to $ \prod_k I_k \subset A $.
• The standard Hilbert $ C^* $-module $ H_{C(X)} = l_2(C(X)) $ is not C*-reflexive if and only if there exists a sequence of pairwise disjoint open sets $ \{U_k\} $ in $ X $ such that $ \prod_k C_0(U_k) \subset C(X) $.
• The $ C^* $-algebra $ A = C(\nu\mathbb{N}) $, consisting of sequences with $ \lim |a_{n+1} - a_n| = 0 $, is C*-reflexive, as shown by contradiction assuming such a product inclusion exists.
• The Higson compactification $ \nu\mathbb{N} $ provides a counterexample where the sufficient condition from earlier work is not necessary, as $ \beta\mathbb{N} \subset \nu\mathbb{N} $ but $ \prod_k C_0(U_k) \not\subset C(\nu\mathbb{N}) $.
• The criterion holds for all countably generated Hilbert $ C^* $-modules over $ C(X) $, due to the Kasparov stabilization theorem reducing the problem to $ H_{C(X)} $.
• The result confirms that $ C(X) $ is C*-reflexive for compact metric spaces $ X $, extending earlier results by Mishchenko and Trofimov.