By James E. Humphreys

The booklet offers an invaluable exposition of effects at the constitution of semisimple algebraic teams over an arbitrary algebraically closed box. After the basic paintings of Borel and Chevalley within the Nineteen Fifties and Nineteen Sixties, extra effects have been acquired over the following thirty years on conjugacy sessions and centralizers of components of such teams

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**Extra info for Conjugacy classes in semisimple algebraic groups**

**Example text**

U*U = I) which is not /). , UU* 1' 0 • • • , an infinite direct sum. n • v = V it) CI V it) CI • • • acts on Ye Thus the "unilateral shift" C:) •• 0 0 0 0 • • • e Yel is a nonunitary isometry which commutes with n • v ;0(A). 11 contains an operator V such that V* V = / 0 V V*. Now some entry of the nonzero m x m operator matrix / — VV is nonzero, hence there is a complex homomorphism ca of pio(A)' which is nonzero at that entry. Defining ii/o) : —> M. , we see that W* W = / but I. This, however, is impossible in M„, a fact easily seen by comparing WW* the trace of / — WW to the trace of / — WW* and noting that a positive matrix with zero trace must be zero.

6, complete. it is irreducible, and the proof is 0 This corollary can be used to prove that every locally compact group has "sufficiently many" irreducible unitary representations. Unfortunately, an adequate discussion of this important application would take us too far afield, and instead we refer the reader to [6]. We can now deduce the theorem of Gelfand and Naimark mentioned at the beginning of the section. 3. Gelfand—Naimark theorem. Every abstract C*-algebra with identity is isometrically *-isomorphic to a C*-algebra of operators.

6. D. Let S be the unilateral shift (cf. D). Show that C*(S) is a GCR algebra and describe its canonical composition series. E. A unilateral weighted shift is an operator defined on an orthonormal base e l , e2,. . by the condition A:e n --■ wnen , 1 , where { w„} is a bounded sequence of nonnegative real numbers. a. Show that if w„ > 0 for every n, then C* (A) is irreducible. b. Show that if w„ > (5 > 0 for every n, then C*(A) cannot be an NGCR algebra. 6. States and the GNS Construction We now want to discuss certain matters relating to the existence of representations of C*-algebras, and how one goes about constructing them.