C*-Algebras by Jacques Dixmier
By Jacques Dixmier
Virtually four-fifths of this booklet bargains with the learn of C*-algebras, and the most effects due, between others, to Fell, Glimm, Kadison, Kaplansky, Mackey and Segal are expounded. due to the quantity of fabric gathered on unitary representations of teams, the latter pages of the ebook are dedicated to a quick account of a few facets of this topic, rather because the conception of teams offers the most fascinating examples of C*-algebras. the speculation of C*-algebras continues to be increasing speedily, yet this paintings is still a transparent and available advent to the basics of the topic.
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1323]. 8. Let A be a unital C*-algebra, N the set of normal elements of A, x E N and V a neighbourhood of 0 in C. There exists a neighbourhood U of x in N such that for every Y E U, Sp Y C Sp x + V and Sp x C Sp Y + V. . 9. Let A be a unital C*-algebra, H a Hilbert space and c/J a linear map of A into 2(H) such that c/J(A+) C 2(Ht. Then, for every her› mitian x in A, we have c/J(x 2) ;a. c/J(X)2. . 10. Let A and B be unital C*-algebras. (a) Let p: A~B be a bijective linear map such that p(x*) = p(x)* and p(x n ) =p(x)" for X hermitian and n an integer >0.
3. Let A be an involutive Banach algebra with an approximate identity (u;), 7T a non-degenerate representation of A in H, ~ an element of Hand f the positive form defined by 7T and f Then Iitli = (~ Ig). 10). It follows that if A is the involutive algebra obtained by adjoining an identity to A, and j and iT are the canonical extensions of f and 7T to A, then j(x) = (iT(x)~ Ig) for each x EA. In particular, if 7T is the identical representation, assumed to be non› degenerate, of a sub-C*-algebra A of 2(H) which does not contain I, then the canonical extension to A = A + C .
Let A be a unital Banach algebra endowed with an involution such that Ilx*xll = Ilx*II’llxll for each x E A. Then A is a C*-algebra. , . 6. Problem: does a unital C*-algebra A of dimension > 1, whose only closed two-sided ideals are 0 and. A possess any projections other than 0 and I? . 7. Let A be a unital C*-algebra, and x an element of A which does not have a left inverse. Then x*x is not invertible in the sub-C*-algebra B generated by I and x*x, so there exist Yt, Y2, ... 1).