John Duncan

University of Arkansas | U of A · Department of Mathematical Sciences. Retired

We continue our investigation of the real space H of Hermitian matrices in $${M_n}(\mathbb{C})$$ with respect to norms on $${\mathbb{C}^n}$$ . We complete the commutative case by showing that any proper real subspace of the real diagonal matrices on $${\mathbb{C}^n}$$ can appear as H . For the non-commutative case, we give a complete solution when...

We investigate the real space H of Hermitian matrices in $M_n(\mathbb{C})$ with respect to norms on $\mathbb{C}^n$ . For absolute norms, the general form of Hermitian matrices was essentially established by Schneider and Turner [Schneider and Turner, Linear and Multilinear Algebra (1973), 9–31]. Here, we offer a much shorter proof. For non-absolute...

We introduce an alternative description of the McAlister monoid, MX, on a set X, and show that, for a field F, the convolution algebra FMX /FΘ is not primitive when X is uncountable. We then investigate a superalgebra of FMX /FΘ and, when F = C, an associated C*-algebra, and show that these algebras are primitive if and only if |X| = ω1.

94.36 Wheels within wheels - Volume 94 Issue 531 - Victor Bryant, John Duncan

We investigate minimal, compact, square-closed subsets of the unit circle by identifying them with subsets of [0,1] and with sets of infinite words on two symbols; in particular, such sets arising from substitution maps. We consider problems on finiteness, recognizability, square roots, density and measure. Properties of the free semigroup on two s...

Let $S$ be the semigroup with identity, generated by $x$ and $y$, subject to $y$ being invertible and $yx=xy^2$. We study two Banach algebra completions of the semigroup algebra $\mathbb{C}S$. Both completions are shown to be left-primitive and have separating families of irreducible infinite-dimensional right modules. As an appendix, we offer an a...

Let S be the semigroup with generators $s_j ,s_j^* $(j = 1,2,..., N) and defining relations $s_j^* s_j = 1$, $s_j^* s_k = \theta (j \ne k)$, where 1 is the identity and θ the zero. We determine the spectrum of the average of these generators in l₁(S)/ℂθ, showing it to be an ellipse together with its interior. The major and minor diameters of the el...

Let Ea(u,v) be the extremal algebra determined by two hermitians u and v with u² = v² = 1. We show that: Ea(u,v) = {f=gu:f,g ε C(T)}, where T is the unit circle; Ea(u,v) is C^*-equivelant to C^*(G), where G is the infinite dihedral group; most of the hermitian elements k od Ea(u,v) have the property that k<...

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Given double-struck S sign ⊂ ℕ, let double-struck S sign̂ be the set of all positive integers m for which h m is hermitian whenever h is an element of a complex unital Banach algebra A with h n hermitian for each n ∈ double-struck S sign. We attempt to characterize when (i) double-struck S sign̂ = ℕ, or (ii) double-struck S sign̂ = double-struck S...

Minimum geometric conditions must exist to provide an ample amount of preview sight distance (PVSD) for comfortable and safe traffic operations. The PVSD concept is based on the assumption that the driver views or previews the roadway surface and other cues that lie ahead to obtain the information needed for vehicular control and guidance. The driv...

Synopsis We investigate algebras associated with a (discrete) Clifford semigroup S =∪ { G e : e ∈ E {. We show that the representation theory for S is determined by an enveloping Clifford semigroup UC(S) =∪ { G x : x ∈ X } where X is the filter completion of the semilattice E. We describe the representation theory in terms of both disintegration th...

There are various algebras which may be associated with a discrete group G . In particular we may consider the complex group ring ℂ G , the convolution Banach algebra l ¹ (G), the enveloping C *-algebra C *( G ) of l ¹ ( G ), and the reduced C *-algebra determined by the completion of l ¹ ( G ) under the left regular representation on l ² ( G ). Th...

Synopsis By using the Halmos-Wallen description of power partial isometries on Hilbert space, we give a complete description of all monogenic inverse semigroups,ℐ. We also describe the full C* -algebra C*ℐ and the reduced C*-algebra C*(ℐ) with particular emphasis on the case of the free monogenic inverse semigroup ℑℐ t .

Let A be a complex unital Banach algebra. An element u ∈ A is a norm unitary if (For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u ∈ A , we have Sp( u )⊃Γ, where Sp( u ) denotes the spectrum of u and Γ denotes the unit circle in C . If Sp( u )≠Γ we may suppose, by r...

Synopsis If G is a group, then G is amenable as a semigroup if and only if l ¹ ( G ), the group algebra, is amenable as an algebra. In this note, we investigate the relationship between these two notions of amenability for inverse semigroups S . A complete answer can be given in the case where the set E s of idempotent elements of S is finite. Some...

Let be a C *-algebra acting on the Hilbert space H and let be the self-adjoint elements of . The following characterization of commutativity is due to I. Kaplansky (see Dixmier [3, p. 58]).

Numerical Ranges II is a sequel to Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras written by the same authors and published in this series in 1971. The present volume reflects the progress made in the subject, expanding and discussing topics under the general headings of spatial and algebra numerical ranges and fu...

Recall from § 12 that a Banach star algebra is a complex Banach algebra A with an involution * satisfying, for all x, y∈A,α∈ℂ, (i) (x+y)*=x* + y* (ii) (αx)*=α* x* (iii) x**=x (iv) (xy)* =y* x*.

In this section A will denote an algebra. Definitions and results will usually be stated only for left ideals; with the obvious changes, they apply to right ideals.

A will denote a complex Banach algebra with unit. As usual, a complex polynomial in one variable is said to be monic if the coefficient of the term of highest degree is 1. We denote by Pn the set of all complex monic polynomials of degree n.

Throughout this book the symbol F will be used to denote a field that is either the real field ℝ or the complex field ℂ.

In this section A denotes an algebra over F and we consider the purely algebraic theory of irreducible left A-modules.

Let X, Y,Z be normed linear spaces over the same field F. A mapping ø: X × Y→Z is said to be bilinear if (i) for each y∈ Y, the mapping x→ø(x,y) is linear (ii) for each x∈ X, the mapping y→ø(x,y) is linear.

The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a whole. The authors develop various applications, in particular to the study of Banach algebras whe...

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