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Corner embeddings into algebras of compact operators in $K$-theory

Bernhard Burgstaller

TL;DR

The paper develops an algebraic analogue of corner embeddings into algebras of compact operators, proving invertibility in a $K$-theory–like category $\Lambda$ whenever a functional extension $U:{\mathcal E}\to A^n$ exists. It establishes permanence properties for changing coefficient algebras under approximate units, and shows that corner embeddings remain invertible under composition. For finite groups $G$, the authors upgrade non-equivariant functional extensions to equivariant ones, yielding invertibility results for canonical corner embeddings with $G$-actions. Collectively, these results extend the GK^G framework and ensure faithfulness and Morita-type equivalences carry over to very special GK^G-theory, preserving the core structure of $GK^G$-theory and its relation to $KK$-theoretic stability. The work strengthens the unprojection techniques in algebraic $K$-theory-like settings and clarifies how corner embeddings interact with coefficient changes and group actions.

Abstract

We show that in $K$-theory-like categories many corner embeddings into a discrete algebra of compact operators are invertible, and consequently functors on splitexact algebraic $KK$-theory are faithful if and only if they are faithful on the subcategory generated by the homomorphisms.

Corner embeddings into algebras of compact operators in $K$-theory

TL;DR

The paper develops an algebraic analogue of corner embeddings into algebras of compact operators, proving invertibility in a -theory–like category whenever a functional extension exists. It establishes permanence properties for changing coefficient algebras under approximate units, and shows that corner embeddings remain invertible under composition. For finite groups , the authors upgrade non-equivariant functional extensions to equivariant ones, yielding invertibility results for canonical corner embeddings with -actions. Collectively, these results extend the GK^G framework and ensure faithfulness and Morita-type equivalences carry over to very special GK^G-theory, preserving the core structure of -theory and its relation to -theoretic stability. The work strengthens the unprojection techniques in algebraic -theory-like settings and clarifies how corner embeddings interact with coefficient changes and group actions.

Abstract

We show that in -theory-like categories many corner embeddings into a discrete algebra of compact operators are invertible, and consequently functors on splitexact algebraic -theory are faithful if and only if they are faithful on the subcategory generated by the homomorphisms.
Paper Structure (6 sections, 11 theorems, 62 equations)

This paper contains 6 sections, 11 theorems, 62 equations.

Table of Contents

  1. introduction
  2. Invertible corner embeddings
  3. Change of coefficient algebra
  4. Composition of corner embeddings
  5. compact groups equivariance
  6. conclusions

Key Result

Lemma 2.8

Functional homomorphisms and functional extensions, respectively, $(U_i , U_i^* )_{i \in I}$ ($I$ any set) are closed under taking direct sums, the external tensor products and compositions, that is, one has functional homomorphisms and functional extensions, respectively, $(\bigoplus_{i \in I} U_i

Theorems & Definitions (29)

  • Definition 2.1: Functional modules
  • Definition 2.2: Direct sum of functional modules
  • Definition 2.3: Compact operators
  • Definition 2.4: Corner embedding
  • Definition 2.5: Change of coefficient algebra of functional modules
  • Definition 2.6: Functional module homomorphism
  • Definition 2.7: Functional extension
  • Lemma 2.8
  • Lemma 2.9
  • proof
  • ...and 19 more