[Paper Review] Full extensions and approximate unitary equivalences

Huaxin Lin|ArXiv.org|Jan 19, 2004

Advanced Operator Algebra Research18 references4 citations

TL;DR

This paper establishes that for unital separable amenable C*-algebras A and C*-algebras C with specific infinite properties, two full monomorphisms from A to C are approximately unitarily equivalent if and only if they represent the same element in the KL group KL(A,C). For non-unital σ-unital C*-algebras B with M(B)/B satisfying certain conditions, full essential extensions of A by B are approximately unitarily equivalent precisely when they induce the same KL class in KL(A,M(B)/B), and the set of such classes forms a group isomorphic to KL(A,M(B)/B) when A satisfies the Universal Coefficient Theorem.

ABSTRACT

Let $A$ be a unital separable amenable \CA and $C$ be a unital \CA with certain infinite property. We show that two full monomorphisms $h_1, h_2: A o C$ are approximately unitarily equivalent if and only if $[h_1]=[h_2]$ in $KL(A,C).$ Let $B$ be a non-unital but $σ$-unital \CA for which $M(B)/B$ has the certain infinite property. We prove that two full essential extensions are approximately unitarily equivalent if and only if they induce the same element in $KL(A, M(B)/B).$ The set of approximately unitarily equivalence classes of full essential extensions forms a group. If $A$ satisfies the Universal Coefficient Theorem, it is can be identified with $KL(A, M(B)/B).$

Motivation & Objective

• To classify full monomorphisms and essential extensions of C*-algebras up to approximate unitary equivalence.
• To determine when the KL group KL(A, M(B)/B) classifies full essential extensions of a unital separable amenable C*-algebra A by a σ-unital C*-algebra B.
• To show that under the Universal Coefficient Theorem, the group of approximate unitary equivalence classes of full essential extensions is isomorphic to KL(A, M(B)/B).
• To extend classification results beyond stable C*-algebras, where KK1 theory fails to detect unitary equivalence.

Proposed method

• Use of the Busby invariant to represent essential extensions as monomorphisms τ: A → M(B)/B.
• Application of the Universal Coefficient Theorem (UCT) to relate K-theory and extension classes.
• Employment of property (P1), (P2), and (P3) on M(B)/B to ensure the existence of full monomorphisms and approximate unitary equivalence.
• Leverage the concept of 'purely large' extensions and 'approximate absorbing' extensions to characterize unitary equivalence.
• Use of ultrapower techniques and commutant analysis in the corona algebra to control approximate unitary equivalence.
• Constructive proof via embedding A into O2 and using stable equivalence in M(B)/B to build full monomorphisms with prescribed KL class.

Experimental results

Research questions

• RQ1 When are two full monomorphisms h1, h2: A → C approximately unitarily equivalent for a unital separable amenable C*-algebra A and a unital C*-algebra C with infinite properties?
• RQ2 Under what conditions on B is the group of approximate unitary equivalence classes of full essential extensions of A by B isomorphic to KL(A, M(B)/B)?
• RQ3 Can the KL group KL(A, M(B)/B) classify full essential extensions when B is non-stable and σ-unital, even if KK1(A,B) fails to do so?
• RQ4 Is every full essential extension approximately absorbing, and when does this imply approximate unitary equivalence?
• RQ5 To what extent do properties (P1), (P2), and (P3) on M(B)/B ensure that full extensions are approximately unitarily equivalent if they induce the same KL class?

Key findings

• Two full monomorphisms h1, h2: A → C are approximately unitarily equivalent if and only if [h1] = [h2] in KL(A,C), provided A is unital, separable, amenable, and C has the required infinite properties.
• For non-unital σ-unital C*-algebras B such that M(B)/B satisfies (P1), (P2), and (P3), two full essential extensions are approximately unitarily equivalent iff they induce the same element in KL(A, M(B)/B).
• The set of approximate unitary equivalence classes of full essential extensions forms a group, which is isomorphic to KL(A, M(B)/B) when A satisfies the Universal Coefficient Theorem.
• When B is stable, KK1(A,B) classifies unitary equivalence classes of full essential extensions, but this fails in general for non-stable B.
• There exist non-stable, non-unital σ-unital C*-algebras B for which M(B)/B satisfies (P1), (P2), and (P3), allowing the same classification via KL theory.
• Full extensions may be approximately absorbing without being purely large, showing that approximate absorption is a strictly weaker condition than purity in the sense of Elliott and Kucerovsky.