C*-Algebras and Operator Theory [Hardcover]

Gerard J. Murphy (Author)

Book Description

Publication Date: September 11, 1990 | ISBN-10: 0125113609 | ISBN-13: 978-0125113601 | Edition: 1

This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.

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This book introduces the reader, graduate student, and non-specialist alike to a lively and important area of mathematics. By its careful and detailed presentation, the book enables the reader to approach the contemporary literature with confidence. A plentiful number of exercises and the choice of topics, which reflect current research interests, distinguish this book from other texts. In addition to the basic theorems of operator theory, including the spectral theorem, the Gefland-Naimark theorem, the double communtant theorem, and the Kaplanski density theorem, some major topics covered by this text are: K-theory, tensor products, and representation theory of C*-algebras.

Product Details

• Hardcover: 296 pages
• Publisher: Academic Press; 1 edition (September 11, 1990)
• Language: English
• ISBN-10: 0125113609
• ISBN-13: 978-0125113601
• Product Dimensions: 9.3 x 6.3 x 1 inches
• Shipping Weight: 1.3 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,271,761 in Books (See Top 100 in Books)

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15 of 16 people found the following review helpful:

5.0 out of 5 stars Excellent, modern introduction to C*-algebras, September 8, 2002

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Dr. Lee D. Carlson (Baltimore, Maryland USA) - See all my reviews
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This review is from: C*-Algebras and Operator Theory (Hardcover)

The use of C*-algebras in operator theory is known as a "soft" technique, in contrast to the "hard" techniques that use deep results from analysis. The blending of algebra, topology, measure theory, and analysis to study operators has resulting in breathtaking advances, and this trend continues. Applications to physics, especially quantum field theory, has also resulted in fruitful collaborations between mathematicians and physicists. The author has given an introduction to C*-algebras in this book from the purely mathematical standpoint, but those interested in applications can gain much from the reading of this book.

Spectral theory is introduced in chapter 1, with a definition of Banach algebras immediately given on the first page. There are many advantages is presenting spectral theory in this general context, and the author illustrates these advantages throughout the chapter. For Banach algebras with a unit, Gelfand's theorem, giving the non-emptiness of the spectrum, is proven. The author also discusses the Gelfand representation, that says essentially that abelian Banach algebras act like continuous functions. He then restricts his attention to compact and Fredholm operators, and discusses their index theory. It is here that the reader can see the origin of the idea of a compact operator acting as a "zero element" in algebras of operators. This is readily apparent in the definition of the Calkin algebra, which is the quotient algebra modulo the compact operators.
By defining an involution on a Banach algebra, the author introduces C*-algebras in chapters 2 and 3, and proves the famous Gelfand-Naimark theorem. The latter allows one to view any C*-algebra as a collection of bounded operators on a separable Hilbert space. In addition, the spectral theorem for normal operators is proven. These two theorems give one confidence in the power of the theory of C*-algebras to study operators, but their nonconstructive nature sometimes is not of much use for calculating explicitly the spectra or to find the Hilbert space that serves as a representative example in the Gelfand-Naimark result.The author also introduces the concept of a hereditary C*-subalgebras, which is a generalization of the concept of an idea, and, like the algebra of compact operators, are always simple if they are subalgebras of simple C*-algebras. Also, the author introduces the important class of Toeplitz operators. These are intimately connected with "hard analysis" and have important applications in physics and cryptography. The author also uses them, via the consideration of the Toeplitz algebra, to introduce concepts in K-theory.

The theory of von Neumann algebras, or W*-algebras as they are sometimes called, is discussed in chapter 4. His viewpoint of them is characteristically modern, as essentially a noncommutative measure theory. This viewpoint meshes will with current research in the field of noncommutative geometry. Proofs of the double commutant theorem and the Kaplansky density theorem are given. The presence of the weak operator topology makes these objects of primary interest to applications in quantum physics, as it is this topology which is physically relevant. The famous "type" characterization of projections in von Neumann algebras is given in an addendum to the chapter.

The representation theory of C*-algebras is considered in chapter 5. The author shows that topological irreducibility is equivalent to algebraic irreducibility for C*-algebras. The reader can see the role that ideals, especially the "primitive" ideals, play in the representation theory. The author also discusses CCR algebras, but he calls them "liminal" algebras. The theory of liminal algebras is of upmost importance in applications to quantum physics (physicists still call them CCR algebras).

Chapter 6 is an introduction to the construction of C*-algebras using direct limits and tensor products of given C*-algebras. And here again, the physicist reader will find a useful class of algebras, namely the AF-algebras, which are used heavily in mathematical statistical mechanics. Nuclear C*-algebras, which are the most well-behaved class under the operation of tensor product, are discussed briefly.

The last chapter of the book is the most interesting, for it deals with the K-theory of C*-algebras. The Brown-Douglas-Fillmore theory was briefly mentioned in an addendum to chapter 2. This theory could be considered a precursor to latter work on K-theory of operator algebras. The author explains the origin of the K-groups K0(A) and K1(A) assigned to a C*-algebra, and how they can be used to study some properties of A. The K-groups are constructed by first forming the set P(A) consisting of the union of all projections in the collection of n by n matrices over an arbitrary *-algebra A. Then for a unital *-algebra A, a notion of stable equivalance is defined for elements of P(A). The reader familiar with the K-theory of vector bundles will see the similarity in this definition. Here the two projections are stably equivalent if they are equivalent under the direct sum of the n x n identity operator. The equivalance classes under stability are then enveloped via the Grothendieck group, giving the K-group K0(A). Then the K0-group shown to be a complete isomorphism invariant for unital AF-algebras, i.e. two unital AF-algebras are *-isomorphic iff there is a unital order isomorphism between their K0-groups. Three fundamental results in K-theory are then discussed: "weak exactness", which gives an exact sequence of K0-groups given an exact sequence of C*-algebras; "homotopy invariance", which shows that mappings between the K0-groups of two C*-algebras are equal if the mappings between the C*-algebras are homotopic; "continuity": which gives a notion of continuity for the K0-functor. Then after defining a notion of stability for the K0-functor, the author proves an analog of Bott periodicity. This involves of course the construction of the K1-group, which is done in terms of the suspension of the C*-algebra, in complete analogy with the vector bundle case.