# IJPAM: Volume 52, No. 3 (2009)

**A DUALITY THEOREM FOR INOCULATED**

GRADED FRACTAL BUNDLES

VS. CUNTZ ALGEBRAS AND THEIR CENTRAL EXTENSIONS

GRADED FRACTAL BUNDLES

VS. CUNTZ ALGEBRAS AND THEIR CENTRAL EXTENSIONS

Institute of Physics

University of ódz

Pomorska 149/153, ódz, PL-90-236, POLAND

Institute of Mathematics

Polish Academy of Sciences ódz Branch

Banacha 22, PL-90-238 ódz, POLAND

e-mail: jlawryno@uni.lodz.pl

High School of Business

Kolejowa 22, Radom, PL-26-600, POLAND

e-mail: gosianmk@poczta.onet.pl

Department of Mathematics

College of Humanities and Sciences

Nihon University

Sakurajosui 3-25-40, Setagaya-ku, Tokyo, 156-8550, JAPAN

e-mail: osuzuki@cssa.chs.nihon-u.ac.jp

**Abstract.**In a recent paper (2003) two of us (Julian awrynowicz 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 of the bundle, starting from a fixed -th embranchment of , is replaced
by a branch of another fractal of the bundle. We say that is inoculated at its -th
embranchment by a branch of . Another kind of inoculation is when a branch of is added
to as an extra branch at the -th embranchment of . 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
-extensions of Cuntz -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