By Joseph C. Varilly, Hector Figueroa, Jose M. Gracia-Bondia

"In accumulating the fabric in a coherent shape, the authors have played a priceless provider and this publication will doubtless open noncommutative geometry to a much wider audience.... [A] huge variety of metaphorical and mathematical comments embedded within the text...highlight the plethora of connections among the cloth of the e-book and different parts of mathematics."

—**Mathematical Reviews**

"The fabric is accumulated from an immense variety of sources.... it all is punctiliously labored out and serves as a hugely recommendable advent to the very important box of noncommutative geometry."

—**Zentralblatt Math**

"The variety of the booklet is full of life, exempt of the stereotypes one encounters really frequently within the mathematical literature. The authors know the way to inform tales and revel in doing it. The pleasant tone will be of a few support for an hypothetical unexperienced reader faced with virtually seven-hundred pages of quite tricky and mild mathematics."

**—Journal of Operator Theory**

"The current e-book is a scientific direction in noncommutative differential geometry and operator thought, with purposes to guantum physics. Its subject matters conceal C*-algebras, vector bundles and C*-modules, K-theory, Fredholm operators, Clifford algebras, Spin representations, noncommutative integration and differential calculus, spectral triples and Connes spin manifold theorem. As purposes, noncommutative tori, quantum idea and Kreimer-Connes-Moscovici algebras are mentioned. The booklet could be worthwhile to all mathematicians and mathematical physicits who desire to find out about noncommutative geometry and tis ramifications."

**---Matheamtica Bohemica**

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**Additional info for Elements of Noncommutative Geometry**

**Example text**

Moreover, it can be shown [446] that all the elements of G(H) are linearly independent inH. Inparticular, whenH = IFG, G(H) is the original group G. 26. The antipode of a Hopf algebra is a unital algebra antihomomorphism. Proof. Consider the maps p, T : H ® H - H given by p(a ® b) := S(ab) and T(a ® b) := S(b)S(a). To prove the lemma, it is enough to show that p is a left convolution inverse and that T is a right convolution inverse for the multiplication m: H ® H H. 34) yields m * T(a ® b) = m o (m ® T) (Li,j a; ® bj ® a;' ® bj') = Li,j a;bjS(bj')S(a;') =Li a;E(b)S(a;') = E(a)E(b) 1H = E(ab) 1H = U o EH®H(a ® b).

And if cjJ -l/J is also positive, then cjJ = l/); for we may suppose that Ais unital, and then it is enough to notice that II cjJ - l/J II = (cjJ - l/J) ( 1) = II cjJ II - lll/J II = 0. A linear functional T: A- Cis called tracial if T(ab) = T(ba) for all a, b E A. 20. A positive linear functional of norm one is called a state of the C* -algebra. If A is unital, any state satisfies cjJ (1) = 1. A state cjJ is called faithful if a ~ 0 and cjJ(a) = 0 imply a = 0. A (normalized) trace on A is a nontrivial tracial state.

14. The two constructions are quite similar, although not strictly parallel. Wehave presented them in the form most suitable for our purposes, tobe revealed in Chapter 3. LA C* -algebra basics We collect here, for the reader's convenience, several facts and theorems about C*-algebras as background for the main text. There are many good LA C*-algebra basics 27 textbooks on this subject: we recommend [129, 137,183,266,352,366,481], in no particular order. 16. 18) for all elements a, b of the algebra; this condition guarantees continuity of the product.