Periodic distributions on \(C^*\)-algebras (Q1915540)

We apply the classical theory of (periodic) distributions founded by L. Schwartz to \(C^*\)-algebras. For a \(C^*\)-dynamical system \((A, \alpha, T^n)\) of a \(C^*\)-algebra \(A\) with action \(\alpha\) of \(n\)-dimensional torus \(T^n\), we define a \(C^*\)-left (resp. right) periodic distribution \(\xi\) as a continuous left (resp. right) \(B\)-module map from \({\mathcal D}_\alpha (A)\) to \(B\), where \(B\) is the fixed point \(C^*\)-subalgebra of \(A\) under \(\alpha\) and \({\mathcal D}_\alpha (A)\) is the Fréchet subalgebra of all smooth elements of \(A\). We extend derivations on \({\mathcal D}_\alpha (A)\) induced by the action \(\alpha\) to the \(C^*\)-distributions \({\mathcal D}'_{\alpha\ell} (A)\) so that all elements of \(A\) can be differentiable. Fourier series for \(C^*\)-distributions are studied. We then construct a non-commutative version of Gelfand triplet as the following quadruplet of left \({\mathcal D}_\alpha (A)\) and right \(B\)-modules \[ {\mathcal D}_\alpha (A) \subset L^2_{\alpha r} (A) \subset L^{2\prime}_{\alpha \ell} (A) \subset {\mathcal D}'_{\alpha \ell} (A), \] where \(L^2_{\alpha r} (A)\) denotes the Hilbert \(C^*\)-right \(B\)-module associated to \(\alpha\) and \(L^{2\prime}_{\alpha \ell} (A)\) is its dual. We finally compute the above quadruplet for the Cuntz algebra \({\mathcal O}_\infty\) with canonical action of the torus.