ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES. 1. Introduction
|
|
- Kelly Alannah Hunter
- 5 years ago
- Views:
Transcription
1 ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES TAKASHI ITOH AND MASARU NAGISA Abstract We descrbe the Haagerup tensor product l h l and the extended Haagerup tensor product l eh l n terms of Schur product maps, and show that l h l B(l 2 ) (resp l eh l B(l 2 )) concdes wth c 0 h c 0 B(l 2 ) (resp c 0 eh c 0 B(l 2 )) For C*-algebras A,B, t s shown that A h B = A eh B f and only f A or B s fnte-dmensonal Introducton For Hlbert spaces H and K, we let B(H, K) and K(H, K) denote the bounded operators and the compact operators of H to K An operator space X on H s a subspace of B(H) = B(H, H) whch s endowed wth norms to each n m matrces M n,m (X) over X as a subspace of M n,m (B(H)) = B(H m, H n ) We allow to use the notaton M I,J (B(H)) = B(H J, H I ) for arbtrary ndex sets I and J Let X and Y be operator spaces The Haagerup tensor product of X and Y s the completon of the algebrac tensor product X Y by the norm n u h = nf{ [a,, a n ] t [b,, b n ] u = a b X Y, n N, a X, b Y }, and s denoted by X h Y [] We also recall the extended Haagerup tensor product X eh Y An element u of X eh Y s represented by the followng formal sum: u = I a b, where a = [a ] I M,I (X), b = t [b ] I M I, (Y ) (n other words, a = I a a /2 <, b = I b b /2 <
2 for a X and b Y ) We apprecate ths formal sum as the blnear form on X Y as follows: u(f, g) = I a (f)b (g) for f X, g Y For ths element u X eh Y, ts norm s defned by u eh = nf{ a b u = I a b, a M,I (X), b M I, (Y )} Then we can realze X eh Y as a subspace of the dual operator space (X h Y ) [6] In [7], the authors studed the Schur product on B(H) and used the extended Haagerup tensor product to descrbe the property of Schur product maps Effros and Ruan has shown that X h Y s (completely sometrcally) embedded to X eh Y [6] We wll be concerned wth the dfference between the Haagerup tensor product and the extended Haagerup tensor product, snce t s essental to deal wth Schur product maps derved from (possbly unbounded) operators The Schur product map on B(l 2 ) s a normal l -bmodule map, where l s a maxmal abelan subalgebra of B(l 2 ) and s dentfed wth the bounded sequences on N (cf [7]) As a deep result concernng (normal) bmodule maps, we often refer to the followng theorem by Blecher and Smth n [2]: f M s a von Neumann algebra, then M w h M s completely somorphc to the completely bounded M -bmodule maps of K(H) to B(H) denoted by CB M (K(H), B(H)), where w h concdes wth eh n ths settng In secton 2, we study the dfference between l h l and l eh l from the vew pont of Schur product and characterze them n terms of Schur product maps Moreover we characterze c 0 h c 0 and c 0 eh c 0 n terms of Schur product maps, where c 0 s the complex sequences on N tends to 0 As a result for Schur product maps derved from bounded operators, we show that l h l B(l 2 ) (resp l eh l B(l 2 )) concdes wth c 0 h c 0 B(l 2 ) (resp c 0 eh c 0 B(l 2 )) In secton, we ntroduce some notons (rght-compact, weakly rghtcompact, left-compact, weakly left-compact) for whch dstngush the Haagerup tensor product from the extended Haagerup tensor product for operator spaces As a man result n ths secton, for C*-algebras A, B, t s shown that A h B = A eh B f and only f A or B s fnte-dmensonal 2
3 2 l h l and l eh l Let X and Y be operator spaces and X Y the algebrac tensor product of X and Y For a = [a,, a n ] M,n (X), b = t [b,, b n ] M n, (Y ) and α = [α j ] M n (C), we denote n a b X Y by a b, and n,j= α ja b j by aα b Proposton If u X Y, then u h = nf{ a α b u = aα b X Y, n N, α M n (C), a M,n (X), b M n, (Y )} Proof It follows from u h = nf{ a n b u = a n b} nf{ a α b u = aα b} nf{ aα b u = aα b} nf{ a b u = a b} = u h Proposton 2 If u X h Y, then u h = nf{ a α b α K(l 2 ), a M, (X), b M, (Y ), u = α j a b j (= aα b) converges n X h Y },j= = nf{max λ a b (λ ) c 0, a M, (X), b M, (Y ), u = λ a b converges n X h Y } Proof Suppose that u X h Y wth u h < To prove the frst equalty, t suffces to show that there exst a = [a, a 2, ] M, (X) wth a <, α = [α j ] K(l 2 ) wth α < and b = t [b, b 2, ] M, (Y ) wth b < such that k α j a b j,j= converges to u n X h Y when k tends to Gven ε = u h > 0 Then we can choose a sequence {u n } X Y, whch converges to u, satsfyng that u n h < ε and u n+
4 u n h < 2 n ε (n ), u 0 = 0 If we put t n = u n+ u n, then t turns out k t n u h = u k+ u h 0 (k ) n=0 For t n X Y, there exst v n M,l(n) (X), β n M l(n) and w n M l(n), such that t n = v n β n w n wth β n = (n 0), v n w n < 2 n ε(n ), v 0 w 0 < ε and v n = w n It follows that t n h v n w n < n=0 Then we can choose an ncreasng sequence {c n } R such that c n >, lm c n =, c n v n w n < n n=0 Now we put a() = c v, α = β /c and b() = c w Then we have k k u k+ = v n β n w n = a(n)α n b(n) n=0 n=0 n=0 α 0 = [ a(0) a() a(k) ] α b(0) b(), α k b(k) and [a(0), a(),, a(k)], t [b(0), b(),, b(k)] <, α k 0 (k ) If we defne a n X, b n Y and α K(l 2 ) by the followng relaton: [a(0), a(),, a(k)] = [a, a 2,, a l(0)+l()+ +l(k) ] [b(0), b(),, b(k)] = [b, b 2,, b l(0)+l()+ +l(k) ] α = α k, k=0 then we can get the frst equalty For the above α K(l 2 ), we can take untares u k, v k M l(k) such that λ α k = u k k =0 l()+ λ k =0 l()+2 4 λ k =0 l() v k (k = 0,,2, )
5 If we put U = u k, V = k=0 λ λ v k, Λ = 2 k=0 λ, then we can get Λ = max λ, au M, (X) and a = au, V b M, (Y ) and b = V b, for any a M, (X) and b M, (Y ) By the fact we can get the second equalty aα b = auλv b = (au)λ (V b), By the above proof, we also get the followng fact: X h Y = {aα b α K(l 2 ), a M, (X), b M, (Y )} = { λ a b (λ ) c 0, a M, (X), b M, (Y )} Let H be a separable Hlbert space, {f } a completely orthonormal system of H and {e j },j= a system of matrx unts of B(H) defned by e j ξ = (ξ f j )f, ξ H We can naturally dentfy the bounded sequences l on N wth the maxmal abelan subalgebra of B(H) generated by {e } We denote by CB l (K(H), B(H)) the l -bmodule completely bounded maps of K(H) to B(H) Then there exsts completely sometrc somorphsm between l eh l and CB l (K(H), B(H)) by the followng: for a b l eh l, < a b > CB l (K(H), B(H)) s defned by < a b > (k) = a kb for k K(H) [2] By the l -bmodularty of < x > for x l eh l, there exsts a scalar x j satsfyng that < x > (e j ) = x j e j,, j =,2, 5
6 Then we can defne an nfnte dmensonal matrx [x] = [x j ],j=, and also dentfy [x] wth a lnear map from c c (N) to l as follows: for ξ = [ξ, ξ 2, ] c c (N), [x]ξ = [ x j ξ j, x 2j ξ j, ], j= where ξ = [ξ, ξ 2, ] c c (N) means that ξ n = 0 for suffcently large n Clearly c c (N) s contaned n l 2 and the mage of c c (N) by [x] s not necessarly contaned n l 2 If [x] can be extended to B(l 2 ) (resp K(l 2 )), then we wrte x (l eh l ) B (resp x (l eh l ) K) We also use the followng notaton: for any subspace S of l eh l, j= S B = (l eh l ) B S, S K = (l eh l ) K S Lemma x l eh l f and only f there exst ξ, η l 2 ( =,2, ) such that sup{ ξ, η } < and x j = (ξ η j ) Proof For x l eh l, note that there exst countable sequences {a } and {b } n l such that [a, a 2, ], t [b, b 2, ] < and x = a b Then we defne ξ = [a (), a 2 (), ] and η = [b (), b 2 (), ] for any N Clearly we have ξ, η l 2, and sup{ ξ, η } max{ [a, a 2, ], t [b, b 2, ] }, < x > (e j ) = = a k e j b k k= a k ()b k (j)e j = (ξ η j )e j k= Conversely, for ξ = [ξ (), ξ (2), ], η = [η (), η (2), ] l 2 wth sup{ ξ, η } < and x j = (ξ η j ), we defne a = [ξ (), ξ 2 (), ] and b = t [η (), η 2 (), ] 6
7 Then we have [a, a 2, ], t [b, b 2, ] sup{ ξ, η } and x = a b Lemma 4 x l h l f and only f there exst β K(l 2 ), ξ, η l 2 ( =,2, ) such that sup{ ξ, η } < and x j = (βξ η j ) Proof By Proposton 2, for gven ε > 0 and x l h l, there exst [a, a 2, ] M, (l ), t [b, b 2, ] M, (l ) and [α j ] K(H) satsfyng [a, a 2, ] t [b, b 2, ] < and [α j ] < x h + ε such that x =,j= α ja b j If ξ = [a (), a 2 (), ], η = [b (), b 2 (), ] and β = [β j ] where β j = α j, then t s clear that sup { x, η } < Thus we have < x > (e j ) = s,t α st a s e j b t = s,t α st a s ()b t (j)e j = (βξ η j )e j Conversely, for gven ξ = [ξ (), ξ (2), ], η = [η (), η (2), ] l 2 and β = [β j ] K(l 2 ), we put a = [ξ (), ξ 2 (), ], b = [η (), η 2 (), ] l and α = [α j ] K(l 2 ) where α j = β j Then we have, for any postve nteger N, [a, a 2,, a N ], t [b, b 2,, b N ] sup{ ξ, η } < For an element x n = [ a a 2 b α ] α n a n b 2 l l, α n α nn b n 7
8 we have x n+k x n = [ ] a a n+k 0 0 α,n+ α,n+k b 0 0 α n+, α n+,n+ b n+k α n+k, α n+k,n+k By the compactness of α and Proposton, lm x n+k x n h = 0 n for any postve nteger k Thus we have that the sequence {x n } converges to x n l h l Snce c 0 s a C*-subalgebra of l, we can see c 0 h c 0 as a subspace of l h l Lemma 5 x c 0 h c 0 f and only f there exst ξ, η l 2 ( =,2, ) such that lm ξ = lm η = 0 and x j = (ξ η j ) Proof By Proposton 2, for gven ε > 0 and x c 0 h c 0, there exst [a, a 2, ] M, (c 0 ), t [b, b 2, ] M, (c 0 ) and [α j ] K(H) satsfyng [a, a 2, ] t [b, b 2, ] < and [α j ] < x h + ε such that x =,j= α ja b j We put ξ = [a (), a 2 (), ], η = [b (), b 2 (), ] and β = [β j ] where β j = α j Then we have and, by the fact a, b c 0, x j = (βξ η j ) lm ξ (j) = lm η (j) = 0 for any j N Ths means that {ξ }, {η } l 2 weakly converges to 0 We can choose β, β 2 K(l 2 ) such that β = β 2 β Then we have lm β ξ = lm β 2 η = 0 and x j = (βξ η j ) = (β ξ β 2 η j ) 8
9 Conversely suppose that lm ξ = lm η = 0 We may assume that ξ < c for all N Then, for any ε > 0, we can choose a number N such that where ξ < c for all and ξ < ε f > N, { ξ (j) = ξ (j) f j < N 2ξ (j) otherwse Clearly we have lm ξ = 0 Applyng ths argument to {ξ } repeatedly, we can choose = n(0) < n() < n(2) < and {ζ } l 2 such that ζ (j) = 2 k ξ (j) f n(k) j < n(k + ), ζ < c for all and ζ < 2 k f > n(k) We put a = [ζ (), ζ 2 (), ], b = [η (), η 2 (), ] and Then we have a, b, (λ ) c 0, and Thus we have λ = 2 k f n(k) < n(k + ) [a, a 2, ], t [b, b 2, ] sup{ ζ, η } < x = λ a b c 0 h c 0 Lemma 6 x c 0 eh c 0 f and only f there exst ξ, η l 2 ( =,2, ) such that lm ξ = lm η = 0 (weakly) and x j = (ξ η j ) Proof By the fact c 0 eh c 0 l eh l and the proof of Lemma, we can choose ξ, η l 2 ( =,2, ) satsfyng that and sup{ ξ, η } <, x j = (ξ η j ) lm ξ j() = lm η j () = 0 ( =,2, ) j j Ths means that {ξ }, {η } weakly converges to 0 9
10 Conversely, snce ξ, η converges to 0 weakly, they are unformly bounded As n the proof of Lemma, f we put a (j) = ξ j () and b (j) = η j (), then we have a, b c 0, [a, a 2, ] <, t [b, b 2, ] < and x = a b c 0 eh c 0 From the proof of the prevous lemmas, we have another representatons of norms for the (extended) Haagerup tensor product on c 0 c 0 and l l Remark 7 () For x l eh l, x eh = nf{sup ξ η j x j = (ξ η j ), ξ, η j l 2 } j (2) For x l h l, x h = nf{sup ξ η j β x j = (βξ η j ), ξ, η j l 2, β K(l 2 )} j () For x c 0 eh c 0, x eh = nf{sup ξ η j x j = (ξ η j ), ξ, η j l 2, ξ 0, η j 0 weakly} j (4) For x c 0 h c 0, x h = nf{sup ξ η j x j = (ξ η j ), ξ, η j l 2, ξ 0, η j 0 strongly} j Theorem 8 () (c 0 h c 0 ) B = (l h l ) B (2) (c 0 eh c 0 ) B = (l eh l ) B Proof () It s clear that (c 0 h c 0 ) B (l h l ) B Let x (l h l ) B By Lemma 4, there exst β K(l 2 ), ξ, η l 2 ( =,2, ) such that sup{ ξ, η } < and x j = (βξ η j ) We choose β, β 2 K(l 2 ) such that β = β 2 β, that s, and we may assume that x j = (β ξ β 2 η j ), Range(β ) span{β 2 η j j N}, and Range(β 2 ) span{β ξ j j N} 0
11 It s suffcent to show that Assume that lm β ξ = lm β 2 η = 0 lm sup β ξ > 0 Then there exst δ > 0 and a subsequence {n(k)} such that β ξ n(k) > δ for k =,2, Snce sup ξ <, we may also assume that {ξ n(k) } weakly converges to some ξ 0 l 2 By the compactness of β, we have lm β ξ n(k) β ξ 0 = 0 Thus t turns out β ξ 0 0 We can choose j 0 such that Then there exsts K N such that (β ξ 0 β 2 η j0 ) 0 x n(k),j0 = (β ξ n(k) β 2 η j0 ) > (β ξ 0 β 2 η j0 ) for k > K 2 Ths contradcts to [x] = [x j ] B(l 2 ) (2) For x (l eh l ) B, that s, [x] B(l 2 ), we can choose α = [ξ j ] and β = [η j ] n B(l 2 ) such that Remarkng the fact we defne for all Then we have [x] = αβ and α = β = [x] /2 x j = k ξ k η kj, a = [ξ, ξ 2, ], b = [η, η 2, ] l 2 c 0 [a, a 2, ], t [b, b 2, ] [x] /2 < and x = a b c 0 eh c 0 Corollary 9 Let x l eh l and lm sup k x (k),j(k) > 0 for some njecton N k ((k), j(k)) N N Then x does not belong to c 0 h c 0 Moreover, f x satsfes an addtonal condton [x] B(l 2 ), then x does not belong to l h l
12 Example 0 () Let x =,j= ( λ λ j ) t e e j l eh l, where λ s are postve real and t s real Then we have ( λ λ ) t ( λ λ 2 ) t [x] = ( λ 2 λ ) t ( λ 2 λ 2 ) t / B(l 2 ), x j = ( λ ) t = ( λ j 0 λ t 0 λ t and x j = Ths means x / (l eh l ) B, x / c 0 eh c 0 (by Lemma 6) and x l h l (by Lemma 4) (2) Let x = k= e k e k c 0 eh c 0 Snce 0 [x] = 0 B(l 2 ) then we have x / l h l (by Corollary 9) () (l eh l ) K (l h l ) B By Lemma 4, t s clear that (l eh l ) K (l h l ) B We consder the followng nfnte dmensonal matrx: p = j 0 ) Snce p s an nfnte dmensonal projecton, p does not belong to K(l 2 ) If we put ξ = [,0,0,0, ] ξ 2 = ξ = [0,, 0,0 ] 2 ξ 4 = ξ 5 = ξ 6 = [0,0, 2, 0, ]
13 and ξ n = η n (n =,2, ), then ξ n, η n l 2 satsfy Ths means that lm n ξ n = lm n η n = 0 and p = [(ξ η j )] ((l h l ) B) ((l eh l ) K) c φ (4) Let a = b = [, 2,, n, ] c 0 Then x = a b c 0 h c 0 and [x j ] = / B(l 2 ) By the above argument, we can get the followng dagram of nclusons: (c 0 h c 0 ) K (c 0 h c 0 ) B (c 0 eh c 0 ) B = = = (l eh l ) K (l h l ) B (l eh l ) B c 0 h c 0 c 0 eh c 0 l h l l eh l X h Y and X eh Y Let X be an operator space We call X rght-compact (resp leftcompact) f M,I (X) = M,I (X)K(l 2 (I)) (resp M I, (X) = K(l 2 (I)) M I, (X)) If X s rght-compact, then, for any a = [a ] I M,I (X), there exst b = [b ] I M,I (X) and α = [α j ],j I K(l 2 (I)) such that a = bα (a j = b α j ) I We also call X weakly rght-compact (resp weakly left-compact) f we have, for any a = [a ] M,I (X) (resp a = [a ] M I, (X)), that { I a 0} s countable and lm a = 0 Lemma If X s a rght-compact (resp left-compact) operator space, then X s weakly rght-compact (resp weakly left-compact)
14 Proof Snce α = [α j ],j I K(l 2 (I)), we have that {(, j) I I α j 0} s countable and lm j α j = 0 For b = [b ] I M,I (X), {b I} s bounded So we have for any a = bα, that { I a 0} s countable and lm a = 0 As a typcal example of rght-compact operator spaces, we can get the followng: Lemma 2 Let X be an operator space on a Hlbert space H If X pb(h) for some fnte-dmensonal projecton p B(H), then X s rght-compact In partcular, any fnte-dmensonal C*-algebra s left- and rghtcompact Proof We assume that dm ph = n < Let a = [a ] I M,I (X), e, I a a < We can consder a a as an element of M n (C), so we put a a = (α jk ) (j, k =,2,, n) By the postvty of a a, we have 0 sup α jj a a j n I I Ths mples that s countable Remarkng the fact we have I 0 = { I α jj > 0 for some j} a a n sup α jj, j n a a a a n n α jj < We can choose a sequence of postve numbers λ such that λ > 0, λ 0 ( ), a a < Then we have j= a = [a ] I = [ a λ ] I [δ j λ ],j I M I (X)K(l 2 (I)), 4 λ 2 I
15 where δ j means Kronecker s symbol Lemma Let {a } be a sequence of bounded operators on a Hlbert space H wth a a < Suppose that there exst sequences {u }, {ξ } of unt vectors n H such that (a u ξ ) > for N Then, for any ε > 0, there exsts a number 0 such that s nfnte { N (a 0 u ξ ) < ε} Proof Let ε > 0 Suppose that {j N (a u j ξ j ) < ε} s fnte for all N If we omt the fnte set we may assume that If we omt agan the fnte set we may assume that { N (a u ξ ) < ε}, (a u ξ ) >, (a 2 u 2 ξ 2 ) >, (a u 2 ξ 2 ) ε { N > 2, (a 2 u ξ ) < ε}, (a u ξ ) ε, (a 2 u ξ ) ε Usng ths argument repeatedly, we may assume that, for any n, Then we have (a u n ξ n ) ε, (a 2 u n ξ n ) ε,, (a n u n ξ n ) ε n n (n )ε 2 (a u n ξ n ) 2 u n 2 a ξ n 2 n (a a ξ n n ξ n ) a a Ths contradcts to the assumpton a a < Therefore we can get a number 0 requred n the statement 5
16 Lemma 4 Let X be an operator space on a Hlbert space H If X s not weakly rght-compact, then there exst a sequence {a } of X, sequences {u }, {ξ } of unt vectors n H and some constant K such that () a a < (2) < a < K () < (a u ξ ) < K (4) (a k u j ξ j ) for k j Kk Proof Snce X s not weakly left-compact, we can choose a sequence {a } of X such that a a < and { a } s not convergent to 0 Then we may assume that < a < K for any and some constant K We choose sequences {u }, {ξ } of unt vectors n H satsfyng (a u ξ ) > for all N Usng Lemma, we can choose a subsequence {n(k)} k= such that (a n(k) u n(j) ξ n(j) ) < for k < j K k If we replace {a n(k) } wth {a }, then we can get the condtons () () and (4) for k < j We consder a sequence {a, u, ξ } of trplets By the calculaton (a u j ξ j ) 2 a ξ j 2 = ( a a ξ j ξ j ) we have a a <, lm (a u j ξ j ) = 0 for any j Choosng a subsequence of {a, u, ξ }, we may assume that (a k u j ξ j ) < K k for k > j Thus we can get the condtons () (4) 6
17 Lemma 5 Let α > β > 0 If sequences {a k }, {b k } of vectors n C m satsfy the followng condtons: then (a k b k ) > α and (a k b l ) < β for k l, sup{ a k (), b k () =,, m, k =,2, } = Proof We assume that sup{ a k (), b k () =,, n, k =,2, } s fnte By the compactness, we can choose a par of convergent subsequences {a n(k) }, {b n(k) } Then we have But ths contradcts to lm k (a n(k) b n(k+) ) = lm k (a n(k) b n(k) ) α lm sup (a n(k) b n(k+) ) β k Theorem 6 Let X and Y be operator spaces Then we have () X h Y = X eh Y f X s rght-compact or Y s left-compact (2) X s weakly rght-compact or Y s weakly left-compact f X h Y = X eh Y Proof () We assume that X s rght-compact For any s X eh Y, there exst a = [a ] M,I (X) and b = t [b ] M I, (Y ) such that s = a b = I a b By the assumpton, there exst c M,I (X) and α K(l 2 (I)) such that a = cα So we have s = a b = cα b X h Y Ths means that X h Y = X eh Y When Y s left-compact, we can also have X h Y = X eh Y by the same argument (2) Let X (resp Y ) be an operator space on H (resp K) We assume that X s not weakly rght-compact and Y s not weakly leftcompact By Lemma 4, we can choose a sequence {a } of X (resp a sequence {b } of Y ), sequences {u }, {ξ } of unt vectors n H (resp sequences {v }, {η } of unt vectors n K) and some constant K satsfyng 7
18 that a a <, b b <, < a, b < K, < (a u ξ ), (b η v ) < K and (a k u j ξ j ), (b k η j v j ) < K k for k j We defne s X eh Y, ϕ k X and ψ k Y as follows: s = Then we have a b, ϕ k ( ) = ( u k ξ k ), ψ k ( ) = ( η k v k ) s(ϕ k, ψ k ) = = ϕ k (a )ψ k (b ) (a u k ξ k )(b η k v k ) (a k u k ξ k )(b k η k v k ) k (a u k ξ k )(b η k v k ) 9 > 8, K2 and, for j k, s(ϕ j, ψ k ) = = ϕ j (a )ψ k (b ) (a u j ξ j )(b η k v k ) (a u j ξ j )(b η k v k ) K + j K + < k K2 8
19 Suppose that X eh Y = X h Y, then s belongs to X h Y We can choose m t = x y X h Y and s t h < Snce ϕ j = ψ k =, that s, Then we have and, for j k, s(ϕ j, ψ k ) t(ϕ j, ψ k ) <, s(ϕ j, ψ k ) m ϕ j (x )ψ k (y ) < m ϕ k (x )ψ k (y ) > 7 m ϕ j (x )ψ k (y ) < 4 Ths contradcts to the boundedness of { ϕ k (x ), ψ k (y ) m, k N} by Lemma 5 We are done Remark 7 The row Hlbert space H r s rght-compact and s not weakly left-compact and the column Hlbert space H c s left-compact and s not weakly rght-compact Then t s clear that (cf[5]) H r h H c = H r eh H c, H c h H r H c eh H r Corollary 8 Let A and B be C*-algebras Then the followng assertons are equvalent: () A h B = A eh B, (2) A or B s fnte dmensonal Proof We have already shown that every fnte-dmensnal C*-algebra s rght-compact and left-compact n Lemma 2 It s suffcent to show that every nfnte-dmensonal C*-algebra s nether weakly rghtcompact nor weakly left-compact 9
20 Suppose that A s nfnte dmensonal Snce the maxmal abelan *-subalgebras n A s nfnte dmensonal, there exst self-adjont elements {a n } A such that a n = and a a j = 0 f j Then we have a 2 = a a = a a < and { a } does not converge to 0 Ths means that A s nether weakly rght-compact nor weakly left-compact References [] D P Blecher and V I Paulsen, Tensor products of operator spaces, J Funct Anal 99, (99), pp [2] D P Blecher and R R Smth, The dual of the Haagerup tensor product, J London Math Soc 45, (992), pp [] E G Effros and A Kshmoto, Module maps and Hochschld-Johnson cohomology, Indana Math J 6, (987), pp [4] E G Effros and Z -J Ruan, A new approach to operator spaces, Canad Math Bull 4, (99), pp 29 7 [5] E G Effros and Z -J Ruan, Self-dualty for the Haagerup tensor product and Hlbert space factorzatons, J Funct Anal 00, (99), pp [6] E G Effros and Z -J Ruan, Operator convoluton algebras: An approach to Quantum groups, preprnt [7] T Itoh and M Nagsa, Schur products and module maps on B(H), Publ RIMS Kyoto Unv 6, (2000), pp [8] R R Smth, Completely bounded module maps and the Haagerup tensor product, J Funct Anal 02, (99), pp Department of Mathematcs, Gunma Unversty, Gunma 7-850, Japan E-mal address: toh@edugunma-uacjp Department of Mathematcs and Informatcs, Chba Unversty, Chba , Japan E-mal address: nagsa@mathschba-uacjp 20
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
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
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationMath 101 Fall 2013 Homework #7 Due Friday, November 15, 2013
Math 101 Fall 2013 Homework #7 Due Frday, November 15, 2013 1. Let R be a untal subrng of E. Show that E R R s somorphc to E. ANS: The map (s,r) sr s a R-balanced map of E R to E. Hence there s a group
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationA CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS
Journal of Mathematcal Scences: Advances and Applcatons Volume 25, 2014, Pages 1-12 A CHARACTERIZATION OF ADDITIVE DERIVATIONS ON VON NEUMANN ALGEBRAS JIA JI, WEN ZHANG and XIAOFEI QI Department of Mathematcs
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationAppendix B. Criterion of Riemann-Stieltjes Integrability
Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for
More informationOn Finite Rank Perturbation of Diagonalizable Operators
Functonal Analyss, Approxmaton and Computaton 6 (1) (2014), 49 53 Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: http://wwwpmfnacrs/faac On Fnte Rank Perturbaton of Dagonalzable
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationNOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules
NOTES FOR QUANTUM GROUPS, CRYSTAL BASES AND REALIZATION OF ŝl(n)-modules EVAN WILSON Quantum groups Consder the Le algebra sl(n), whch s the Le algebra over C of n n trace matrces together wth the commutator
More informationAn Introduction to Morita Theory
An Introducton to Morta Theory Matt Booth October 2015 Nov. 2017: made a few revsons. Thanks to Nng Shan for catchng a typo. My man reference for these notes was Chapter II of Bass s book Algebrac K-Theory
More informationOn C 0 multi-contractions having a regular dilation
SUDIA MAHEMAICA 170 (3) (2005) On C 0 mult-contractons havng a regular dlaton by Dan Popovc (mşoara) Abstract. Commutng mult-contractons of class C 0 and havng a regular sometrc dlaton are studed. We prove
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationDIFFERENTIAL SCHEMES
DIFFERENTIAL SCHEMES RAYMOND T. HOOBLER Dedcated to the memory o Jerry Kovacc 1. schemes All rngs contan Q and are commutatve. We x a d erental rng A throughout ths secton. 1.1. The topologcal space. Let
More informationP.P. PROPERTIES OF GROUP RINGS. Libo Zan and Jianlong Chen
Internatonal Electronc Journal of Algebra Volume 3 2008 7-24 P.P. PROPERTIES OF GROUP RINGS Lbo Zan and Janlong Chen Receved: May 2007; Revsed: 24 October 2007 Communcated by John Clark Abstract. A rng
More informationAli Omer Alattass Department of Mathematics, Faculty of Science, Hadramout University of science and Technology, P. O. Box 50663, Mukalla, Yemen
Journal of athematcs and Statstcs 7 (): 4448, 0 ISSN 5493644 00 Scence Publcatons odules n σ[] wth Chan Condtons on Small Submodules Al Omer Alattass Department of athematcs, Faculty of Scence, Hadramout
More informationGoogle PageRank with Stochastic Matrix
Google PageRank wth Stochastc Matrx Md. Sharq, Puranjt Sanyal, Samk Mtra (M.Sc. Applcatons of Mathematcs) Dscrete Tme Markov Chan Let S be a countable set (usually S s a subset of Z or Z d or R or R d
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationWeek 2. This week, we covered operations on sets and cardinality.
Week 2 Ths week, we covered operatons on sets and cardnalty. Defnton 0.1 (Correspondence). A correspondence between two sets A and B s a set S contaned n A B = {(a, b) a A, b B}. A correspondence from
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationSTEINHAUS PROPERTY IN BANACH LATTICES
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT STEINHAUS PROPERTY IN BANACH LATTICES DAMIAN KUBIAK AND DAVID TIDWELL SPRING 2015 No. 2015-1 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookevlle, TN 38505 STEINHAUS
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationOn the smoothness and the totally strong properties for nearness frames
Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo
More informationLecture 7: Gluing prevarieties; products
Lecture 7: Glung prevaretes; products 1 The category of algebrac prevaretes Proposton 1. Let (f,ϕ) : (X,O X ) (Y,O Y ) be a morphsm of algebrac prevaretes. If U X and V Y are affne open subvaretes wth
More informationIdeal Amenability of Second Duals of Banach Algebras
Internatonal Mathematcal Forum, 2, 2007, no. 16, 765-770 Ideal Amenablty of Second Duals of Banach Algebras M. Eshagh Gord (1), F. Habban (2) and B. Hayat (3) (1) Department of Mathematcs, Faculty of Scences,
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationA Note on \Modules, Comodules, and Cotensor Products over Frobenius Algebras"
Chn. Ann. Math. 27B(4), 2006, 419{424 DOI: 10.1007/s11401-005-0025-z Chnese Annals of Mathematcs, Seres B c The Edtoral Oce of CAM and Sprnger-Verlag Berln Hedelberg 2006 A Note on \Modules, Comodules,
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationLinear Algebra and its Applications
Lnear Algebra and ts Applcatons 4 (00) 5 56 Contents lsts avalable at ScenceDrect Lnear Algebra and ts Applcatons journal homepage: wwwelsevercom/locate/laa Notes on Hlbert and Cauchy matrces Mroslav Fedler
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationINVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS
INVARIANT STABLY COMPLEX STRUCTURES ON TOPOLOGICAL TORIC MANIFOLDS HIROAKI ISHIDA Abstract We show that any (C ) n -nvarant stably complex structure on a topologcal torc manfold of dmenson 2n s ntegrable
More informationThe Pseudoblocks of Endomorphism Algebras
Internatonal Mathematcal Forum, 4, 009, no. 48, 363-368 The Pseudoblocks of Endomorphsm Algebras Ahmed A. Khammash Department of Mathematcal Scences, Umm Al-Qura Unversty P.O.Box 796, Makkah, Saud Araba
More informationALGEBRA SCHEMES AND THEIR REPRESENTATIONS
ALGEBRA SCHEMES AND THEIR REPRESENTATIONS AMELIA ÁLVAREZ, CARLOS SANCHO, AND PEDRO SANCHO Introducton The equvalence (Carter dualty) between the category of topologcally flat formal k-groups and the category
More informationErrata to Invariant Theory with Applications January 28, 2017
Invarant Theory wth Applcatons Jan Drasma and Don Gjswjt http: //www.wn.tue.nl/~jdrasma/teachng/nvtheory0910/lecturenotes12.pdf verson of 7 December 2009 Errata and addenda by Darj Grnberg The followng
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationMATH Homework #2
MATH609-601 Homework #2 September 27, 2012 1. Problems Ths contans a set of possble solutons to all problems of HW-2. Be vglant snce typos are possble (and nevtable). (1) Problem 1 (20 pts) For a matrx
More informationSUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) M(B) := # ( B Z N)
SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) S.BOUCKSOM Abstract. The goal of ths note s to present a remarably smple proof, due to Hen, of a result prevously obtaned by Gllet-Soulé,
More informationALGEBRA SCHEMES AND THEIR REPRESENTATIONS
ALGEBRA SCHEMES AND THEIR REPRESENTATIONS AMELIA ÁLVAREZ, CARLOS SANCHO, AND PEDRO SANCHO Introducton The equvalence (Carter dualty) between the category of topologcally flat formal k-groups and the category
More informationGraph Reconstruction by Permutations
Graph Reconstructon by Permutatons Perre Ille and Wllam Kocay* Insttut de Mathémathques de Lumny CNRS UMR 6206 163 avenue de Lumny, Case 907 13288 Marselle Cedex 9, France e-mal: lle@ml.unv-mrs.fr Computer
More information10-801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014)
0-80: Advanced Optmzaton and Randomzed Methods Lecture : Convex functons (Jan 5, 04) Lecturer: Suvrt Sra Addr: Carnege Mellon Unversty, Sprng 04 Scrbes: Avnava Dubey, Ahmed Hefny Dsclamer: These notes
More informationThe lower and upper bounds on Perron root of nonnegative irreducible matrices
Journal of Computatonal Appled Mathematcs 217 (2008) 259 267 wwwelsevercom/locate/cam The lower upper bounds on Perron root of nonnegatve rreducble matrces Guang-Xn Huang a,, Feng Yn b,keguo a a College
More informationDeriving the X-Z Identity from Auxiliary Space Method
Dervng the X-Z Identty from Auxlary Space Method Long Chen Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA 92697 chenlong@math.uc.edu 1 Iteratve Methods In ths paper we dscuss teratve
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationDirichlet s Theorem In Arithmetic Progressions
Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons,
More informationa b a In case b 0, a being divisible by b is the same as to say that
Secton 6.2 Dvsblty among the ntegers An nteger a ε s dvsble by b ε f there s an nteger c ε such that a = bc. Note that s dvsble by any nteger b, snce = b. On the other hand, a s dvsble by only f a = :
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationarxiv: v1 [math.ca] 31 Jul 2018
LOWE ASSOUAD TYPE DIMENSIONS OF UNIFOMLY PEFECT SETS IN DOUBLING METIC SPACE HAIPENG CHEN, MIN WU, AND YUANYANG CHANG arxv:80769v [mathca] 3 Jul 08 Abstract In ths paper, we are concerned wth the relatonshps
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationGELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n
GELFAND-TSETLIN BASIS FOR THE REPRESENTATIONS OF gl n KANG LU FINITE DIMENSIONAL REPRESENTATIONS OF gl n Let e j,, j =,, n denote the standard bass of the general lnear Le algebra gl n over the feld of
More informationMath 217 Fall 2013 Homework 2 Solutions
Math 17 Fall 013 Homework Solutons Due Thursday Sept. 6, 013 5pm Ths homework conssts of 6 problems of 5 ponts each. The total s 30. You need to fully justfy your answer prove that your functon ndeed has
More informationCorrespondences and groupoids
Proceedngs of the IX Fall Workshop on Geometry and Physcs, Vlanova la Geltrú, 2000 Publcacones de la RSME, vol. X, pp. 1 6. Correspondences and groupods 1 Marta Macho-Stadler and 2 Moto O uch 1 Departamento
More informationEXTENSIONS OF STRONGLY Π-REGULAR RINGS
EXTENSIONS OF STRONGLY Π-REGULAR RINGS H. Chen, K. Kose and Y. Kurtulmaz ABSTRACT An deal I of a rng R s strongly π-regular f for any x I there exst n N and y I such that x n = x n+1 y. We prove that every
More informationAnti-van der Waerden numbers of 3-term arithmetic progressions.
Ant-van der Waerden numbers of 3-term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The ant-van der Waerden number, denoted by aw([n], k), s the smallest
More informationVolume 18 Figure 1. Notation 1. Notation 2. Observation 1. Remark 1. Remark 2. Remark 3. Remark 4. Remark 5. Remark 6. Theorem A [2]. Theorem B [2].
Bulletn of Mathematcal Scences and Applcatons Submtted: 016-04-07 ISSN: 78-9634, Vol. 18, pp 1-10 Revsed: 016-09-08 do:10.1805/www.scpress.com/bmsa.18.1 Accepted: 016-10-13 017 ScPress Ltd., Swtzerland
More informationA note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 7, Number 2, December 203 Avalable onlne at http://acutm.math.ut.ee A note on almost sure behavor of randomly weghted sums of φ-mxng
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationDiscrete Mathematics. Laplacian spectral characterization of some graphs obtained by product operation
Dscrete Mathematcs 31 (01) 1591 1595 Contents lsts avalable at ScVerse ScenceDrect Dscrete Mathematcs journal homepage: www.elsever.com/locate/dsc Laplacan spectral characterzaton of some graphs obtaned
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationFixed points of IA-endomorphisms of a free metabelian Lie algebra
Proc. Indan Acad. Sc. (Math. Sc.) Vol. 121, No. 4, November 2011, pp. 405 416. c Indan Academy of Scences Fxed ponts of IA-endomorphsms of a free metabelan Le algebra NAIME EKICI 1 and DEMET PARLAK SÖNMEZ
More informationCONJUGACY IN THOMPSON S GROUP F. 1. Introduction
CONJUGACY IN THOMPSON S GROUP F NICK GILL AND IAN SHORT Abstract. We complete the program begun by Brn and Squer of charactersng conjugacy n Thompson s group F usng the standard acton of F as a group of
More informationLecture 5 Decoding Binary BCH Codes
Lecture 5 Decodng Bnary BCH Codes In ths class, we wll ntroduce dfferent methods for decodng BCH codes 51 Decodng the [15, 7, 5] 2 -BCH Code Consder the [15, 7, 5] 2 -code C we ntroduced n the last lecture
More informationProjective change between two Special (α, β)- Finsler Metrics
Internatonal Journal of Trend n Research and Development, Volume 2(6), ISSN 2394-9333 www.jtrd.com Projectve change between two Specal (, β)- Fnsler Metrcs Gayathr.K 1 and Narasmhamurthy.S.K 2 1 Assstant
More informationGeometry of Müntz Spaces
WDS'12 Proceedngs of Contrbuted Papers, Part I, 31 35, 212. ISBN 978-8-7378-224-5 MATFYZPRESS Geometry of Müntz Spaces P. Petráček Charles Unversty, Faculty of Mathematcs and Physcs, Prague, Czech Republc.
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introducton to Cluster Algebra Ivan C.H. Ip Updated: Ma 7, 2017 3 Defnton and Examples of Cluster algebra 3.1 Quvers We frst revst the noton of a quver. Defnton 3.1. A quver s a fnte orented
More informationMATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1
MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationSPECTRAL PROPERTIES OF IMAGE MEASURES UNDER THE INFINITE CONFLICT INTERACTION
SPECTRAL PROPERTIES OF IMAGE MEASURES UNDER THE INFINITE CONFLICT INTERACTION SERGIO ALBEVERIO 1,2,3,4, VOLODYMYR KOSHMANENKO 5, MYKOLA PRATSIOVYTYI 6, GRYGORIY TORBIN 7 Abstract. We ntroduce the conflct
More informationHomework 1 Lie Algebras
Homework 1 Le Algebras Joshua Ruter February 9, 018 Proposton 0.1 Problem 1.7a). Let A be a K-algebra, wth char K. Then A s alternatve f and only f the folowng two lnear) denttes hold for all a, b, y A.
More informationSELECTED SOLUTIONS, SECTION (Weak duality) Prove that the primal and dual values p and d defined by equations (4.3.2) and (4.3.3) satisfy p d.
SELECTED SOLUTIONS, SECTION 4.3 1. Weak dualty Prove that the prmal and dual values p and d defned by equatons 4.3. and 4.3.3 satsfy p d. We consder an optmzaton problem of the form The Lagrangan for ths
More informationGames of Threats. Elon Kohlberg Abraham Neyman. Working Paper
Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18-023 Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18-023 Copyrght 2017
More informationSemilattices of Rectangular Bands and Groups of Order Two.
1 Semlattces of Rectangular Bs Groups of Order Two R A R Monzo Abstract We prove that a semgroup S s a semlattce of rectangular bs groups of order two f only f t satsfes the dentty y y, y y, y S 1 Introducton
More informationSolutions HW #2. minimize. Ax = b. Give the dual problem, and make the implicit equality constraints explicit. Solution.
Solutons HW #2 Dual of general LP. Fnd the dual functon of the LP mnmze subject to c T x Gx h Ax = b. Gve the dual problem, and make the mplct equalty constrants explct. Soluton. 1. The Lagrangan s L(x,
More informationPROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 7, July 1997, Pages 2119{2125 S (97) THE STRONG OPEN SET CONDITION
PROCDINGS OF TH AMRICAN MATHMATICAL SOCITY Volume 125, Number 7, July 1997, Pages 2119{2125 S 0002-9939(97)03816-1 TH STRONG OPN ST CONDITION IN TH RANDOM CAS NORBRT PATZSCHK (Communcated by Palle. T.
More informationDIFFERENTIAL FORMS BRIAN OSSERMAN
DIFFERENTIAL FORMS BRIAN OSSERMAN Dfferentals are an mportant topc n algebrac geometry, allowng the use of some classcal geometrc arguments n the context of varetes over any feld. We wll use them to defne
More informationwhere a is any ideal of R. Lemma 5.4. Let R be a ring. Then X = Spec R is a topological space Moreover the open sets
5. Schemes To defne schemes, just as wth algebrac varetes, the dea s to frst defne what an affne scheme s, and then realse an arbtrary scheme, as somethng whch s locally an affne scheme. The defnton of
More informationPerfect Competition and the Nash Bargaining Solution
Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange
More informationON SEPARATING SETS OF WORDS IV
ON SEPARATING SETS OF WORDS IV V. FLAŠKA, T. KEPKA AND J. KORTELAINEN Abstract. Further propertes of transtve closures of specal replacement relatons n free monods are studed. 1. Introducton Ths artcle
More informationA combinatorial problem associated with nonograms
A combnatoral problem assocated wth nonograms Jessca Benton Ron Snow Nolan Wallach March 21, 2005 1 Introducton. Ths work was motvated by a queston posed by the second named author to the frst named author
More informationMath 594. Solutions 1
Math 594. Solutons 1 1. Let V and W be fnte-dmensonal vector spaces over a feld F. Let G = GL(V ) and H = GL(W ) be the assocated general lnear groups. Let X denote the vector space Hom F (V, W ) of lnear
More informationarxiv: v4 [math.ac] 20 Sep 2013
arxv:1207.2850v4 [math.ac] 20 Sep 2013 A SURVEY OF SOME RESULTS FOR MIXED MULTIPLICITIES Le Van Dnh and Nguyen Ten Manh Truong Th Hong Thanh Department of Mathematcs, Hano Natonal Unversty of Educaton
More informationLECTURE V. 1. More on the Chinese Remainder Theorem We begin by recalling this theorem, proven in the preceeding lecture.
LECTURE V EDWIN SPARK 1. More on the Chnese Remander Theorem We begn by recallng ths theorem, proven n the preceedng lecture. Theorem 1.1 (Chnese Remander Theorem). Let R be a rng wth deals I 1, I 2,...,
More informationBayesian epistemology II: Arguments for Probabilism
Bayesan epstemology II: Arguments for Probablsm Rchard Pettgrew May 9, 2012 1 The model Represent an agent s credal state at a gven tme t by a credence functon c t : F [0, 1]. where F s the algebra of
More informationw ). Then use the Cauchy-Schwartz inequality ( v w v w ).] = in R 4. Can you find a vector u 4 in R 4 such that the
Math S-b Summer 8 Homework #5 Problems due Wed, July 8: Secton 5: Gve an algebrac proof for the trangle nequalty v+ w v + w Draw a sketch [Hnt: Expand v+ w ( v+ w) ( v+ w ) hen use the Cauchy-Schwartz
More informationA Radon-Nikodym Theorem for Completely Positive Maps
A Radon-Nody Theore for Copletely Postve Maps V P Belavn School of Matheatcal Scences, Unversty of Nottngha, Nottngha NG7 RD E-al: vpb@aths.nott.ac.u and P Staszews Insttute of Physcs, Ncholas Coperncus
More informationOn a ρ n -Dilation of Operator in Hilbert Spaces
E extracta mathematcae Vol. 31, Núm. 1, 11 23 (2016) On a ρ n -Dlaton of Operator n Hlbert Spaces A. Salh, H. Zeroual PB 1014, Departement of Mathematcs, Sences Faculty, Mohamed V Unversty n Rabat, Rabat,
More informationEigenvalues of Random Graphs
Spectral Graph Theory Lecture 2 Egenvalues of Random Graphs Danel A. Spelman November 4, 202 2. Introducton In ths lecture, we consder a random graph on n vertces n whch each edge s chosen to be n the
More informationn-strongly Ding Projective, Injective and Flat Modules
Internatonal Mathematcal Forum, Vol. 7, 2012, no. 42, 2093-2098 n-strongly Dng Projectve, Injectve and Flat Modules Janmn Xng College o Mathematc and Physcs Qngdao Unversty o Scence and Technology Qngdao
More information