By William Arveson

This ebook offers an advent to C*-algebras and their representations on Hilbert areas. now we have attempted to offer merely what we think are the main simple rules, as easily and concretely as shall we. So at any time when it's handy (and it always is), Hilbert areas develop into separable and C*-algebras develop into GCR. this tradition most likely creates an influence that not anything of price is understood approximately different C*-algebras. after all that's not precise. yet insofar as representations are con cerned, we will aspect to the empirical incontrovertible fact that to this present day not anyone has given a concrete parametric description of even the irreducible representations of any C*-algebra which isn't GCR. certainly, there's metamathematical proof which strongly means that nobody ever will (see the dialogue on the finish of part three. 4). sometimes, while the belief in the back of the facts of a common theorem is uncovered very basically in a distinct case, we turn out merely the exact case and relegate generalizations to the routines. In influence, we've got systematically eschewed the Bourbaki culture. we've additionally attempted take into consideration the pursuits of a number of readers. for instance, the multiplicity concept for regular operators is contained in Sections 2. 1 and a couple of. 2. (it will be fascinating yet no longer essential to comprise part 1. 1 as well), while an individual attracted to Borel constructions might learn bankruptcy three individually. bankruptcy i'll be used as a bare-bones advent to C*-algebras. Sections 2.

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**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.