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Author Bhattacharyya, Tirthankar ♦ Misra, Gadadhar
Source CiteSeerX
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Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Contractive Homomorphism ♦ Linear Transformation ♦ Arveson Dilation Theorem ♦ Max Space ♦ Recent Example ♦ C2 C2 ♦ Planar Alge-bra ♦ Two-dimensional Hilbert Space ♦ Finite-dimensional Hilbert Space ♦ Certain Special Class ♦ Direct Integral ♦ Operator Space ♦ Suitable Functional Calculus
Abstract Abstract. We consider contractive homomorphisms of a planar alge-bra A(Ω) over a finitely connected bounded domain Ω ⊆ C and ask if they are necessarily completely contractive. We show that a homomor-phism ρ: A(Ω) → B(H) for which dim(A(Ω) / ker ρ) = 2 is the direct integral of homomorphisms ρT induced by operators on two-dimensional Hilbert spaces via a suitable functional calculus ρT: f 7 → f(T), f ∈ A(Ω). It is well known that contractive homomorphisms ρT induced by a linear transformation T: C2 → C2 are necessarily completely con-tractive. Consequently, using Arveson’s dilation theorem for completely contractive homomorphisms, one concludes that such a homomorphism ρT possesses a dilation. In this paper, we construct this dilation explic-itly. In view of recent examples discovered by Dritschel and McCullough, we know that not all contractive homomorphisms ρT are completely contractive even if T is a linear transformation on a finite-dimensional Hilbert space. We show that one may be able to produce an example of a contractive homomorphism ρT of A(Ω) which is not completely contractive if an operator space which is naturally associated with the problem is not the MAX space. Finally, within a certain special class of contractive homomorphisms ρT of the planar algebra A(Ω), we con-struct a dilation. 1.
Educational Role Student ♦ Teacher
Age Range above 22 year
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Education Level UG and PG ♦ Career/Technical Study
Learning Resource Type Article