Noncommutative Lp modules

By Marius Junge, David Sherman. 2003

62 citations recorded by the source corpus.

We construct classes of von Neumann algebra modules by considering ``column sums" of noncommutative L^p spaces. Our abstract characterization is based on an L^{p/2}-valued inner product, thereby generalizing Hilbert C*-modules and representations on Hilbert space. While the (single) representation theory is similar to the L^2 case, the concept of L^p bimodule (p not 2) turns out to be nearly trivial.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/419515e9e78790ebb28a6a9d6b0b8c1146e093bf

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In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

BMO spaces associated with semigroups of operators (2011, background): “t have to obtain a larger space that the interpolation couple BMO,Lpbelongs to. 5.2.1. Lp-Hilbert module. We will need the following deﬁnition and lemma of Lp-Hilbert module due to Junge/Sherman (see [JS05]). For p = ∞ these spaces are well-known through the GNS construction for a completely positiv”
Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, methodology): “NONCOMMUTATIVE BURKHOLDER/ROSENTHAL INEQUALITIES II: APPLICATIONS 25 Indeed, P is a the projection onto a submodule (see [Lan] for more details, see also [JS] for a general treatment without assuming N∗ separable based on [Pas]).”
Rosenthal Type inequalities for free chaos. (2005, background): “29], there exists a normal right B-module map u: A → C∞(B) satisfying (16) E(x∗y) = X∞ k=1 uk(x)∗uk(y) = u(x)∗u(y) for all x,y∈ A, where ukstands for the k-th coordinate of u. Note that, according to [13, 16], this map canonically extends to Lp(A). On the other hand, recalling that amalgamation give”
Noncommutative Riesz transforms -- a probabilistic approach (2008, background): “Moreover, uk is a Nk right module map with complemented range (see [Jun02] and for the non-separable case [JS05]).”
Noncommutative martingale deviation and Poincaré type inequalities with applications (2012, unknown): “ we want to remove separability, we can use the fact that every Hilbert right module over N embeds isometrically in a self-dual module. Indeed, this follows from [45]. Let us sketch the approach from [33]. Consider the L1/2(N) module Y = X⊗N L1(N). Then the antilinear dual Y∗ is self-dual and obviou”
Smooth Fourier multipliers on group von Neumann algebras (2010, unknown): “t(a ∗ ja ′ k)b ′ k. When p= ∞, Lp(L(G)⊗¯S ψ,t L(G)) is the closure of algebraic tensors with respect to the strong operator topology determined by the seminorms ϕ(hz,ziψ,t) with ϕ∈ L(G)∗. We refer to [36, 54] for the following facts i) L∞(L(G)⊗¯S ψ,t L(G)) is the dual space of L1(L(G)⊗¯S ψ,t L(G)). ”