Martin E. Walter

Algebraic structures determined by 3 by 3 matrix geometry

Using a 3 by 3 matrix trick we show that multiplication (an algebraic structure) in a C*-algebra A is determined by the geometry of the C*-algebra of the 3 by 3 matrices with entries from A, M 3 (A). This is an example of an algebra-geometry duality which, we claim, has applications.

On the norm of a Schur-product

Abstract If A ∘ X is the Schur product of n × n matrices A and X , then we study estimates on the norm of the map X ↦ A ∘ X , where X has the norm it acquires as a linear operator on a complex n -dimensional Hilbert space.

A duality between locally compact groups and certain Banach algebras

We make precise the following statements: B(G), the Fourier-Stieltjes algebra of locally compact group G, is a dual of G and vice versa. Similarly, A(G), the Fourier algebra of G, is a dual of G and vice versa. We define an abstract Fourier (respectively, Fourier-Stieltjes) algebra; we define the dual group of such a Fourier (respectively, Fourier-Stieltjes) algebra; and we prove the analog of the Pontriagin duality theorem in this context. The key idea in the proof is the characterization of translations of B(G) as precisely those isometric automorphisms Φ of B(G) which satisfy ∥ p − eiθΦp ∥2 + ∥ p + eiθΦp ∥2 = 4 for all θ ∈ R and all pure positive definite functions p with norm one. One particularly interesting technical result appears, namely, given x1, x2 ϵ G, neither of which is the identity e of G, then there exists a continuous, irreducible unitary representation π of G (which may be chosen from the reduced dual of G) such that π(x1) ≠ π(e) and π(x2) ≠ π(e). We also note that the group of isometric automorphisms of B(G) (or A(G)) contains as a (“large”) .closed, normal subgroup the topological version of Burnsides “holomorph of G.”

Isomorphisms of hypergroups

Let K1, K2 be locally compact hypergroups. It is shown that every isometric isomorphism between their measure algebras restricts to an isometric isomorphism between their L1-algebras. This result is used to relate isometries of the measure algebras to homeomorphisms of the underlying locally compact spaces.

Semigroups of Positive Definite Functions and Related Topics

Every locally compact group, G, has defined on it a semigroup of continuous, positive definite functions,P(G).This semigroup, additionally equipped with a normalized, partially ordered, convex structure is a complete invariant of the underlying group. This semigroup has an identity and we investigate what it means to differentiate in the classical calculus sense at this identity. This leads us to the concept of a semiderivation. We are also naturally led to consider the cohomology of continuous, unitary representations of G, as well as the “screw functions” of J. von Neumann and I. J. Schoenberg, a Levy-Khinchin formula, and a characterization of groups with property (T).

The dual group of the Fourier-Stieltjes algebra

We announce the definition of the dual group, GB(^, of the Fourier-Stieltjes algebra, B(G), of a locally compact group G; and we state four main theorems culminating in the result that GB(G) is a locally compact topological group which is topological^ isomorphic to G. This result establishes an explicit dual relationship between a group and its Fourier-Stieltjes algebra. Moreover, this result extends naturally the notion of Pontriagin duality to the case of noncommutative groups. 1. We shall adopt the notation and assume familiarity with the results of [3], [4]. We recall from [3], [4] that each <f> e Aut(£(G)), the isometric algebra automorphisms of the Banach algebra B(G), can be written in the form

Off-diagonal matrix coefficients are tangents to state space: Orientation and C*-algebras

Any non-commutative C * -algebra A, e.g., two by two complex matrices, has at least two associative multiplications for which the collection of positive linear functionals is the same. Alfsen and Shultz have shown that by selecting an orientation for the state space K of A, i.e., the convex set of positive linear functionals of norm one, a unique associative multiplication for A is determined. We give a simple method for describing this orientation.

A Complete Invariant of a Locally Compact Group

We will begin our discussion with G, a finite group; although our main result is true for the general case that G is a locally compact group. We write out the multiplication table of G, whose elements are the set {g 1, g 2,..., g n } in “symmetric” fashion. Thus,