MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples of C algebras. In order to prove that they are non-somorphc, we need several results whch are of nterest n ther own rght. 1. Techncal Lemmas Ths secton contans two techncal lemmas about approxmatng projectons. Lemma 1 If p and q are projectons n a C algebra A such that p q < 1, then p and q are equvalent n A, p q,.e there exsts an element u A such that uu = p and u u = q. P roof. We have p pqp = p(p q)p p p q p = p q < 1. Hence pqp s nvertble n the C algebra pap wth nverse =0 (p pqp) (where (p pqp) 0 = p, the dentty of pap). As pqp s postve, there exsts the postve element x = (pqp) 1/2 pap. Let u = x 1 pq. Then we have uu = x 1 pqpx 1 = x 1 x 2 x 1 = p. Usng p = x 2 pqp = x 2 qp (as xp = x pap), we obtan (qx 2 q)qpq = q(x 2 qp)q = qpq. By the same arguments as before, we can show that qpq s nvertble n qaq. Multplyng the equaton above by (qpq) 1 from the rght, we obtan q = qx 2 q = (qpx 1 )(x 1 pq) = u u. Lemma 2 Let p A be a projecton. Then there exsts an ε > 0 such that f b p < ε for a selfadjont b A then there exsts a projecton q C (b) such that q p. P roof. Frst observe that b = b p + p b p + p 1 + ε. Hence we obtan b 2 b = b 2 bp + bp p 2 + p b b b p = b p p + p b (3 + ε)ε. Ths mples that λ(1 λ) < 4ε for any λ σ(b). For ε suffcently small (say ε = 1/20), ths mples that λ < 1/3 or 1 λ < 1/3. Let χ be any contnuous functon on the real lne such that χ(λ) = 0 for λ < 1/3 and χ(λ) = 1 for 1 λ < 1/3. Then t follows from spectral calculus that q = χ(b) s a projecton wth q b = χ(λ) λ 1/3, where χ(λ) λ s the supremum of χ(λ) λ on σ(b). It follows that q p q b + b p 1/3 + ε < 1. 2. Some non-somorphc AF algebras A C algebra A s called an AF (almost fnte) algebra f there exsts a sequence of fnte dmensonal C algebras such that the unon A n s dense n A. A 1 A 2 A 3... Theorem 1 (a) Any projecton n an AF algebra A s equvalent to a projecton n a fnte dmensonal algebra A n for some n. (b) Let A and B be AF algebras wth unque traces. If there exsts a projecton p A n for some n such that tr A (p) tr B (q) for all projectons q B n, n N, then A s not somorphc to B. 1
P roof. If p s a projecton n A and ε > 0, we can fnd an element b n some A n such that b p < ε. As also b p < ε, t follows that Re(b) p < ε, where Re(b) = 1 2 (b + b ) s self-adjont. Then clam (a) follows from the prevous lemma after choosng ε suffcently small. For part (b), f there were an somorphsm Φ : A B, t would nduce a trace functonal tr on A from the trace tr B on B by tr (a) = tr B (Φ(a)). By unqueness of the traces, tr would concde wth the trace tr A on A. But then tr B (Φ(p)) = tr A (p) for the projecton Φ(p) B. But as Φ(p) would be equvalent to some projecton q n some B n, by (a), ths would contradct our assumpton. Corollary 1 Let A = =1 M k and B = =1 M m. If there exsts a prme number p o such that p o k, but p o m, then A s not somorphc to B. P roof. If tr s a normalzed trace on A (normalzed means that tr(1) = 1), then the restrcton of tr to the fnte dmensonal subalgebra A n = n =1 M k = M k n s the unque normalzed trace on M k n,.e. the usual sum of dagonal elements dvded by k n (see exercse below). So any two traces on A concde on all of the fnte dmensonal subalgebras A n, and hence they concde on all of A by contnuty. Smlarly, the trace on B s unque. As p o k, we have a projecton p n M k, and hence also n A such that tr(p) = 1/p o. On the other hand, f q s a projecton n B n, tr(q) = j/m n for some j N. As p o m, tr(q) tr(p). Hence A = B, by the prevous Theorem. Exercse : Show that any functonal φ on M n satsfyng φ(ab) = φ(ba) s equal to a multple of the usual trace T r, gven by the sum of the dagonal entres. Ths can be done by calculatng drectly φ(e j ), where the E j are the usual matrx unts. E.g. we have φ(e j ) = φ(e E j ) = φ(e j E ) = 0, f j. The algebras n the theorem belong to one of the smplest classes of AF algebras, also called UHF algebras (unformly hyperfnte algebras). In order to study more complcated AF algebras, we wll need some smple statements about the representaton theory of C algebras. We say that a Hlbert space H s an A-module for the C algebra A f a A acts va a bounded lnear operator ρ(a) on H such that ρ(a ) = ρ(a), the usual adjont of ρ(a) B(H). In order to keep the notaton smple, we may sometmes just deal wth the case that A s a closed subalgebra of B(H). In vew of the GNS constructon, ths s not an essental restrcton. An A module H s called smple f the only submodules of H are H tself and 0. Lemma 3 Let A B(H) be a C algebra. Then (a) If H 1 H s an A-submodules of H, we have the decomposton H = H 1 H1 as A-submodules. (b) If A s a fnte dmensonal C algebra, any fnte dmensonal A module s a drect sum of smple A-modules. P roof. For part (a) we only need to show that f ξ H1 and a A, then also aξ H1. Ths follows from (aξ, η) = (ξ, a η) = 0 for η H 1, as a η H 1. Part (b) follows by nducton on the dmenson on H, usng part (a) n case H s not smple. Inclusons of fnte-dmensonal C algebras We assume that we have an ncluson of fnte dmensonal C algebras A B such that both A and B have the same dentty element 1. We assume A = M d B = M ej. Let V = C d and W j = C e j be smple A- resp B-modules. Then W j, vewed as an A-module, can be wrtten as a drect sum of smple A-modules, by Lemma 3. Let g j be the number of those modules whch are somorphc to V. Comparng dmensons, we get j e j = g j d and G t d = e, 2
where G = (g j ) s called the ncluson matrx for A B, and e = (e j ), d = (d ) are the dmenson vectors of B and A respectvely. We also assume B has a normalzed trace tr. Then tr Mej = w j T r for some non-negatve w j, by the exercse above. We call the vector w = (w j ) the weght vector of the trace tr. We can smlarly defne the weght vector v = (v ) of the restrcton of tr to A. We then also have v = j g j w j and v = Gw. 3. Perron-Frobenus Theorem and more AF -algebras We frst deal wth a useful techncal theorem, the Perron-Frobenus Theorem. To prove t, we frst prove the followng useful proposton. Proposton Let K be a compact set, and let g : K K be a contnuous map. Assume that K has a metrc d for whch d(g(x), g(y)) < d(x, y) for all x y n K. Then n gn (K) = {x} for a sngle pont x K wth g(x) = x. P roof. Let K o = n gn (K). Then K o and, by constructon, we have g(k o ) = n g n+1 (K) = K o,.e. the map g s surjectve on K o. As K o s compact, there exst ponts x, y K such that d(x, y) d(x, y ) for all x, y K o. By surjectvty of g, there exst ponts x 1, y 1 K o such that g(x 1 ) = x and g(y 1 ) = y. But f x y, we would get d(x, y) = d(g(x 1 ), g(y 1 )) < d(x 1, y 1 ) d(x, y), where the frst nequalty follows from our assumpton on d, and the second one from our choce of x and y. Ths shows that x = y. Theorem (Perron-Frobenus) Let G be a k k matrx wth all of ts entres beng postve. Then we have (a) There exsts an egenvector v of G wth all of ts entres beng postve. (b) The ntersecton n N Gn (R k +) s equal to R + v, wth v as n (a). In partcular, v s the unque egenvector of G such that all of ts coordnates are postve, up to postve scalar multples. P roof. Let K be the smplex gven by the vectors x wth x 0 and x = 1. Ths s obvously compact and convex. We defne the map g : K K by g(x) = Gx Gx 1, where x 1 = x. Then g s well-defned on K and does ndeed map nto K (check for yourself!). It follows from Brower s fxed pont theorem that g has a fxed pont x o. Ths means Gx o = Gx o 1 x o. As all the entres of G are postve, so are all the coordnates of ts egenvector x o. Ths fnshes the proof of (a). To prove (b), we consder the smplex K consstng of all vectors x satsfyng x 0 and w t x = 1; here w s an egenvector as n (a) for the matrx G t wth egenvalue λ. Then t follows from the defntons that the map g(x) = 1 λ G(x) defnes a contnuous map from K nto tself. We defne the metrc d on K va d(x, y) = w x y. 3
Usng the egenvector property g jw = λw j, we obtan for x y d(g(x), g(y)) = 1 w G(x y) = 1 w λ λ j < 1 g j w x j y j = λ,j j g j (x j y j ) w j x j y j = d(x, y); here the strct nequalty follows from the fact that w (x y ) = w t (x y) = 1 1 = 0 and from x y. Hence the condtons of the proposton are satsfed, and we have n gn (K) = {x o }. The clam now follows from the fact that G n (R k +) = R + g n (K) for all n N. Corollary The statements of the last theorem also hold for matrces G all of whose entres are non-negatve, and such that G m only has postve entres for some nteger m. P roof. Let v be the Perron-Frobenus egenvector of G n wth egenvalue µ. Then we have G m (Gv) = G(G m v) = µgv. Hence also Gv s an egenvector of G m wth only postve coordnates,.e. t must be a multple of v by unqueness of the Perron-Frobenus egenvector. The other statements of the theorem now follow easly. Theorem Let A be an AF algebra for whch the ncluson matrces for A n A n+1 are all gven by a constant matrx G such that G m only has postve entres for some postve nteger m. Then A has a unque postve trace, whose weght vectors for the fnte dmensonal algebras A n are gven by multples of the Perron-Frobenus egenvector of G. P roof. Let w n be the weght vector for the restrcton of the trace on A n. Then we have w n = G k n w k for all postve ntegers k. It follows from the Perron-Frobenus theorem and ts corollary, that w n k Gk (R k +) s a multple of the Perron-Frobenus vector of G. Ths shows unqueness of the trace. On the other hand, f w 1 s the multple of the Perron-Frobenus vector of G such that d w = 1, where d = (d ) s the dmenson vector of A 1, we defne w n = λ 1 n w 1, where λ s the egenvalue for w 1. Then the restrcton of the trace on A n defned by w n to A n 1 concdes wth the trace defned by w n 1. Hence we obtan a well-defned trace on A n, whch extends to A by contnuty. 4. Examples of C algebras (a) We have already seen nfntely many examples of non-somorphc UHF algebras, all of whch have a unque trace. Further examples come from AF algebras wth constant ncluson matrx. E.g. f we have the ncluson matrx [ ] 2 1 G =, 1 1 t has PF egenvector (λ, 1) t, where λ = (1 + 5)/2 s the golden rato, wth egenvalue λ 2. If A 1 = C C, the weght vector w 1 of the trace s gven by (λ, 1) t /λ 2. It follows for any projecton p A that tr(p) takes a value n the semgroup generated by the elements 1/λ n, n N, for any projecton p A. Semgroup here means all possble sums (wth repettons) of such elements. As the traces of projectons n U HF algebras are always ratonal numbers, t follows from Theorem 1 that ths AF algebra s not somorphc to a UHF algebra. (b) We can defne another example of an AF algebra for whch the ncluson dagram s gven by Pascal s ( ) trangle. Hence A n 2 n = An, where A n, s the full matrx algebra of dmenson. Then t s easy to check that we obtan for each t [0, 1] a postve trace tr on A for whch the weght vector for A n s gven by (t (1 t) n ) 0 n. Hence here we have an AF algebra wth nfntely many dfferent postve traces. Ths AF algebra A can also be realzed n a more conceptual way as a subalgebra of the UHF algebra M 2. Let d be the 2 2 dagonal matrx wth egenvalues 1 and λ, where λ s not a root of unty. We let 4
d act on M 2 va conjugaton,.e. d.( a ) = da d 1. Then show that A s equal to the fxed ponts under ths acton. (c) We fnally gve an example of a C algebra whch does not allow any fnte trace. Let H be a Hlbert space wth orthonormal bass (ξ j ) j N. Fx n N, n > 1. We defne partal sometres s, 1 n by s ξ j = ξ (n 1)j+. Then one can show that s s = 1 and n s s = 1. =1 Let O(n) be the C algebra generated by the s. Then t follows from the relatons above that the only postve functonal tr O(n) satsfyng tr(ab) = tr(ba) for all a, b O(n) s the zero functonal. Indeed, f tr(1) = α, we get α = tr(1) = tr( s s ) = tr(s s ) = nα. Hence α = 0. As we have a a 1 for any postve element a A, t also follows that 0 tr(a) a tr(1) = 0. The general clam now s a consequence of the fact that any element a can be wrtten as a 1 a 2 + (a 3 a 4 ), wth a postve for 1 4. 5