E-theory is a special case of KK-theory

Klaus Thomsen; Vladimir Manuilov

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E-theory is a special case of KK-theory

Vladimir Manuilov, Klaus Thomsen

Operator Algebras · Submitted 9 Sep 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim It is shown that there are natural isomorphisms $E(A,B)=KK(SA,Q(B))=[SA,Q(B)\otimes K]$, where $SA=C_0(0,1)\otimes A$, $[\cdot,\cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B)/B$ is the generalized Calkin algebra and K denotes the $C^*$-algebra of compact operators.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,486 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.13 · mentions code or data availability 0.01.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Let A and B be $C^*$-algebras, A separable, and B $\sigma$-unital and stable. It is shown that there are natural isomorphisms $E(A,B)=KK(SA,Q(B))=[SA,Q(B)\otimes K]$, where $SA=C_0(0,1)\otimes A$, $[\cdot,\cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B)/B$ is the generalized Calkin algebra and K denotes the $C^*$-algebra of compact operators.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let A and B be $C^*$-algebras, A separable, and B $\sigma$-unital and stable.	background 0.99	0.06	0.02	unclear 0.61
s1	It is shown that there are natural isomorphisms $E(A,B)=KK(SA,Q(B))=[SA,Q(B)\otimes K]$, where $SA=C_0(0,1)\otimes A$, $[\cdot,\cdot]$ denotes the set of homotopy classes of -homomorphisms, $Q(B) = M(B)/B$ is the generalized Calkin algebra and K denotes the $C^$-algebra of compact operators.	result 0.98	0.88	0.92	asserted 1.00

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Original abstract

Let A and B be $C^*$-algebras, A separable, and B $\sigma$-unital and stable. It is shown that there are natural isomorphisms $E(A,B)=KK(SA,Q(B))=[SA,Q(B)\otimes K]$, where $SA=C_0(0,1)\otimes A$, $[\cdot,\cdot]$ denotes the set of homotopy classes of *-homomorphisms, $Q(B) = M(B)/B$ is the generalized Calkin algebra and K denotes the $C^*$-algebra of compact operators.

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