C*-Convexity and the Numerical Range

2000 ◽  
Vol 43 (2) ◽  
pp. 193-207 ◽  
Author(s):  
Bojan Magajna
Keyword(s):  
Convex Hull ◽  
Extreme Point ◽  
Von Neumann Algebra ◽  
Numerical Range ◽  
Von Neumann ◽  
Norm Closure

AbstractIf A is a prime C*-algebra, a ∈ A and λ is in the numerical range W(a) of a, then for each ε > 0 there exists an element h ∈ A such that . If λ is an extreme point of W(a), the same conclusion holds without the assumption that A is prime. Given any element a in a von Neumann algebra (or in a general C*-algebra) A, all normal elements in the weak* closure (the norm closure, respectively) of the C*-convex hull of a are characterized.

scholarly journals THE SPECTRAL SCALE AND THE NUMERICAL RANGE

2003 ◽  
Vol 14 (02) ◽  
pp. 171-189 ◽  
Author(s):  
CHARLES A. AKEMANN ◽  
JOEL ANDERSON
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Point Spectrum ◽  
Numerical Range ◽  
Spectral Information ◽  
Von Neumann ◽  
Line Segments ◽  
Information Base ◽  
Arbitrary Operator ◽  
Spectral Scale

Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Let τ be a faithful normal tracial state on N and set b1 = (c + c*)/2 and b2 = (c - c*)/2i. Also write B for the spectral scale of {b1, b2} relative to τ. In previous work by the present authors, some joint with Nik Weaver, B has been shown to contain considerable spectral information about the operator c. In this paper we expand that information base by showing that the numerical range of c is encoded in B also. We begin by proving that the k-numerical range of an arbitrary operator d in B(H) coincides with the numerical range of d when the von Neumann algebra generated by d contains no finite rank operators. Thus, the k-numerical range is not useful for most operators considered here. We next show that the boundary of the numerical range of c is exactly the set of radial complex slopes on B at the origin. Further, we show that points on this boundary that lie in the numerical range are visible as line segments in the boundary of B. Also, line segments on the boundary which lie in the numerical range show up as faces of dimension two in the boundary of B. Finally, when N is abelian, we prove that the point spectrum of c appears as complex slopes of 1-dimensional faces of B.

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

scholarly journals ON PRODUCT SYSTEMS ARISING FROM SUM SYSTEMS

2005 ◽  
Vol 08 (01) ◽  
pp. 1-31 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
R. SRINIVASAN
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebra ◽  
Type I ◽  
Product System ◽  
Von Neumann ◽  
Type Iii ◽  
Product Systems ◽  
Weyl Representation ◽  
Uncountable Family ◽  
Type Spaces

B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and "slightly colored noises", using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples. We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the "Shale map" respects compositions (this settles an old conjecture of K. R. Parthasarathy8). Using this we associate a product system to a sum system. This construction includes the exponential product system of Arveson, as a trivial case, and the type III examples of Tsirelson. By associating a von Neumann algebra to every "elementary set" in [0, 1], in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.

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