Representable E-Theory for C0(X)-Algebras

Abstract

Let X be a locally compact space, and let A and B be C 0 (X)-algebras. We define the notion of an asymptotic C 0 (X)-morphism from A to B and construct representable E-theory groups RE(X; A, B). These are the universal groups on the category of separable C 0 (X)-algebras that are C 0 (X)-stable, C 0 (X)-homotopy-invariant, and half-exact. If A is RKK(X)-nuclear, these groups are naturally isomorphic to Kasparov's representable KK-theory groups RKK(X; A, B). Applications and examples are also discussed. Academic Press

Equivariant E-theory for groupoids acting on C∗-algebras

“…One can construct the E-theory for C 0 ðX Þ-algebras starting with this as definition, as it was done in [26].…”

“…When G is a topological space X ; the group E X ðA; BÞ was also defined in [26] where it is denoted REðX ; A; BÞ (as corresponding to the group RKKðX ; A; BÞ defined by Kasparov [18]) and called accordingly the representable E-theory for C 0 ðX Þ-algebras. In this case a Haar system is just a Radon measure with full support X and there is an isomorphism KðH X ÞCK#C 0 ðX Þ; where K denotes the algebra of compact operators on a separable Hilbert space.…”

E-theory for C[0, 1]-algebras with finitely many singular points

“…In this paper, we obtain information on KK X (A, B) using the E-theory groups E X (A, B), [8], [4]. It is known [8,Theorem 4.7] that E X (A, B) coincides with KK X (A, B) when X is a locally compact Hausdorff space and A is a separable continuous nuclear C 0 (X)-algebra. Furthermore, the fact that E-theory satisfies excision for all extensions of C 0 (X)-algebras enables us to compute the E [0,1] -group for a class of elementary C[0, 1]-algebras using the E-theory of their fibers and the E-theory classes of the connecting maps.…”

“…In this paper, we obtain information on KK X (A, B) using the E-theory groups E X (A, B), [8], [4]. It is known [8,Theorem 4.7] that E X (A, B) coincides with KK X (A, B) when X is a locally compact Hausdorff space and A is a separable continuous nuclear C 0 (X)-algebra.…”

“…is approximately X-equivariant if and only if it satisfies(2) for only those open sets U n in a countable subbasis (U n ) n of the topology of X.If in addition X is a locally compact Hausdorff space, then by [4, Lemma 2.11] ϕ is an approximately X-equivariant morphism if and only if ϕ is an asymptotic C 0 (X)-morphism in the sense of[8, Definition 3.1]. i.e.…”