Von Neumann Algebra VN Research Articles

A probabilistic approach to Hilbert transforms on free group von Neumann algebras

The paper contains a probabilistic proof of the L p L^p -boundedness of the Hilbert transform in the context of a free group von Neumann algebra V N ( F q ) VN(\mathbb {F}_q) . The argument rests on noncommutative version of good- λ \lambda inequalities and yields a tight L log ⁡ L L\log L order of the L p L^p norms as p → 1 + p\to 1^+ and p → ∞ p\to \infty .

How the norm one positive definite functions determine a finite group

We study a finite (discrete) group G through the information we obtain fromP1(G)={〈π(⋅)ξ,ξ〉:π:G→U(H)is unitary,ξ∈H,‖ξ‖=1} of norm one positive definite functions of G, arising from matrix coefficients of any unitary representation π. Below we summarize our results.(a)Knowing P1(G) as a set of functions, when G is a finite abelian group we can determine G≅∏jZpjrj as a direct product of its cyclic subgroups of prime power orders.(b)Knowing P1(G) as a multiplicative semigroup, we can construct the subgroup lattice L(G) of G. With L(G) in stock, we can tell if G is cyclic, simple, perfect, solvable, supersolvable, or nilpotent. When G′ is a finite simple group with P1(G′)≅P1(G) as multiplicative semigroups, we show that G′≅G as groups.(c)Knowing P1(G) as a compact convex set, we can construct the group von Neumann algebra vN(G) of G. Consequently, when G′ is another finite group with P1(G)≅P1(G′) as convex sets, we show that vN(G)≅vN(G′) as von Neumann algebras. In particular, we can tell if G is abelian.

Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

AbstractBuilding on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group and 1-injectivity of as an operator -module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra VN(G) is 1-injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded -module maps on , and give a simpliûed proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

Three-dimensional Gaussian fluctuations of non-commutative random surfaces along time-like paths

We construct a continuous-time non-commutative Markov operator on U(glN) with corresponding morphisms U(glN)→End(L2(U(N)))⊗∞ at any fixed times 0≤t1<t2<…. This is an analog of a continuous-time non-commutative Markov operator on the group von Neumann algebra vN(U(N)) constructed in [16], and is a variant of discrete-time non-commutative random walks on U(glN)[2,9].It is also shown that when restricting to the Gelfand–Tsetlin subalgebra of U(glN), the non-commutative Markov operator matches a (2+1)-dimensional random surface model introduced in [7]. As an application, it is then proved that the moments converge to an explicit Gaussian field along time-like paths. Combining with [7] which showed convergence to the Gaussian free field along space-like paths, this computes the entire three-dimensional Gaussian field. In particular, it matches a Gaussian field from eigenvalues of random matrices [5].

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L2(G)) of bounded linear operators on L2(G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E⊆G is a set of multiplicity if and only if the set E⁎={(s,t)∈G×G:ts−1∈E} is a set of operator multiplicity. Analogous results are established for M1-sets and M0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function ψ:G→C defines a closable multiplier on the reduced C⁎-algebra Cr⁎(G) of G if and only if Schur multiplication by the function N(ψ):G×G→C, given by N(ψ)(s,t)=ψ(ts−1), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L2(G). Similar results are obtained for multipliers on VN(G).

Orthogonally additive polynomials on Fourier algebras

We show that an n-homogeneous polynomial P on the Fourier algebra A(G) of a locally compact group G can be represented in the form P(f)=〈T,fn〉(f∈A(G)) for some T in the group von Neumann algebra VN(G) of G if and only if it is orthogonally additive and completely bounded.

Weak factorizations of operators in the group von Neumann algebras of certain amenable groups and applications

Let G be a compact group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group and A(G × H) be the Fourier algebra of G × H. We prove that the group von Neumann algebra VN(G × H) = A(G × H)* has the weak uniform A(G × H)** factorization property of level b(G). As a corollary we show that A(G × H) is strongly Arens irregular, and the topological centre of UC 2(G × H)* is equal to the Fourier–Stieltjes algebra B(G × H).

On p-approximation properties for p-operator spaces

This paper has a two-fold purpose. Let 1 < p < ∞ . We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on L p -spaces. We then apply these properties to the study of the pseudofunction algebra PF p ( G ) , the pseudomeasure algebra PM p ( G ) , and the Figà–Talamanca–Herz algebra A p ( G ) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C ∗-algebra C λ ∗ ( G ) , the group von Neumann algebra VN ( G ) , and the Fourier algebra A ( G ) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PF p ( G ) , PM p ( G ) , and A p ( G ) , respectively. The p-completely bounded multiplier algebra M cb A p ( G ) plays an important role in this work.

On ideals in the bidual of the Fourier algebra and related algebras

Let G be a compact nonmetrizable topological group whose local weight b ( G ) has uncountable cofinality. Let H be an amenable locally compact group, A ( G × H ) the Fourier algebra of G × H , and UC 2 ( G × H ) the space of uniformly continuous functionals in VN ( G × H ) = A ( G × H ) ∗ . We use weak factorization of operators in the group von Neumann algebra VN ( G × H ) to prove that there exist at least 2 2 b ( G ) left ideals of dimensions at least 2 2 b ( G ) in A ( G × H ) ∗ ∗ and in UC 2 ( G × H ) ∗ . We show that every nontrivial right ideal in A ( G × H ) ∗ ∗ and in UC 2 ( G × H ) ∗ has dimension at least 2 2 b ( G ) .

Harmonic operators: the dual perspective

The study of harmonic functions on a locally compactgroupGhas recently been transferred to a non-commutative setting in two different directions: Chu and Lau replaced the algebra L ∞ (G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L ∞ (G) by the canonical action of a positive definite function σ on VN(G); on the other hand, Jaworski and the first author replaced L ∞ (G) by B(L 2 (G)) to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action of σ on VN(G) can be extended to B(L 2 (G)). We study the corresponding space ˜ Hσ

Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces

Consider a compact locally symmetric space M of rank r , with fundamental group Γ . The von Neumann algebra VN ( Γ ) is the convolution algebra of functions f ∈ ℓ 2 ( Γ ) which act by left convolution on ℓ 2 ( Γ ) . Let T r be a totally geodesic flat torus of dimension r in M and let Γ 0 ≅ Z r be the image of the fundamental group of T r in Γ . Then VN ( Γ 0 ) is a maximal abelian ★ -subalgebra of VN ( Γ ) and its unitary normalizer is as small as possible. If M has constant negative curvature then the Pukánszky invariant of VN ( Γ 0 ) is ∞ .

The Von Neumann Algebra VN(G) of a Locally Compact Group and Quotients of Its Subspaces

AbstractLet VN(G) be the von Neumann algebra of a locally compact group G. We denote by μ the initial ordinal with |μ| equal to the smallest cardinality of an open basis at the unit of G and X = ﹛α ; α &lt; μ﹜.We show that if G is nondiscrete then there exist an isometric *-isomorphism of l∞(X) into VN(G) and a positive linear mapping π of VN(G) onto l∞(X) such that π o = idl∞(X) and and π have certain additional properties. Let UCB((Ĝ)) be the C*–algebra generated by operators in VN(G) with compact support and F(Ĝ) the space of all T∈ VN(G) such that all topologically invariant means on VN(G) attain the same value at T. The construction of the mapping π leads to the conclusion that the quotient space UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) has l∞(X) as a continuous linear image if G is nondiscrete. When G is further assumed to be non-metrizable, it is shown that UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) contains a linear isomorphic copy of l∞(X). Similar results are also obtained for other quotient spaces.

Weakly almost periodic elements in 𝐿_{∞}(𝐺) of a locally compact group

Let G G be a locally compact abelian group with dual group G G acting on L ∞ ( G ) {L_\infty }(G) by pointwise multiplication. We show that if L ∞ ( G ) {L_\infty }(G) contains a nonzero element f f such that O ( f ) = { x ⋅ f : χ ∈ G ^ } O(f) = \left \{ {x \cdot f:\chi \in \hat G} \right \} is relatively compact in the weak (or norm) topology of L ∞ ( G ) {L_\infty }(G) , then G G is discrete. In this case O ( f ) O(f) is relatively compact in the weak or norm topology of L ∞ ( G ) {L_\infty }(G) if and only if f f vanishes at infinity. A related result when G G acts on the von Neumann algebra V N ( G ) VN(G) is also determined.

On the structure of 𝑉𝑁(𝐺) for almost connected groups

We prove that for any almost connected locally compact group G G , the von Neumann algebra VN ( G ) {\text {VN}}\left ( G \right ) , generated by the left regular representation of G G is semifinite.

Invariant means on a class of von Neumann algebras

For G a locally compact group with associated von Neumann algebra VN(G) we prove the existence of an invariant mean on VN(G). This mean is shown to be unique if and only if G is discrete. 1. Definitions and preliminaries. Throughout we follow in general the notation of [3]. Let G be a locally compact group with identity e. Let L 1(G) be the group algebra of G with respect to left Haar measure and let C *(G) be the enveloping C *-algebra of L 1(G). Denote by B(G) the dual space of C *(G). Then B(G) may be realized as the space of all finite linear combinations of continuous positive-definite functions on G. With the dual norm and pointwise multiplication, B(G) is a commutative Banach algebra-the Fourier-Stieltjes algebra of G. The closed subalgebra A(G) generated by elements with compact support is called the Fourier algebra of G. Let P = u e A(G): u is positive definite and llu|l = u(e) = 1}. For x a complex-valued (a.e. defined) function on G, denote by X, x the functions x(g) = x(g 1), x(g) = x(g) where denotes complex conjugation. If / C L1(G), denote by Lf the bounded operator on L2(G) defined by Lfx = / * x (where * denotes convolution). An important property of A(G) is that it is precisely the set of functions of the form x * y where x, y e L2(G) [3, p. 2181. Let VN(G) denote the vonNeumann algebra on L 2(G) generated by the operators Lf where / e L 1(G). Then VN(G) may be regarded as the dual space of A(G) under the map (T, u) (Tx, y) if u e A(G) with ui = y *X. In particular, the w*and weak operator topologies on VN(G) coincide. For u C A(G), T C VN(G) define the operator uT e VN(G) by (uT, v) = T, uv), v C A(G). It follows readily that VN(G) is an A(G)-module and that |luT|| < Ilull 1ITIh. A linear functional m on VN(G) is called a mean if Received by the editors June 28, 1971. AMS 1970 subject classfications. Primary 22D15, 43A15; Secondary 22D25, 43A07, 43A40.