By Lajos Pukanszky

This booklet provides to the nice physique of analysis that extends again to A. Weil and E. P. Wigner at the unitary representations of in the neighborhood compact teams and their characters, i.e. the interaction among classical team idea and glossy research. The teams studied listed below are the hooked up Lie teams of normal sort (not unavoidably nilpotent or semisimple).

Final effects replicate Kirillov's orbit approach; on the subject of teams that could be non-algebraic or non-type I, the strategy calls for significant sophistication. equipment used variety from deep practical research (the conception of C∗-algebras, elements from F. J. Murray and J. von Neumann, and degree thought) to differential geometry (Lie teams and Hamiltonian actions).

This ebook provides for the 1st time a scientific and concise compilation of proofs formerly dispersed during the literature. the result's a powerful instance of the deepness of Pukánszky's work.

Readership: Graduate scholars and examine mathematicians operating in topological teams and Lie teams; theoretical physicists

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Characters of Connected Lie Groups

This publication provides to the good physique of study that extends again to A. Weil and E. P. Wigner at the unitary representations of in the community compact teams and their characters, i. e. the interaction among classical team conception and glossy research. The teams studied listed here are the hooked up Lie teams of common sort (not unavoidably nilpotent or semisimple).

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In fact, J - 1 =SJS = S(M'-Mi1)S = SM'S • (SM^)-1 = M • M'-1- hence J = M[ • M~l. Summing up, we have so far J — M' • M^1 (by definition) and J = M[ • M~l. (iii) We observe next that M and Mi commute. In fact, we have by definition Mi = J " 1 • M'. Then M commutes with M' and, by M = SM'S and J " 1 = SJS, M commutes with J - 1 . Conclusion. So far J = M' • M^1, J = M[ • M _ 1 and M commutes with M x , hence M' commutes with M[. (iv) We conclude from (iii) that M' • M^1 = M[ • M " 1 whence ( M ^ ) - 1 • M ; = _1 M • Mi, and that there is a C positive, selfadjoint, invertible, Crj(L(G))c such that M • M f 1 = C, ( M ' ) " 1 • M{ = C, and S C S = C'1.

2 = (#1)0; this is a connected Lie group, closed in GL(V). Let (Hs,rj) be the universal covering of # 2 . Since Lie(iJ2) = fy = Lie(if), there is a homomorphism from H% onto H, cp say, the kernel T of which is discrete. 2> (v) Let a be some element in A. We define ^2(^)^2 = oh^oT1 (/12 G H2). Let /03(a) be the covering map of ^2(0) (a G #2)- We form £>' - {6; 6 G A such that p2{b)r]{t) = r](t) for all t G T}. We note that 1) D' is algebraic in GL(V) and 2) H2 C £>'. We define D = (£>')()• (vi) 1) Assume that b e D.

Clearly, if 7\ is of class one, and ker(Ti) = ker(T 2 ), then T2, too, is of class one. Similar observations apply in connection with representations of class minus one. Next we recall (cf. [Dal], p. 100) that ker(T) is a primitive ideal of C*(T). In this manner we can speak of a primitive ideal of class one (or minus one). Let us denote by Primi(r) the set of all primitive ideals of class one of T. If x € F-> we put J(x) ~ ker(V(x)); this is an element of Primi(T). This being so we shall show (cf.

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