Noncommutative Geometry
by Alain Connes
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Hardcover (22 November, 1994)
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Reviews (4)

The place to find it!
This is *the book* on noncommutative geometry. Alain Connes created the main parts of the theory, and paved the way for
its many exciting applications. As most would expect, the book
covers operator algebras, quantum theory, and other areas of mathematical physics; --but, in addition, there is a lovely treatment of tiling theory[Penrose tilings, cartwheel tilingsand the like], of the Dixmier trace, of groupoids, and of foliations.-- All new subjects: intriguing and penetrating!

A beautiful subject
Even though detailed proofs are omitted for most of the major results, the book is an excellent overview of a beautiful subject that the author has made substantial contributions to. The subject of noncommutative geometry has recently made its way into theoretical physics, and so a perusal of this book would be of interest to individuals working in string theory or quantum field theory.

The main idea of this book is to generalize measure and operator theory to non-commutative situations. In the usual operator theory, von Neumann algebras serve as a generalization of "classical" measure theory. Commutative von Neumann algebras, or W*-algebras as they are sometimes called, are essentially bounded meausurable functions, and have measure spaces as their dual. These facts and a fine movtivation for the subject appear in the introduction to the book. The author shows with great clarity what is involved in extending measure theory to the non-commutative case. What is most interesting about the extensions is that they involve ideas from quantum physics. In addition, readers familiar with K-theory will see some brilliant uses of it in the book, particularly in the extension of BDF-theory to noncommutative situations, namely the KK-theory of Kasparov. The author also gives a taste of physics applications in the very last section of the book. He shows, interestingly, that when space-time is replaced by a product with a certain finite space, the Lagrangian of quantum electrodynamics becomes that of the Standard Model. Although such "add-ons" to space-time are not uncommon in physics (Kaluza-Klein theories being one example), the author's strategy is unique in its use of bimodules, and gives the three lepton generations.

There are also many other interesting topics as well in the book, such as how to deal with non-Hausdorff quotient spaces using noncommutative C*-algebras, deformation theory and the Kasparov group, the notion of Morita equivalence, leaf spaces of foliations, the E-theory of morphisms of separable C*-algebras, the extension of de Rham cohomology to a noncommutative framework (cyclic cohomology) and its relation to K-theory, the noncommutative torus and the quantum Hall effect.

The book is an excellent source of information on noncommutative geoemtry and with the many references given one can find more detailed proofs. It is a subject that will no doubt continue to make its presence known in mathematics (and physics) in years to come.

Does not stand on its own
I must depart a bit from the previous breathless outpourings about this book. The fact is, it is a whirlwind TOUR (or travellog) of noncommutative geometry, not anything like a handbook of it, or even an atlas of detailed maps of it. I say this because theorems are asserted but almost never proved, no 'problems' are worked, and it is my experience and universally that of all mathematicians/physicists I know, no matter how gifted, that one cannot really understand the subject matter without doing problems! The previous reviewer does hint: "even if you do not know the subject matter" - aye, there's the rub! I appreciate the author's gifts, and I can compass his vision of how useful his approach might be, but between rather trivial points in quantum mechanics to very abstruse theorems in abstract harmonic analysis there is no bridge provided, and the original literature, either by Connes or his predecessor Dixmier, is practically all in French. If you already know this stuff, it might be useful to have all the relevant topics gathered together in one place, but if you don't already know it, you are going to be disappointed. Customers should be aware of this fact before they shell out the bucks. A much better book covering similar ground, but at a more directly physical and more elementary level, is Souriau's: "Structure of Dynamical Systems. A Symplectic View of Physics" Although that work again doesn't have worked exercises. I wonder, is it just the STYLE in French literature nowadays to DISCOURSE about mathematics, instead of DEMONSTRATING it? Rather like a self-fulfillment of the Derrida-Lacan-Latour-deconstructionist position about scientific communities and their provenance, I'd say. ... Read more

Isbn: 012185860X
Sales Rank: 221404
Subjects:  1. Algebra - Linear    2. Algebraic Geometry    3. Differential Equations    4. Geometry - Algebraic    5. Geometry - General    6. Geometry, Algebraic    7. Mathematics    8. Noncommutative rings    9. Science/Mathematics    10. Mathematics / Differential Equations   

Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, Volume 16)
by Viviane Baladi
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Hardcover (01 September, 2000)
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An operator with nine lives.
The "transfer operator" has many incarnations (and many names). It has emerged and re-emerged, over the years, in a rich variety of applications. The basic idea behind it is in fact quite
clear from the familiar and classical Perron-Frobenius theorem for positive matrices. But Baladi's lovelybook deals with the infinite-dimensional case, where the applications are, if possible--, even more striking. The starting point there was the powerful use David Ruelle first made of it in the sixties in his study of phase-transition problems in quantum statistical mechanics. The work leading to the Perron-Frobenius-Ruelle theorem!-- Since then, it has re-emerged in a rich variety of applications in dynamics; both continuous and discrete;-- both experimental and symbolic;-- being central in our (limited) understanding of attractors!It is used both in applications (wavelets, fractals, ...),and in pure math (Zeta functions, and trace formulas, to mention only a few such examples). It is all clearly and wonderfully presented in Baladi's book, which now serves as a lucid introduction to a rich and exciting set of new trends in the research literature. ... Read more

Isbn: 9810233280
Sales Rank: 1491278
Subjects:  1. Chaos (Physics)    2. Chaos Theory (Mathematics)    3. Mathematical Physics    4. Physics    5. Science    6. Science/Mathematics   

An Invitation to C-Algebras (Graduate Texts in Mathematics, Vol 39)
by William Arveson
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Hardcover (01 August, 1976)
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Lasting value.
"Operator algebras! Oh, right! That is Arveson's book, isn't it?"--From a typical conversation I would have with a colleague from the other end of campus.-- Or, an impatient student, I meet in the elevator, wants to quickly get an idea of what it is all about,-- and have fun reading about the material, at the same time: That would be Arveson's lovely little book!-- Sure, there *are* lots of great books in the subject; but they haven't had this same kind of wide impact. The central ideas are very attractively presented: It *is* an invitation! The other books in C*-algebra theory will typically be thicker, and they might be more narrowly focused, --more for the specialists, if you like. The first edition of Arveson's book is from the seventies; but still,-- now many years later, everyone knows "An invitation", and reads it. Authors keep immitating its approach and its style: You see immitators,--authors in other specialties of math writing books entitled "An invitation to ...". But none of the immitations seem to have quite the charm of the original. You can't very well **plan to** write a charming book in math. But when one arrives, we all know it. The subject of Arveson's book started with quantum theory, Hilbert space, spectral theory, and representations of groups and algebras. And, in the half century plus, since its inception, the subject has found applications in a surprisingly wide range of other fields: geometry, K-theory, fundamental physics, symbolic dynamics, and tiling theory, to mention just a few. ... Read more

Isbn: 0387901760
Sales Rank: 675635
Subjects:  1. Algebra - Linear    2. C*-algebras    3. Hilbert space    4. Mathematics    5. Representations of algebras    6. Science/Mathematics   

Theory of Linear Operators in Hilbert Space
by N. I. Akhiezer, I. M. Glazman
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Paperback (16 December, 1993)
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Reviews (2)

A Dover classic.
The spectral theorem of David Hilbert, John von Neumann, and Marshall Stone gives a complete answer to the question of which operators admit a diogonal representation, up to unitary equivalence, and makes the question precise as well. The theorem states that these are the normal operators in Hilbert space. This includes the selfadjoint operators which represent observables in quantum physics, and the more interesting ones are unbounded. Remember the Heisenberg commutation relations do not admit bounded solutions. But there is a mathematical distinction between formally selfadjoint operators (also called symmetric operators) and the selfadjoint ones. It is only the latter to which the spectral theorem applies. The distinction between the two is understood from a pair of indicies (n,m), now called deficiency indices. In some applications they represent
boundary conditions, and when n = m, and the boundary conditions are assigned, the symmetric operator in question has selfadjoint extensions. And we know from von Neumann what they are. A central question in the book concerns the issue of unequal indices. Then selfadjoint extensions do not exist, at least not unless the Hilbert space is enlarged. A central theme in the book is that in case of unequal indices, there is a larger Hilbert space which does in fact admit selfadjoint extensions. The co-authors, along with Naimark, are the authorities on this. Because of applications to PDE theory and to physics, there has been constant interest in the theme right up to the present. Even the current interest, and lively activity, in quantum measurement theory (in connection with quantum information theory) and entanglement brings back to to the fore this old issue around diagonalizing operators by passing to an "enlarged" (or dilated)Hilbert space, or looking for an orthonormal basis in the extended Hilbert space. So the theme of the book is still current.

Such an easy book to read!
This is a great intro to functional analysis. Having taken a graduate course on the subject, I used this as my text. The proofs are very readable and kept clear and simple. You'll see the subject develop before your eyes.One thing: when reading this book on infinite dimensional vector spaces,always try to draw a parallel with the finite dimensional version of thesubject, linear algebra. You'll appreciate the book all the more. For everytheorem relating to a bounded linear operator on Hilbert space, replace theoperator by a matrix on Euclidean n-space.. you'll say "oh yeah! Iremember that from linear algebra!" ... Read more

Isbn: 0486677486
Sales Rank: 295702
Subjects:  1. General    2. Hilbert space    3. Linear operators    4. Mathematics    5. Science/Mathematics    6. Transformations    7. Mathematics / General   

C*-Algebras by Example (Fields Institute Monographs, 6)
by Kenneth R. Davidson
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Hardcover (01 June, 1996)
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We learn from examples!
The theory of C*-algebras grew from applications, such as
quantum theory and group representations. The continued
vitality of the subject owes much to the study of examples, such as the approximately finite-dimensional C*-algebras(also called AF-algebras), the Cuntz algebras, the reduced group-C*-algebras, and the examples from physics, based on Fock space techniques.
More ambitious classification schemes then flow from a closer
study of the examples. A great book for a beginning grad course! ... Read more

Isbn: 0821805991
Sales Rank: 639523
Subjects:  1. C*-algebras    2. Science/Mathematics   

Continuous Analogues of Fock Space (Memoirs of the American Mathematical Society, 409)
by William Arveson
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Paperback (01 July, 1989)
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The start of a trend.
An method from particle physics uses the concept of particle number in a recursive construction of Hilbert space; --the Hilbert space named after Vladimir Fock (a contemporary of Lev Landau and P.A.M. Dirac). But both math physics, and operator algebras, have pressing needs for an analogue this fundamental construction in a continuous context, as opposed to the familiar countable discrete framework. Some examples: semigroups of endomorphisms(quantum dynamics), continuous products(operator algebras), invariants(classification), and index(non-commutative geometry)...The Arveson memoir was the catalyst for
these developments in the 1990ties, and they are still active, as evidenced by math conferences on the subject. ... Read more

Isbn: 0821824724
Sales Rank: 1646286
Subjects:  1. General    2. Mathematics    3. Science/Mathematics    4. Semigroups    5. Semigroups of endomorphisms    6. Tensor products   

Wandering Vectors for Unitary Systems and Orthogonal Wavelets (Memoirs of the American Mathematical Society)
by Xingde Dai, David R. Larson
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Paperback (01 July, 1998)
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A powerful idea!
The term "wandering vector" is from the theory of operators in Hilbert space;-- it means that the vector in question is transformed into a family of orthogonal vectors under a prescribed set of unitary operators. Since wavelets represent
orthogonal functions in the Hilbert space of square integrable functions on R, or on R^n for some n, it would only seem natural to merge the two concepts. But it takes originality to carry through the program: The book represents a success story in the application of operator theory to a current problem (in this case, wavelets) in classical analysis. What came out, for the theory of wavelets, was at first quite unexpected: You could have wavelets, it turned out, that are generated by fewer functions on R, or on R^n, than
had been predicted by the more conventional approach to wavelet theory. One generator is enough! The new wavelets turned out to be localized in frequency domain, and they gave rise to a variety of new trends in wavelet theory, the wavelet sets being just one of them, multiplicity, and the dimension function are others. ... Read more

Isbn: 0821808001
Sales Rank: 2933444
Subjects:  1. Functional analysis    2. Harmonic Analysis    3. Infinity    4. Mathematical Analysis    5. Mathematics    6. Operator theory    7. Science/Mathematics    8. Theory Of Operators    9. Wavelets (Mathematics)   

Wavelets through a Looking Glass (Applied and Numerical Harmonic Analysis)
by Ola Bratteli, Palle Jorgensen, B. Treadway
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Hardcover (12 July, 2002)
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Reviews (6)

A masterpiece in wavelets
This is a very well written book on the theory of wavelets and its applications. The presentation is self-contained and may serve as an introduction for someone who wants to learn about this topic. It also is an excellent book for those who have an advanced graduate degree in applied mathematics since it demonstrates how to truly understand complex concepts. This book gives a general presentation of some recent developments in wavelet theory with an emphasis on techniques that have a geometric and spectral-theoretic flavor. It can be certainly used as a textbook for graduate students as well as as a reference book for the specialists and researchers. A very nice feature of the book are the tutorials at the beginning of each chapter and some sections, which serve as summaries of main ideas and guides through the background ideas and motivation. At the end of every chapter there are plenty of excercises of various level of difficulty. Overall, this is a great book and I recommend it highly to a motivated reader.

Distinguished link between math and computer science
This is an important mathematical reference written in excellent style.Wavelets have found applications in many areas of engineering and CS.The authors provide a detailed, rich and entertaining tour through this relatively young but important field for both math and CS/Eng.Connections are, e.g., made between advanced CS virtual-reality applications such as audio-systems processing, future applications such as quantum computing, and advanced math in functional analysis and operator theory.

A fresh look at wavelets
The Book by Bratteli and Jorgensen is a superb book on wavelet's theory. It is very well written and has new and a fresh point of view on the subject. Although there are several good books on wavelets, the book by Bratteli and Jorgensen covers an important niche that has not been covered before. In particular

1- The book covers the theory of wavelets from the point of view of operators and functional analysis and will appeal to a growing number of pure as well as applied mathematicians interested in the subject.

2- The writing of the book is very appealing: every chapter starts by a tutorial that gives motivation as well as intuition. It is then followed by a very clean mathematical development of the subject, together with many examples, figures, and applications from physics and engineering.A set of nice problems is provided at the end of each chapter. Thus this book can be used as a graduate textbook or for mathematical seminars in mathematics departments.

3- This book can even be used by experts in wavelet theory for learning about recent developments and new perspectives from operator theory and functional analysis.

I highly recommend this book. ... Read more

Isbn: 0817642803
Sales Rank: 566574
Subjects:  1. Applied    2. Functional Analysis    3. Geometry - Algebraic    4. Harmonic Analysis    5. Homotopy Theory    6. Mathematical Analysis    7. Mathematics    8. Science/Mathematics    9. Spectral theory (Mathematics)    10. Topology - General    11. Wavelets (Mathematics)    12. Approximation Theory    13. CS/Numerical Mathematics    14. Communications Theory    15. Harmonic Analysis/Applications    16. Mathematics / Group Theory    17. Signal Processing   

Quantum Symmetries on Operator Algebras (Oxford Mathematical Monographs)
by David E. Evans, Yasuyuki Kawahigashi
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Hardcover (01 August, 1998)
list price: $265.00
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Reviews (1)

The latest developments!
Students and researchers can find here the material which lies at the foundation of the subject, and read about the latest and most exciting developments in this wonderful book.-- Material which would otherwise be difficult to extract from the diverse journal literature in math and in physics. The bibliography and the index are both great as well. The treatment of the material in the book aims at students interested in operator algebras, mathematical physics, [especially statistical mechanics and conformal field theory], and low dimensional topology. The book starts with a very nice treatment of the basics in operator algebras, K-theory, and it then moves on to the applications: eg., the Ising model, subfactors and bimodels, the Jones index [for subfactors] and its applications, topological quantum field theory, and more. ... Read more

Isbn: 0198511752
Sales Rank: 1703108
Subjects:  1. Algebra - Linear    2. Mathematical Physics    3. Mathematics    4. Operator algebras    5. Quantum groups    6. Science    7. Science/Mathematics    8. Symmetry (Physics)    9. Category theory    10. Fields & rings    11. Functional analysis    12. Relativistic quantum mechanics & quantum field theory    13. Statistical physics   

Categories of Operator Modules: Morita Equivalence and Projective Modules (Memoirs of the American Mathematical Society)
by David P. Blecher, Paul S. Muhly, Vern I. Paulsen
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Paperback (01 December, 1999)
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A lovely and well written invitation.
This is a lovely little book. While the subject may be relatively specialized within mathematics, the presentation in the book is lively, and the book is likely to engage readers who are generally interested in math,-- even if they might not, at the outset, recognize the words in the title of the book. And readers who are not already familiar with the subject, are likely to be surprised by the universality, the beauty, and power of the underlying idea. If they thought at first that the writing might be dry, they will instead find a page-turner (--at least relative to other specialized math books). The present authors do take great care in explaining the ideas in plain English. The co-authors are all masterful expositors who engage their readers. They are equally engaging and popular as speakers at confernces and workshops in mathematics. ... Read more

Isbn: 082181916X
Sales Rank: 1966291
Subjects:  1. Algebra - Linear    2. General    3. Hilbert space    4. Mathematics    5. Morita duality    6. Operator algebras    7. Science/Mathematics   

An Introduction to Frames and Riesz Bases
by Ole Christensen
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Hardcover (20 December, 2002)
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Reviews (1)

A new tool!
The subject of the book is a new tool in math, with a host of exciting applications. Of course, the subject has roots in classical ideas from harmonic analysis. But the book covers an explosive and exciting variety of developments since roughly 1990, and it is presented in the form of a graduate text. The basic idea begins with linear algebra, and progresses to expansions in function spaces, and multiresolutions. It will be useful to anyone who wants to learn from scratch about the underlying principles, the new results, and the applications. It is well written. A student of mine picked it up accidentally from my desk, and couldn't put it down. After awhile, he had completely forgotten what he came to see me about. It could have been some of the applications, such as antenna theory, wavelets, time-frequency analysis, uses in Radar, speech processing... ... Read more

Isbn: 0817642951
Sales Rank: 570821
Subjects:  1. Advanced    2. Analytic Mechanics (Mathematical Aspects)    3. Bases (Linear topological spac    4. Bases (Linear topological spaces)    5. Frames (Vector analysis)    6. Geometry - Differential    7. Linear Topological Spaces    8. Mathematics    9. Science/Mathematics    10. Signal processing    11. Topology - General    12. Vector Analysis    13. Mathematics / Mathematical Analysis   

Functional Analysis
by Frigyes Riesz, Bela Sz.-Nagy
Average Customer Review:
Paperback (01 June, 1990)
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Reviews (2)

The standard work, but there are better
This is the "standard" book on the subject. It is referenced everywhere. It has a lot in it. I have not read it cover to cover, just used it for reference, but if you are new to the subject I think Kolmogorov and Fomin looks beeter, and Shilov's books look better too.

Very readable classic
This book is a bargain ... in these days of $100 paperbacks ! The foreign authors, who 1st published this in the early 50s, write in a very readable way as opposed to most US profs. The book starts with an example of a continuous function which is not differentiable and then proves Lebesgue's theorem which tells you when a function does have a derivative.The 2nd part of the book is about Integral equations which again starts with some examples of problems the early 19th century mathematicians solved. Particularly interesting to me was Fredholm's method which was to replace the integral with a series. The book covers all the topics you would expect in a very readable form. ... Read more

Isbn: 0486662896
Sales Rank: 282933
Subjects:  1. Calculus    2. Functional analysis    3. Mathematics    4. Science/Mathematics    5. Mathematics / General   

Mathematics of Classical and Quantum Physics
by Frederick W. Byron, W. Fuller Robert
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Paperback (20 August, 1992)
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Reviews (12)

Terrific book to ramp up and read original papers
If you're tired of reading descriptive physics like Paul Davies' books and want to be able to read the original papers and follow the math, this is the book.It presumes a strong undergraduate math background in some areas but the explanations are clear and the proofs are easy to follow.If you want to read quantum physics, the chapters on vectors and operators will give you the math foundations.

Important Information
This book is not, and I repeat, IS NOT for the inexperienced.This book is a GRADUATE LEVEL TEXT on mathematical physics.If you are an undergraduate student taking a physics class, this book will be of no use to you.I recommend that anyone interested in purchasing this book have a somewhat decent amount of mathematical background.I personally recommend Calculus I-IV, Advanced Calculus and Linear Algebra.
If, though, you have this background, then this book is may just be for you.It is concise, to the point and presents a clear and well written discussion of mathematical physics.
I just felt that before you dive, head first, into the world of mathematical physics, somebody needed to warn you about what you were getting yourself into.

Fantastic
Concise and complete.I find myself coming to this book not only for Physics courses and Physics research, but I find it helpful in most Math courses.This book is a must for anyone in Physics and applied mathematics. ... Read more

Isbn: 048667164X
Sales Rank: 17001
Subjects:  1. Mathematical Physics    2. Physics    3. Quantum Theory    4. Science    5. Science/Mathematics    6. Science / Physics   

Noncommutative Dynamics and E-Semigroups
by William Arveson
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Hardcover (02 March, 2003)
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Reviews (1)

Dynamics, quantum theory, and operator algebras.
The landmark papers by F. Murray and J. von Neumann introduced Rings of Operators (now called von Neumann algebras) as a model for quantum observables, and as a tool in mathematics. Thanks to the conservation laws of physics, one-parameter groups of automorphisms have been used sucessfully as a mathematical model for time evolution in quantum mechanical systems. This begins with work by Eugene Wigner. While the mathematical structure of one-parameter groups is now well understood, thanks to the theorems of Stone and von Neumann, it is only in recent years that systematic results have emerged for the different, but related dissipative or time-irreversible dynamical systems. The dissipative framework includes the current models for quantum information theory, e.g., decoherence due to build-up of errors in quantum channels. This book deals with the mathematics of these dissipative systems. The time parameter is the half-line, and the transformations are now endomorphisms as opposed to automorphisms. The issues include type-classification of the semigroups, and the tools are operator algebraic. A new class of C*-algebras, called spectral algebras, is introduced, and they are of independent interest.

For over a decade, W. B. Arveson and others have pioneered research on the structure of irreversible quantum dynamical systems on von Neumann algebras, and the related operator theory. This book serves as an excellent introduction to the theory. Also included are other areas which have had an impact on the theory, such as Brownian motion, dilation theory, quantum probability, and free probability.

The book is suitable for graduate students and research mathematicians interested in the dynamics of quantum systems and corresponding topics in the theory of operator algebras.
Review by Palle Jorgensen, September 2003. ... Read more

Isbn: 0387001514
Sales Rank: 893366
Subjects:  1. Algebra - General    2. Endomorphisms (Group theory)    3. Group Theory    4. Mathematics    5. Noncommutative algebras    6. Science/Mathematics    7. Semigroups    8. Mathematics / Mathematical Analysis