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Extending cellular cohomology to $ C\sp *$-algebras


Authors: Ruy Exel and Terry A. Loring
Journal: Trans. Amer. Math. Soc. 329 (1992), 141-160
MSC: Primary 46L80; Secondary 19K56, 46M20, 58A10, 58G12
DOI: https://doi.org/10.1090/S0002-9947-1992-1024770-7
MathSciNet review: 1024770
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Abstract: A filtration on the $ K$-theory of $ {C^*}$-algebras is introduced. The relative quotients define groups $ {H_n}(A),n \geq 0$, for any $ {C^*}$-algebra $ A$, which we call the spherical homology of $ A$. This extends cellular cohomology in the sense that

$\displaystyle {H_n}(C(X)) \otimes {\mathbf{Q}} \cong {H^n}(X;{\mathbf{Q}})$

for $ X$ a finite CW-complex. While no extension of cellular cohomology which is derived from a filtration on $ K$-theory can be additive, Morita-invariant, and continuous, $ {H_n}$ is shown to be infinitely additive, Morita invariant for unital $ {C^*}$-algebras, and continuous in limited cases.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1024770-7
Keywords: $ {C^*}$-algebras, homology, $ K$-theory, filtration, determinant
Article copyright: © Copyright 1992 American Mathematical Society

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