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Register a new userHilbert C*-systems for actions of the circle group
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Added by
Hellmut Baumg\\\"artel
Publication date
2000
fields
Physics
and research's language is
English
Authors
H. Baumgaertel
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The paper contains constructions of Hilbert systems for the action of the circle group $T$ using subgroups of implementable Bogoljubov unitaries w.r.t. Fock representations of the Fermion algebra for suitable data of the selfdual framework: ${cal H}$ is the reference Hilbert space, $Gamma$ the conjugation and $P$ a basis projection on ${cal H}.$ The group $C({spec} {cal Z}to T)$ of $T$-valued functions on ${spec} {cal Z}$ turns out to be isomorphic to the stabilizer of ${cal A}$. In particular, examples are presented where the center ${cal Z}$ of the fixed point algebra ${cal A}$ can be calculated explicitly.
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We prove a conjecture of Ambrus, Ball and Erd{e}lyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form sum_{k=1}^n f(d(z,z_k)), where $f:[0,pi]to [0,infty]$ is non-increasing and strictly convex and $d(z,w)$ denotes the geodesic distance between $z$ and $w$ on the circle.
Given a C*-algebra $A$, a discrete abelian group $X$ and a homomorphism $Theta: Xto$ Out$A$ defining the dual action group $Gammasubset$ aut$A$, the paper contains results on existence and characterization of Hilbert ${A,Gamma}$, where the action is given by $hat{X}$. They are stated at the (abstract) C*-level and can therefore be considered as a refinement of the extension results given for von Neumann algebras for example by Jones [Mem.Am.Math.Soc. 28 Nr 237 (1980)] or Sutherland [Publ.Res.Inst.Math.Sci. 16 (1980) 135]. A Hilbert extension exists iff there is a generalized 2-cocycle. These results generalize those in [Commun.Math.Phys. 15 (1969) 173], which are formulated in the context of superselection theory, where it is assumed that the algebra $A$ has a trivial center, i.e. $Z=C1$. In particular the well-known ``outer characterization of the second cohomology $H^2(X,{cal U}(Z),alpha_X)$ can be reformulated: there is a bijection to the set of all $A$-module isomorphy classes of Hilbert extensions. Finally, a Hilbert space representation (due to Sutherland in the von Neumann case) is mentioned. The C*-norm of the Hilbert extension is expressed in terms of the norm of this representation and it is linked to the so-called regular representation appearing in superselection theory.
We derive the complete asymptotic expansion in terms of powers of $N$ for the geodesic $f$-energy of $N$ equally spaced points on a rectifiable simple closed curve $Gamma$ in ${mathbb R}^p$, $pgeq2$, as $N to infty$. For $f$ decreasing and convex, such a point configuration minimizes the $f$-energy $sum_{j eq k}f(d(mathbf{x}_j, mathbf{x}_k))$, where $d$ is the geodesic distance (with respect to $Gamma$) between points on $Gamma$. Completely monotonic functions, analytic kernel functions, Laurent series, and weighted kernel functions $f$ are studied. % Of particular interest are the geodesic Riesz potential $1/d^s$ ($s eq 0$) and the geodesic logarithmic potential $log(1/d)$. By analytic continuation we deduce the expansion for all complex values of $s$.
We show that if $G_1$ and $G_2$ are non-solvable groups, then no $C^{1,tau}$ action of $(G_1times G_2)*mathbb{Z}$ on $S^1$ is faithful for $tau>0$. As a corollary, if $S$ is an orientable surface of complexity at least three then the critical regularity of an arbitrary finite index subgroup of the mapping class group $mathrm{Mod}(S)$ with respect to the circle is at most one, thus strengthening a result of the first two authors with Baik.
We generalize the concept of coherent states, traditionally defined as special families of vectors on Hilbert spaces, to Hilbert modules. We show that Hilbert modules over $C^*$-algebras are the natural settings for a generalization of coherent states defined on Hilbert spaces. We consider those Hilbert $C^*$-modules which have a natural left action from another $C^*$-algebra say, $mathcal A$. The coherent states are well defined in this case and they behave well with respect to the left action by $mathcal A$. Certain classical objects like the Cuntz algebra are related to specific examples of coherent states. Finally we show that coherent states on modules give rise to a completely positive kernel between two $C^*$-algebras, in complete analogy to the Hilbert space situation. Related to this there is a dilation result for positive operator valued measures, in the sense of Naimark. A number of examples are worked out to illustrate the theory.
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