 

Sameer Chavan
C*algebras generated by spherical hyperexpansions view print


Published: 
August 19, 2013 
Keywords: 
DruryArveson mshift, subnormality, spherical hyperexpansivity, spherical Cauchy dual, Toeplitz algebra, boundary representation 
Subject: 
Primary 47A13, 47B37, 46L05; Secondary 47B20, 46E20 


Abstract
Let T be a spherical completely hyperexpansive mvariable
weighted shift on a complex, separable Hilbert space H
and let T^{s} denote its spherical Cauchy dual. We
obtain the hyperexpansivity analog of the structure theorem of
OlinThomson for the C*algebra C*(T) generated by T, under
the natural assumption that T^{s} is commuting. If, in
addition, the defect operator I  T_{1}T*_{1}  ...  T_{m}T*_{m} is
compact then we ensure exactness of the sequence of C*algebras
0 → C(H)
→ C*(T) →
C(σ_{ap}(T)) → 0,
where C(H) stands for the ideal of compact operators on H, and π : C*(T) → C(σ_{ap}(T))
is the unital *homomorphism defined by
π(T_{i})= z_{i} (i=1, ..., m). This unifies and generalizes the
results of Coburn, 1973/74 and Arveson, 1998. We further
illustrate our results by exhibiting a one parameter family
F of spherical completely hyperexpansive 2tuples
T_{νλ} acting on P^{2}(μ_{λ}) (1 ≦ λ ≦
2), where dμ_{λ}:= dν_{λ} dσ,
ν_{λ} is a probability measure on [0, 1], and σ
is the normalized surface area measure on the unit sphere ∂B. Interestingly, within the family F, the
Szegö 2shift T_{ν1} and the DruryArveson
2shift T_{ν2} occupy the extreme positions. We would like to
emphasize that T_{νλ} is unitarily equivalent to the
multiplication operator tuples in P^{2}(μ_{λ}) if and only
if λ =1.


Author information
Indian Institute of Technology Kanpur, Kanpur 208016, India
chavan@iitk.ac.in

