Group Theory

Download e-book for kindle: Abelian Groups and Modules by K.M. Rangaswamy, David Arnold

By K.M. Rangaswamy, David Arnold

ISBN-10: 0824797892

ISBN-13: 9780824797898

Includes the lawsuits of a global convention on abelian teams and modules held lately in Colorado Springs. provides the most recent advancements in abelian teams that experience facilitated cross-fertilization of latest concepts from assorted components reminiscent of the illustration idea of posets, version concept, set thought, and module concept.

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For each α ∈ , let Pα denote the orthogonal projection of H(π ) onto Kα . Define U : H(π ) → ⊕α∈ Kα by U ξ = (Pα ξ )α∈ , for each ξ ∈ H(π ). It is a routine application of the definitions to check that U is a unitary map intertwining π with ⊕α∈ π Kα . For any two representations π and σ of G, let HomG (π, σ ) = {T ∈ B(H(π ), H(σ )) : T π (x) = σ (x)T , for all x ∈ G}, where B(H(π), H(σ )) denotes the space of bounded linear operators from H(π ) to H(σ ). The elements of HomG (π, σ ) are called intertwining operators of π and σ .

A function ϕ : G → C is called positive definite if n αi αj ϕ(xj−1 xi ) ≥ 0, i,j =1 for any x1 , . . , xn ∈ G, α1 , . . , αn ∈ C, and n ∈ N. 57 Let ϕ be a bounded continuous function on G. Then the following are equivalent. (i) ϕ is positive definite. (ii) ϕ is of positive type, that is, G (f ∗ ∗ f )(x)ϕ(x)dx ≥ 0 for all f ∈ L1 (G). 6 C ∗ -algebras and weak containment of representations It will turn out that C ∗ -algebras play a major role in later chapters of this book, not only because associated with every locally compact group is its group C ∗ -algebra (see below).

Thus, for any x ∈ G, ⊥ π(x)ξ = π K (x)ξ + π K (x)ξK⊥ . We then say that π is decomposed as the direct sum of the two subrepresen⊥ ⊥ tations π K and π K and write π = π K ⊗ π K . More generally, we make the following definition. 30 Let be a nonempty set. For each α ∈ , let Hα be a Hilbert space and let πα be a representation of G with H(πα ) = Hα . For (ξα )α∈ ∈ ⊕α∈ Hα , the Hilbert space direct sum, let (⊕α∈ πα )(x) ((ξα )α∈ ) = (πα (x)(ξα ))α∈ , for all x ∈ G. Then ⊕α∈ πα is called the direct sum of the πα , α ∈ .

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Abelian Groups and Modules by K.M. Rangaswamy, David Arnold


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