On super-KMS functionals and entire cyclic cohomology
Let (\({\mathcal A},\Gamma,\alpha_ t,d)\) be a quantum algebra consisting of a (i) \({\mathbb{Z}}_ 2\)-graded, unital \(C^*\)-algebra \({\mathcal A}={\mathcal A}_+\oplus {\mathcal A}_-\) and an automorphism \(\Gamma\) of \({\mathcal A}\) given by \(a=a_++a_-\to a^{\Gamma}=a_+-a_-;\) (ii) a group action \(\alpha\) (a homomorphism of \({\mathbb{R}}\) into Aut \({\mathcal A}\) even under the grading in the sense \(\alpha_ t(a)^{\Gamma}=\alpha_ t(a^{\Gamma});\) (iii) an even derivation \(D=-i(d/dt)\alpha_ t|_{t=0}\) with domain including the norm dense subalgebra \({\mathcal A}_{\infty}\) of elements a in \({\mathcal A}\) for which the map \(t\to \alpha_ t(a)\) has an extension to an entire function \(z\to a(z);\) (iv) a so-called super (odd) derivation d of \({\mathcal A}\) with dense domain D(d) with the properties \((da)^{\Gamma}=-da^{\Gamma}\) and \(d(ab)=(da)b^{\Gamma}+a^{\Gamma}db\); and (v) the square root property \((d^ 2a)^{\Gamma}=d^ 2a^{\Gamma}\) and \(d^ 2(ab)=(d^ 2ab)=(d^ 2a)b+a(d^ 2b).\) Let \({\mathcal C}^ n({\mathcal A})\) denote the space of \((n+1)\)-linear functionals on A which are continuous with respect to the Sobolev norm \(\| a\|_*=\| a\| +\| da\|\). Let \({\mathcal C}({\mathcal A})\) be the set of sequences \(\{f_ n\}\) with \(f_ n\in {\mathcal C}^ n({\mathcal A})\) such that \(n^{1/2}\| f_ n\|_*\to 0\). Then boundary operators b and B (creation and annihilation respectively) are defined on the even and odd parts \({\mathcal C}_+({\mathcal A})\) and \({\mathcal C}_-({\mathcal A})\) of \({\mathcal C}({\mathcal A})\) under the grading and the boundary operator \(\delta =b+B\) is used to define an entire cyclic cohomology. A continuous linear functional \(\omega\) on \({\mathcal A}\) is said to have the super KMS (or sKMS) property with respect to the quantum algebra (\({\mathcal A},\Gamma,\alpha,d)\) if, for every a, b in \({\mathcal A}_{\alpha}\) and every \(z\in {\mathbb{R}}\), \(\omega (\alpha_ z(a)b)=\omega (b^{\Gamma}\alpha_{z+i}(a))\) and if \(\omega \circ d=0\) on D(d). Then the authors show that every sKMS functional \(\omega\) satisfies \(\omega \circ \alpha_ t=\omega\) on \({\mathcal A}\) for all \(t\in {\mathbb{R}}\) and \(\omega \circ \alpha_ z=\omega\) on \({\mathcal A}_{\alpha}\) for all \(z\in {\mathbb{C}}\). Also \(\omega (adb(z))=\omega ((db)^{\Gamma}a(1-z))\) for all \(a,b\in {\mathcal A}_{\alpha}\) and \(z\in {\mathbb{C}}.\) The authors then show that the Chern character \(\tau_ n(a_ 0,...,a_ n)\) defined by the authors in a prior paper and associated to a super-KMS functional \(\omega\) by D. Kastler satisfies the entire growth condition \(| \tau_ n(a_ 0,...,a_ n)| \leq (| \omega | (1)/n!)\prod^{n}_{j=0}\| a_ j\|_*.\) This means that a super-KMS functional defines a cocycle in the centre cyclic cohomology for quantum algebras.
centre cyclic cohomology for quantum algebras
scientific article; zbMATH DE number 4132915