- Levene, Rupert H.

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# Levene, Rupert H.

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Levene, Rupert H.

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Levene, Rupert H.

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- PublicationNorms of idempotent Schur multipliersLet D be a masa in B(H) where H is a separable Hilbert space. We find real numbers Î·0 < Î·1 < Î·2 < Â· Â· Â· < Î·6 so that for every bounded, normal D-bimodule map Î¦ on B(H), either kÎ¦k > Î·6 or kÎ¦k = Î·k for some k âˆˆ {0, 1, 2, 3, 4, 5, 6}. When D is totally atomic, these maps are the idempotent Schur multipliers and we characterise those with norm Î·k for 0 â‰¤ k â‰¤ 6. We also show that the Schur idempotents which keep only the diagonal and superdiagonal of an n Ã— n matrix, or of an nÃ—(n+ 1) matrix, both have norm 2 n+1 cot(Ï€ 2(n+1) ), and we consider the average norm of a random idempotent Schur multiplier as a function of dimension. Many of our arguments are framed in the combinatorial language of bipartite graphs.
70 - PublicationGrothendieck's inequality in the noncommutative Schwartz spaceIn the spirit of Grothendieck's famous inequality from the theory of Banach spaces, we study a sequence of inequalities for the noncommutative Schwartz space, a FrÃ©chet algebra of smooth operators. These hold in nonoptimal form by a simple nuclearity argument. We obtain optimal versions and reformulate the inequalities in several different ways.
310ScopusÂ© Citations 2 - PublicationCompletely bounded norms of right module mapsIt is well-known that if T is a Dmâ€“Dn bimodule map on the mÃ—n complex matrices, then T is a Schur multiplier and kTkcb = kTk. If n = 2 and T is merely assumed to be a right D2-module map, then we show that kTkcb = kTk. However, this property fails if m â‰¥ 2 and n â‰¥ 3. For m â‰¥ 2 and n = 3, 4 or n â‰¥ m2 we give examples of maps T attaining the supremum C(m, n) = sup{kTkcb : T a right Dn-module map on Mm,n with kTk â‰¤ 1}, we show that C(m, m2) = âˆš m and succeed in finding sharp results for C(m, n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on K(H) which is not completely bounded.
237ScopusÂ© Citations 2 - PublicationCommutants of weighted shift directed graph operator algebrasWe consider non-selfadjoint operator algebras L(G, Î») generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs G. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for L(G, Î») in the case of the single vertex graph with two edges and a suitable choice of left weight function Î».
246ScopusÂ© Citations 4 - PublicationSchur idempotents and hyperreflexivityWe show that the set of Schur idempotents with hyperreflexive range is a Boolean lattice which contains all contractions. We establish a preservation result for sums which implies that the weak* closed span of a hyperreflexive and a ternary masa-bimodule is hyperreflexive, and prove that the weak* closed span of finitely many tensor products of a hyperreflexive space and a hyperreflexive range of a Schur idempotent (respectively, a ternary masa-bimodule) is hyperreflexive.
307ScopusÂ© Citations 1 - PublicationPrivate algebras in quantum information and infinite-dimensional complementarityWe introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.
328ScopusÂ© Citations 11