By Andreescu T., Feng Z.

Show description

Read or Download 101 Problems in Algebra From the Training of the USA IMO Team (Enrichment Series, Volume 18) PDF

Best algebra books

Mathematik für Wirtschaftswissenschaftler 2: Lineare Algebra, Funktionen mehrerer Variablen

Mathematik gehört zu den Grundfächern für jeden Studierenden der Wirtschafts- und Sozialwissenschaften. Er benötigt Kenntnisse der research, der Linearen Algebra sowie der Funktionen einer und mehrerer Variablen. Das zweibändige Taschenbuch, hervorgegangen aus Vorlesungen des Autors an der Universität Regensburg, stellt den Studienstoff sehr anschaulich dar, unterstützt durch eine Vielzahl von Beispielen und Abbildungen.

Extra resources for 101 Problems in Algebra From the Training of the USA IMO Team (Enrichment Series, Volume 18)

Sample text

Let a, b be positive elements of A. It shall be shown that a + b is positive. Upon replacing a, b by suitable multiples, it suffices to show that e + a + b is invertible. Since e + a and e + b are invertible, we may define c := a(e + a)−1 , d := b(e + b)−1 . The Rational Spectral Mapping Theorem yields rλ (c) < 1 and rλ (d) < 1. The preceding proposition then gives rλ (cd) < 1, so e − cd is invertible. We now have (e + a)(e − cd)(e + b) = e + a + b, so that e + a + b is invertible. 8). ÈÖÓÔÓ× Ø ÓÒº Let A be a Hermitian Banach ∗-algebra.

Then ∆ Co (Ω) is homeomorphic to Ω. Indeed, for each ω ∈ Ω, consider the evaluation εω at ω given by εω : Co (Ω) → C f → εω (f ) := f (ω). Then the map ω → εω is a homeomorphism Ω → ∆ Co (Ω) , and f (εω ) = f (ω) for all f ∈ C(Ω) and all ω ∈ Ω. Thus, upon identifying εω ∈ ∆ Co (Ω) with ω ∈ Ω, the Gelfand transformation on Co (Ω) becomes the identity map. Before going to the proof, we note the following corollary. 12). ÓÖÓÐÐ Öݺ Two locally compact Hausdorff spaces Ω, Ω′ are homeomorphic whenever the C*-algebras Co (Ω), Co (Ω′ ) are isomorphic as ∗-algebras.

This makes that α belongs to the unit circle: α = e it for 48 2. THE GELFAND TRANSFORMATION some t ∈ R/2πZ. For an integer k, we obtain τ (δk ) = τ (δ1 ) This implies that for a ∈ ℓ 1 (Z), we have k = e ikt . a(k) e ikt . a(τ ) = k∈Z Conversely, by putting for some t ∈ R/2πZ a(k) e ikt τ (a) := k∈Z (a ∈ A), a multiplicative linear functional τ on ℓ 1 (Z) is defined. 1), we have: (ab)(k) e ikt = τ (ab) = k∈Z = l∈Z a(l) b(m) e ikt k∈Z l+m=k a(l) e ilt · b(m) e imt m∈Z = τ (a) · τ (b). As a multiplicative linear functional is determined by its value at δ1 , it follows that ∆(ℓ 1 Z) can be identified with R/2πZ.

Download PDF sample

Rated 4.07 of 5 – based on 34 votes