[Paper Review] Complete isometries into C*-algebras

David P. Blecher, Damon M. Hay|ArXiv.org|Mar 18, 2002

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper characterizes complete isometries between C*-algebras by generalizing Holsztynski's classical result on isometries between C(K) spaces. It shows that a linear map $ T: A \to B $ is a complete isometry if and only if it is completely contractive and admits a reducing $\wedge$-compression of the form $ u\pi(\cdot) $, where $ \pi $ is a *-homomorphism and $ u $ is a partial isometry in $ B^{**} $, with the key insight that the *-homomorphism component determines the isometry, while the complementary part plays a minimal role.

ABSTRACT

We give various characterizations of into (not necessarily onto) complete isometries between $C^*$-algebras, generalizing a classical result of Holsztynski. Our results are related to a natural embedding of the noncommutative Shilov boundary in a second dual.

Motivation & Objective

• To generalize Holsztynski's characterization of isometries between commutative C*-algebras to noncommutative C*-algebras.
• To identify necessary and sufficient conditions for a linear map between C*-algebras to be a complete isometry.
• To clarify the role of the noncommutative Shilov boundary and second dual embeddings in characterizing complete isometries.
• To extend the theory to nonselfadjoint operator algebras, with applications to triple systems and operator space theory.

Proposed method

• Introduce the concept of a reducing $\wedge$-compression of a map $ T: A \to B $, defined via the canonical embedding $ B \to B^{**} $.
• Use the universal representation of $ B $ to represent $ T $ as a block-diagonal operator matrix in $ B(H_u) $, decomposing it into a *-homomorphism part and a complementary part.
• Show that $ T $ is a complete isometry if and only if it is completely contractive and admits a reducing $\wedge$-compression of the form $ u\pi(\cdot) $, where $ \pi $ is a *-homomorphism and $ u $ is a partial isometry in $ B^{**} $.
• Analyze the structure of such compressions using projections in $ B^{**} $, and relate them to quotients of $ B $ by left ideals when $ T $ is unital.
• Establish that for unital maps, the partial isometry $ u $ can be taken as 1, and the complementary part $ S $ is completely positive.
• Use examples involving $ \ell^\infty_n $ and $ M_n $ to demonstrate that the existence of a reducing projection for the complementary part is not guaranteed, even for unital complete isometries.

Experimental results

Research questions

• RQ1 What characterizes a linear map between C*-algebras as a complete isometry, especially when not surjective?
• RQ2 How does the noncommutative Shilov boundary and the second dual $ B^{**} $ influence the structure of complete isometries?
• RQ3 Can Holsztynski’s commutative result on isometries be generalized to noncommutative C*-algebras in a precise and intrinsic way?
• RQ4 What role does the *-homomorphism component play in determining whether a map is a complete isometry, and how does the complementary part affect this?
• RQ5 Under what conditions can a complete isometry be decomposed into a *-homomorphism and a completely positive part in the second dual?

Key findings

• A linear map $ T: A \to B $ between C*-algebras is a complete isometry if and only if it is completely contractive and admits a reducing $\wedge$-compression of the form $ u\pi(\cdot) $, where $ \pi $ is a *-homomorphism and $ u $ is a partial isometry in $ B^{**} $.
• For unital maps, the partial isometry $ u $ can be taken as 1, and the complementary part $ S $ is completely positive, generalizing a result of Choi and Effros.
• The structure of the map is determined primarily by the *-homomorphism component, while the complementary part plays a minimal role in ensuring the isometric property.
• The paper constructs an example of a unital complete isometry $ \Psi: M_n \to B(H) $ for infinite-dimensional $ H $ such that no projection $ p $ exists for which $ (1-p)\Psi(\cdot) $ is a *-homomorphism unless $ p = 1 $, in which case the map is zero.
• This example shows that the existence of a reducing projection for the complementary part is not guaranteed, even in the unital case, and that the noncommutative structure obstructs a direct analogue of the commutative case.
• The theory extends naturally to nonselfadjoint operator algebras, with applications to triple systems and operator space theory, as demonstrated in related work.