IJPAM: Volume 52, No. 3 (2009)

A DUALITY THEOREM FOR INOCULATED
GRADED FRACTAL BUNDLES
VS. CUNTZ ALGEBRAS AND THEIR CENTRAL EXTENSIONS

Julian \Lawrynowicz$^1$, Ma\lgorzata Nowak-Kepczyk$^2$, Osamu Suzuki$^3$
$^1$Institute of Physics
University of \Lódz
Pomorska 149/153, \Lódz, PL-90-236, POLAND
$^1$Institute of Mathematics
Polish Academy of Sciences \Lódz Branch
Banacha 22, PL-90-238 \Lódz, POLAND
e-mail: [email protected]
$^2$High School of Business
Kolejowa 22, Radom, PL-26-600, POLAND
e-mail: [email protected]
$^3$Department of Mathematics
College of Humanities and Sciences
Nihon University
Sakurajosui 3-25-40, Setagaya-ku, Tokyo, 156-8550, JAPAN
e-mail: [email protected]


Abstract.In a recent paper (2003) two of us (Julian \Lawrynowicz and Osamu Suzuki) and K. Nôno have dealt with the duality problem for fractals of the flower type and branch type. In various problems of complex analysis and physics of condensed phase we need, however, graded fractal bundles of the branch type with the property that a branch of a fractal $\Sigma _1$ of the bundle, starting from a fixed $n$-th embranchment of $\Sigma _1$, is replaced by a branch of another fractal $\Sigma _2$ of the bundle. We say that $\Sigma _1$ is inoculated at its $n$-th embranchment by a branch of $\Sigma _2$. Another kind of inoculation is when a branch of $\Sigma _2$ is added to $\Sigma _1$ as an extra branch at the $n$-th embranchment of $\Sigma _1$. Analogous situation can be imagined in the case of graded fractal bundles of the flower type. An example of inoculated fractal is given, refering to the periodicity in the case of graded fractal bundles related to complex and quaternionic structures.

We prove the existence of duality between inoculated graded fractal bundles of the flower type and branch type, which is called flower-branch duality (Theorem 1). Next, we introduce in our context a concept of central $C^*$-extensions of Cuntz $C^*$-algebras and make a Fock representation on an inoculated graded fractal bundle of the branch type (Theorem 2). We prove the corresponding duality theorem between the representations of Cuntz algebras and their central extensions (Theorem 3). Finally it is suggested how the duality theorems can be applied to several topics in complex analysis and physics of condensed phase.

Received: January 28, 2008

AMS Subject Classification: 81R25, 32L25, 53A50, 15A66

Key Words and Phrases: Cuntz algebra, bilinear form, quadratic form, fractal

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 3