IJPAM: Volume 52, No. 3 (2009)
GRADED FRACTAL BUNDLES
VS. CUNTZ ALGEBRAS AND THEIR CENTRAL EXTENSIONS






University of

Pomorska 149/153,


Polish Academy of Sciences

Banacha 22, PL-90-238

e-mail: jlawryno@uni.lodz.pl

Kolejowa 22, Radom, PL-26-600, POLAND
e-mail: gosianmk@poczta.onet.pl

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