\(K\)-groups of \(C^*\)-algebras deformed by actions of \(\mathbb{R}^ d\)
Earlier [Mem. Am. Math. Soc., to appear] the author has shown, how to deform the original product on the dense \(*\)-subalgebra \(A^ \infty\) of smooth vectors in \(C^*\)-algebra \(A\) with the action \(\alpha\) of \(\mathbb{R}^ d\) upon \(A\). The new associative product is defined by \[ a\times_ J b= \iint \alpha_{Ju}(a) \alpha_ v(b)\exp(2\pi i(u\cdot v)), \] where \(J\) is an arbitrary skew-symmetric operator on \(\mathbb{R}^ d\), \(a,b\in A^ \infty\) and the integral is defined as an oscillatory integral. (In other papers, the author shows how to construct some quantum groups by this process.) The main result of the present paper is that the (usual) topological \(K\)- groups \(K_ j A_ J\) and \(K_ j A\) are isomorphic for \(j= 0,1\). The main step in the proof consists in showing that \(A_ J\) is a generalized fixed-point algebra for a suitable action -- this implies that \(A_ J\) is strongly Morita equivalent to a certain crossed product algebra for an action of \(\mathbb{R}^ d\). Then Connes' Thom isomorphism theorem is invoked to conclude the proof. The main theorem applies to show that the quantum versions of various spaces have the same \(K\)-groups as the original spaces.
scientific article; zbMATH DE number 515248