第36卷第3期 2010年7月 曲阜师范 Journal of Qufu 大学学报 Normal University V0136 No3 July 2010 On the Spectnml of Some Pproducts of Operators in Free Products WANG Liguang (School of Mathematical Sciences，Qufu Normal University，273 165，Qufu，Shandong，PRC) Abstract：In this note，we compute the spectrum of some products of operators in the full free product M2(C) c (C)ofM2(C) (C)(over C)using elementary techniques of operator algebras Key words：free product；spectrum； 一homomorphism；state CLC number：O17525；O17526 Document code：A Article ID：1001-5337(2010)03-0005-04 1 Introduction Brown measure was introduced by Brown L G in 1 983 for nonnormal elements in a finite von Neumann a1 gebra with respect to a fixed normal faithful tracial stateRecentlyHaagerup and Schuhz【 J proved a remarkable result which states that if the support of Brown measure of an operator in a typel factor which contains more than two points，then the operator has a nontrivial hyperinvariant subspace affiliated with the type I1 l factorIn general cases，the computation of Brown measures of non-norm operators ale nontrivia1The first essential result was given by Haagerup and LarsenIn3，Haagerup and Lalen computed the spectrum and Brown measure of Rdiagonal operators in a finite von Neumann algebra，in terms of the distribution of its radial partBrown measures of some nonnormal and nonRdiagonal operators are computed by Biane and Lehner in4Fang，Hadwin and Ma computed the spectrum and the Brown spectrum(the support of Browns measure)of operator AB for normal matri ces A，BME(C) The purpose of this short note is to compute the spectrum of some operators of the form AB in the full free prod- uct M2(C) (C)of (C)with M2(C)，where A and B ale matrices in M2(C)For the definitions and properties of full free product，we refer to6and7We also refer to8for some applications of free prod uctsFor basic theory of operator algebras we refere to9 In this paper，we always denote by o-(A)the spectrum of an operator A in a unital C 一algebra ，that is， or(A)=tC： do not have twosided inverse in where I is the unit of 2 Computation of the spectrum We first recall the following definition of full free product of two C 一algebras(see6and7) Received data：200911-09 Foundation item：This work was partially supported bythe NSF of China(No1062603 1)，the Scientific Research Fund of the Shandong Provin cial Education Department(JOSLI15)and Qufu Normal University(ydo71o) Autobiography：Wang Liguang，male，1974-，PhD，associate ss0r：Research field：Operator algebra；Emall：wangliguang()510163 Com 一 曲阜师范大学学报(自然科学版) =。一一、I J， ， 2010晕 DefiIIi舶n 21 The unital full free produet。f the unital C 一algebras and ver C is a unita1 C*-alge- bm 。quiPP。d with unita1 embedding ： j such that(i)the set I( 1)u 2( )is nonTl dense in ；卸d(ii)if is a unital c _hom。m。rphism from i int。aunital C algebra for 1，2，then there is a unital C 一hom。m。rphisms from t0 satisfying咖= oar for ：1，2 supp。 。 ： (C)Den。te by = ：I= fhe full fre product of with We want to caJculate the spectrum o-(AB)of A and We also need the following resuIt O11 22 Suppose is a unital C 一algebraand戈 Then 盯( )=u (7r( )：仃： 一 刀( unitalhom。m。I1)hism Pr。of GiVen any _horn。morphism ： 一 (。 If 隹 ( )，then has an inverse ins and therefl01e 仃( )has an inVerse (s)in ( Henee隹 (仃( )Thus (霄( )c ( )for all爿ch。 momorphism仃： 一 ( Therefore u (仃( )： ： ( unital肆：-homomorphismc叮(x) By出e Gelfand-Neumark The。rerrl ，there is a faithful representation p of on a HilbeIt space en ： 叶p( )is a isomorphism，hence we have ( )： (p( )This shows tha ( )c u (7r( )：7r： 一 ( unita1：l= homomorphism and completes the proof Le岫a 23 Let e a unital C*-algebra and A ，A e JSupp。se ( n。rm)an (A )If z -t，then to-(A) N。te n，_A A in n。imThus if隹 (A)，then tlA is invertib】e。 S。wben is big e n。 gh，t A would be invertibleThis means that t 甓 (A )which is a c。ntradiction Theorem 2 ) and Then ( )=l Ou：eC ，JI e Il=1 Proof supp0se 7r： _+ is a unita1 一isom。rphismLet仃l：7rl露and 7r2：7l驴Then since repre一 。ntati。“of a c 一algebra is the direct sum of irreducible representati。ns and ineducible representati。ns of M(C 1 are unnary equ Valen 。the identity representati。n，we may assume that there are unitaries Ul and such that 7rl(A)U = 7r ( ) = Thus for A=( we may assume (A B 删= ： ， where is a closed linear subspace。f H and is the ohogonal c伽p1e珊em jn4wth and are isomorphicThen 、IIl， 第3期 王利广：自由积中算子乘积的谱 7 仃( B) 7r(A)7rz(B) ( )垒 Let =f。 ? T-hen 5，。=( )and s一 抖 sin w 叫 U- Lemma 25 or(B1)Cw(B1)c w(B) (BI)=：e lI e II=1 c：e II e lI=1 c W(r2( )c ( ) Since ( )is closed，we haVe W(B1)C (曰)By spectral theory we know that or(B1)=：rap(B1)u orap( ) GiVen Aor印( 1)，there existe in with ll e Il=1 such that l1日le 一A en Il-4)Thus II-0S。I一AI and-+AThis implies that A and 。 (B1) cW(B1)。Therefore we have or(BI)CW(B1)This comp