New York Journal of Mathematics
Volume 19 (2013) 511-531


Sameer Chavan

C*-algebras generated by spherical hyperexpansions

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Published: August 19, 2013
Keywords: Drury-Arveson m-shift, subnormality, spherical hyperexpansivity, spherical Cauchy dual, Toeplitz algebra, boundary representation
Subject: Primary 47A13, 47B37, 46L05; Secondary 47B20, 46E20

Let T be a spherical completely hyperexpansive m-variable weighted shift on a complex, separable Hilbert space H and let Ts denote its spherical Cauchy dual. We obtain the hyperexpansivity analog of the structure theorem of Olin-Thomson for the C*-algebra C*(T) generated by T, under the natural assumption that Ts is commuting. If, in addition, the defect operator I - T1T*1 - ... - TmT*m is compact then we ensure exactness of the sequence of C*-algebras
0 → C(H) → C*(T) → C(σap(T)) → 0,
where C(H) stands for the ideal of compact operators on H, and
π : C*(T) → C(σap(T))
is the unital *-homomorphism defined by π(Ti)= zi (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 2-tuples Tνλ acting on P2λ) (1 ≦ λ ≦ 2), where dμλ:= dνλ dσ, νλ 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 2-shift Tν1 and the Drury-Arveson 2-shift Tν2 occupy the extreme positions. We would like to emphasize that Tνλ is unitarily equivalent to the multiplication operator tuples in P2λ) if and only if λ =1.

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Indian Institute of Technology Kanpur, Kanpur- 208016, India