Covariant representations of subproduct systems

Viselter, Ami

doi:10.1112/plms/pdq047

Cited by 23 publications

(38 citation statements)

References 44 publications

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“…In fact, one also looks at subproduct systems over more general semigroups, and it is useful to allow fibers that are Hilbert W*-correspondences, and not just Hilbert spaces, but such generality is beyond the scope of the present work. Subproduct systems give rise to a class of natural operator algebras, and in recent years these algebras have been investigated by several researchers [9,23,25,26,35,43,84,85]. We will now explain how algebras of bounded analytic functions on homogeneous varieties are operator algebras associated with subproduct systems, and indicate points of intersection with previous works.…”

Section: Corollary 92mentioning

confidence: 99%

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Salomon

Shalit

Shamovich

2018

Trans. Amer. Math. Soc.

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We study algebras of bounded, noncommutative (nc) analytic functions on nc subvarieties of the nc unit ball. Given an nc variety V in the nc unit ball B d , we identify the algebra of bounded analytic functions on V -denoted H ∞ (V) -as the multiplier algebra Mult H V of a certain reproducing kernel Hilbert space H V consisting of nc functions on V. We find that every such algebra H ∞ (V) is completely isometrically isomorphic to the quotient H ∞ (B d )/J V of the algebra of bounded nc holomorphic functions on the ball by the ideal J V of bounded nc holomorphic functions which vanish on V. In order to demonstrate this isomorphism, we prove that the space H V is an nc complete Pick space (a fact recently proved -by other methods -by Ball, Marx and Vinnikov).We investigate the problem of when two algebras H ∞ (V) and H ∞ (W) are (completely) isometrically isomorphic. If the variety W is the image of V under an nc analytic automorphism of B d , then H ∞ (V) and H ∞ (W) are completely isometrically isomorphic. We prove that the converse holds in the case where the varieties are homogeneous; in general we can only show that if the algebras are completely isometrically isomorphic, then there must be nc holomorphic maps between the varieties (in the case d = ∞ we need to assume that the isomorphism is also weak- * continuous).We also consider similar problems regarding the bounded analytic functions that extend continuously to the boundary of B d and related norm closed algebras; the results in the norm closed setting are somewhat simpler and work for the case d = ∞ without further assumptions.Along the way, we are led to consider some interesting problems on function theory in the nc unit ball. For example, we study various versions of the Nullstellensatz (that is, the problem of to what extent an ideal is determined by its zero set), and we obtain perfect Nullstellensatz in both the homogeneous as well as the commutative cases.Analogues of the Nevanlinna-Pick interpolation on the noncommutative ball first appeared in [21] and [12]. More general noncommutative versions of the classical interpolation and realization results appeared recently in the works of Agler and M c Carthy [5] and Ball, Marx and Vinnikov [14,15], who also introduced a generalization of reproducing kernel Hilbert spaces to the free setting.Our first goal in this work is to show that the full Fock space is a noncommutative reproducing kernel Hilbert space (nc RKHS), and its algebra of multipliers is, on the one hand, the algebra of bounded functions on the noncommutative ball (such that the multiplier norm and the supremum norm coincide), and, on the other hand, that this algebra coincides with the WOT closed algebra considered by Arias-Popescu and Davidson-Pitts. With this identification in hand, our second goal is to show that several results of [14] in the case of the noncommutative ball follow from established operator algebraic techniques and results, in particular, the complete Pick property of the noncommutative kernel of the full Fock space.W...

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Section: Corollary 92mentioning

confidence: 99%

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Salomon

Shalit

Shamovich

2018

Trans. Amer. Math. Soc.

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“…See [21, p. 193, (2)] or [17, Example 2.9] for details.The original definition of subproduct systems ([23, Definitions 1.1, 6.2]) is for the context of von Neumann algebras. We require its adaptation to the C * -setting of [26]. Definition 1.5.…”

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confidence: 99%

Cuntz–pimsner Algebras for Subproduct Systems

Viselter

2012

Int. J. Math.

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In this paper we generalize the notion of Cuntz-Pimsner algebras of C * -correspondences to the setting of subproduct systems. The construction is jus-Definition 1.1 ([17, Definition 2.1]). Let M , N be C * -algebras. A (right) Hilbert C * -module E over M is called an N −M (C * -) correspondence if it is also equipped with a left N -module structure, implemented by a * -homomorphism ϕ : N → L(E); that is, a · ζ := ϕ(a)ζ for a ∈ N , ζ ∈ E. We say that E is faithful if ϕ is faithful and essential if ϕ(N )E is total in E. If N = M , we say that E is a (C * -) correspondence over M . Example 1.2. Every Hilbert space is a C * -correspondence over C. Example 1.3. If M is a C * -algebra and α is an endomorphism of M , we write α M for the C * -correspondence that is equal to M as sets, with the obvious right M -module structure, and left M -action given by ϕ(a)b := α(a)b for a, b ∈ M .Example 1.4. Every quiver (directed graph) possesses an associated C * -correspondence. See [21, p. 193, (2)] or [17, Example 2.9] for details.The original definition of subproduct systems ([23, Definitions 1.1, 6.2]) is for the context of von Neumann algebras. We require its adaptation to the C * -setting of [26]. Definition 1.5. A family X = (X(n)) n∈Z + of C * -correspondences over a C * -algebra M is called a (standard) subproduct system if X(0) = M and for all n, m ∈ Z + , X(n+m) is an orthogonally-complementable sub-correspondence of X(n)⊗X(m). This implies, in particular, that X(n) is essential for each n ∈ N.Example 1.6. If E is an essential C * -correspondence over M , the product system X E , defined by X E (n) := E ⊗n for each n ∈ Z + , is trivially a subproduct system. Example 1.7 ([23, Example 1.3]). Fix a Hilbert space H, and let X(n) := H n (the n-fold symmetric tensor product of H) for every n. The resulting family X satisfies the requirements of Definition 1.5. It is called the symmetric subproduct system over H, and denoted by SSP H . Specifically, we put SSP d := SSP C d for d ∈ N and SSP ∞ := SSP ℓ 2 (N) .The reader is urged to consult [23] for many other interesting examples of subproduct systems. Given a subproduct system X, we shall use the following notation throughout the paper. Set E := X(1). The X-Fock space is the sub-correspondence F X := n∈Z + X(n)

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“…This work also connects to the ongoing effort to understand operator algebras arising from subproduct systems (see [13,14,23,24,37,38,39]). If we restrict attention to homogeneous ideals, then the algebras studied in this paper are precisely the algebras arising from commutative subproduct systems over N, with finite dimensional Hilbert spaces as fibres.…”

Section: Introduction Notation and Preliminariesmentioning

confidence: 69%

Essential normality, essential norms and hyperrigidity

Kennedy

Shalit

2015

Journal of Functional Analysis

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Abstract. Let S = (S 1 , . . . , S d ) denote the compression of the dshift to the complement of a homogeneous ideal I of C[z 1 , . . . , z d ].Arveson conjectured that S is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary.The unital norm closed algebra B I generated by S 1 , . . . , S d modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on V := Z(I) ∩ B d , where Z(I) is the variety corresponding to I. Consequently, the essential norm of an element in B I is equal to the sup norm of its Gelfand transform, and the C*-envelope of B I is identified as the algebra of continuous functions on V ∩ ∂B d , which means it is a complete invariant of the topology of the variety determined by I in the ball.Motivated by this determination of the C*-envelope of B I , we suggest a new, more qualitative approach to the problem of essential normality. We prove the tuple S is essentially normal if and only if it is hyperrigid as the generating set of a C*-algebra, which is a property closely connected to Arveson's notion of a boundary representation.We show that most of our results hold in a much more general setting. In particular, for most of our results, the ideal I can be replaced by an arbitrary (not necessarily homogeneous) invariant subspace of the d-shift.2010 Mathematics Subject Classification. 47A13, 47L30, 46E22.

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Covariant representations of subproduct systems

Cited by 23 publications

References 44 publications

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Cuntz–pimsner Algebras for Subproduct Systems

Essential normality, essential norms and hyperrigidity

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