Removing the quadratik-ring restriction in splitexact equivariant algebraic GK^G-theory

Determine whether, after omitting the generalized inverse corner embeddings from the definition of splitexact equivariant algebraic GK^G-theory (as in Definition 3.2 of Burgstaller–Garkusha [gk]), the theory can be formulated on the category of all algebras or rings rather than being restricted to quadratik rings. Specifically, establish that the technical requirements used in [gk]—injectivity of the multiplication map A → Hom_A(A,A), availability of the adjoint G-action on Hom_A(A,A) induced by multiplication in A, and occasional use of approximate units—can be satisfied without imposing the quadratik-ring hypothesis, thereby allowing GK^G-theory to encompass all algebras and rings.

Background

The paper studies a splitexact, homotopy-invariant, stable, equivariant algebraic version of KK-theory (GK^G-theory) defined via generators and relations, and proves that in the very special setting—where only ordinary corner embeddings are declared invertible—GK^G(,A) coincides with classical equivariant K-theory in the sense of Phillips (and with Cuntz’s kk for locally convex algebras in the non-equivariant case).

In prior work [gk], the object class was restricted to quadratik rings to ensure certain technical properties, including injectivity of the multiplication map and well-defined adjoint G-actions on module endomorphism algebras. In this note, the authors discuss implications of omitting generalized inverse corner embeddings and conjecture that, under such omission, the restriction to quadratik rings may be unnecessary, potentially broadening GK^G-theory to all algebras or rings.