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Let  X  be a compact space and  M  be a factor of type      I I ∞  \mathrm {II}_\infty       acting on a separable Hilbert space. Let      K M ( X ) {K_M}(X)       denote the Grothendieck group generated by the semigroup of isomorphism classes of  M -vector bundles over  X , and, if  X  is also metric, let      Ext M ⁡  ( X ) \operatorname {Ext}^M(X)       denote the group of equivalence classes of extensions of      C ( X ) C(X)       relative to  M . We show that      K M ( X ) {K_M}(X)       is the direct sum of the even-dimensional Čech cohomology groups of  X , and that      Ext M ⁡  ( X ) \operatorname {Ext}^M(X)       is the direct product of the odd-dimensional Čech homology groups of  X .

$K$-theory and $K$-homology relative to a ${\rm II}\sb{\infty }$-factor