MATHEMATICS: CONCEPTS, AND FOUNDATIONS Vol. II - Operator Theory and Operator Algebra - H. Kosaki OPERATOR THEORY AND OPERATOR ALGEBRA

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1 OPERATOR THEORY AND OPERATOR ALGEBRA H. Kosai Graduate School of Mathematics, Kyushu Universy, Japan Keywords: Hilbert space, linear operator, compact operator, polar decomposion, spectral decomposion, spectrum, dilation, invariant subspace, unilateral shift, Toeplz operator, operator monotone function, C -algebra, type I C -algebra, nuclear C -algebra, purely infine C -algebra, K-theory, von Neumann algebra, modular theory, AFD factors, Jones index, free probabily Contents 1. Hilbert space. Bounded linear operator.1. Compact Operator.. Miscellaneous Operators.3. Polar Decomposion and Spectral Decomposion.4. Spectrum 3. Operator theory 3.1. Dilation Theory 3. Generalization of Normaly 3.3 Toeplz Operator 3.4 Operator Inequalies 4. Operator algebra 4.1. C -algebra Type I C -algebra Nuclear C -algebra Operator Algebra K-theory Purely Infine C -algebra 4.. von Neumann Algebra Basic Theory 4... Modular Theory and Structure of Type III Factors Classification of AFD Factors Index Theory 4..5 Free Probabily Theory Glossary Bibliography Biographical Setch Summary A Hilbert space is a Banach space whose norm comes from an inner product, and is the most natural infine-dimensional generalization of the Euclidean space. Operators on a Hilbert space appear in many places, and may be viewed as matrices of infine size. One can add and multiply them, and furthermore the -operation (extending the notion of adjoint matrix) can be introduced due to the presence of an inner product. Operator

2 theory studies individual operators while the theory of operator algebras deals wh - algebras of operators. C -algebras and von Neumann algebras are particularly important classes of such -algebras. In this chapter, some selected topics on operator theory, C - algebras, and von Neumann algebras are explained. 1. Hilbert Space A pre-hilbert space means a linear space H (usually over C) equipped wh an inner product, satisfying (i) (, ξη) H H ξη, C is linear in the first variable ξ and conjugate linear in the second variableη, (ii) ξξ, 0 and ξξ, = 0 iff ξ = 0. By virtue of (ii) we can set ξ = ξ, ξ ( 0 ), the norm of ξ. The Cauchy-Schwarz inequaly ξη, ξ η guarantees that is indeed a norm on H : (i) λξ = λ ξ forλ C, (ii) ξ+ η ξ + η (Triangle inequaly), (iii) ξ = 0 iff ξ = 0. Wh this norm a pre-hilbert space H becomes a normed linear space so that d (, ξη) = ξ η defines a metric: a sequence {} ξ n= 1, converges to ξ when lim n d( ξ n, ξ ) = 0. When this metric is complete (every Cauchy sequence is convergent), H is called a Hilbert space. In other words, H is a Hilbert space when H equipped wh the associated norm is a Banach space. The norm (from, as above) satisfies ( ) ξ+ η + ξ η = ξ + η (Parallelogram law) 3 1 ξη, = i ξ+ i η (Polarization identy) 4 = 0 wh i = 1. The parallelogram law is of fundamental importance in handling Hilbert spaces, and actually characterizes Hilbert spaces. In fact, when the norm of a Banach space H satisfies the parallelogram law, the right side of the polarization identy gives rise to an inner product whose associated norm is. The n -dimensional vectors C n (wh the ordinary vector operations) form a Hilbert space wh the inner product n ξη, = ξη = 1 for = ( 1,,..., n ), = ( 1,,..., n) C. ξ ξ ξ ξ η η η η n n The associated norm is ξ = ξ = 1, the Euclidean distance. More interesting (infine-dimensional) examples are as follows:

3 (i) H = L ( R; dx), which is the space of measurable functions f ( x) on R satisfying the square integrabily condion f( x) dx <, where dx is the Lebesgue measure on the real line R and functions are being identified if they differ only on a null set. The linear structure here is given by the point-wise sum of functions, and f, g = f( x ) g ( x ) d x gives rise to an inner product. (ii) Any measure space gives rise to a Hilbert space as in (i). One important special case is N equipped wh the counting measure: = { the space of sequences ξ = { ξ } = 1,... satisfying =1 ξ < } wh the inner product ξη, ξη. = = 1 (iii) Let H be the set of analytic functions f on the open dis D= { z C; z < 1} satisfying 1 1 f lim π iθ = f( e ) dθ < 1 π r 0. r This space can be naturally identified as a closed subspace of the L -space L ( T; dθ π ) over the torus T= D(as will be explained shortly). Hence, H self is a Hilbert space, nown as the Hardy space. A family { e α } α Λ in H is called an orthogonal system when e α = 1 and e, e = 0 forα β. Furthermore, when linear combinations of e α s form a dense α β subspace, is called an orthogonal basis. In many practical suations Hilbert spaces are separable (i.e., possessing a dense sequence) so that separabily will be assumed in the rest of our discussion. Then, the above index set Λ is (at most) countable, and hence an e = orthogonal basis is actually a sequence { n} n 1,,... (or a fine sequence { } when dim H < ). Wh such a basis an arbrary element ξ H can be e n n= 1,,..., m expressed as ξ e e ξ e e = 1 = n n = 1, ( lim, ) wh ξ e = 1,,.... {, } Consequently, infine-dimensional (separable) Hilbert spaces are all isomorphic to, and in particular the above L ( R; dx) is also isomorphic to as a Hilbert space. Let { } e ( L ( T; dθ / π )) n n Z int be the (exponential) orthonormal basis defined by en ( e ) = e ( e T). The expression f = Z ae n n nfor f L ( T; dθ / π ) wh a = f, e is nothing but the Fourier series expansion. Then the above Hardy space n n

4 H can be identified wh the closed subspace generated by { e n } n = 0,1,,.... More precisely, for f H radial lims Fe ( ) = lim f( re ) exist almost everywhere (a.e.) on the r 1 torus T. Then the radial lim function F( e ) ss in L ( T; dθ / π ) and indeed falls into the above mentioned closed subspace, i.e., = 0 Fe ( ) = ae n n wh f = ( a ) n. The analytic function f H is = 0 recovered from F by maing use of the Poisson integral, and also the power series expansion 0 f() z = a = n z is valid on D. n 1 A linear functional ϕ : H C is called bounded if ϕ = sup{ ϕ( ξ) ; ξ H 1 } <, where H1 = { ξ H; ξ 1} is the un ball. Let H (the dual space of H ) be the set of all bounded linear functionals on H. Wh the obvious linear structure and the above norm, the dual space H is a Banach space. For a general Banach space the fact that the dual H contains an abundance of elements is not so obvious and in fact follows form the Hahn-Banach theorem. However, for the Hilbert space case is easy. Indeed any η H induces the bounded linear functional ϕ : ξ ξη, C wh ϕ η = η. The Riesz theorem asserts that every element in the dual this way, which plays an important role in many places.. Bounded Linear Operator n H arises in A linear operator on a Hilbert space H means a linear map T : H H. A linear operator T is continuous iff is bounded, i.e. T = sup Tξ <. Many natural linear operators (such as differential operators on function spaces) are unbounded, and very beautiful and useful theories on unbounded operators are nown. However, we will mainly deal wh bounded linear operators, and they will be simply called operators in what follows. The quanty T is referred to as the operator norm of T. It is indeed a norm on the set B( H) of all (bounded linear) operators wh the obvious linear structure, and B( H) is a Banach space. For operators T, S B( H) we have TS T S so that B( H) is actually a Banach algebra wh the un 1, the identy operator. Linear operators on a Banach space can be also considered, but the theory of operators on a Hilbert space is much richer than the theory of those on a Banach space. What maes the former so is the adjoint operation. Namely, for T B( H) there is a unique operator S (denoted byt, the adjoint of T) satisfying ξ H 1 Tξ, η = ξ, Sη for each ξ, η H (by virtue of the Riesz theorem), which maes B( H) a -Banach algebra (i.e., a Banach algebra wh an involution satisfying T = T ).

5 When dim H= n <, B( H) is the space of n n-matrices. Hence an operator can be regarded as a matrix of infine size, and one naturally tries to generalize what is nown for matrices to the operator setting (although such generalizations are impossible in many cases at least in a straight-forward fashion). Besides the norm topology induced by, the strong operator topology and wea operator topology are often considered and useful. They are the linear topologies determined by the following families of seminorms: { p ()} wh p ( T ) = Tξ (strong operator topology), ξ ξ H ξ { p ()} wh p, ( T) = Tξ, η (wea operator topology). ξη, ξη, H ξη.1. Compact Operator An operator T B( H) is called a compact operator if TH 1 is relatively compact in the norm topology ( TH 1 is automatically closed when H is a Hilbert space). The set K=K ( ( H)) of all compact operators is a norm closed two-sided ideal (actually the only non-trivial one) in B( H ). For compact operators some results for matrices can be generalized in the most straight-forward fashion. For example, as a generalization of diagonalization by a unary matrix, one can show the following: if a compact operator T is normal (i.e.,tt = T T ), then one can find an orthogonal basis { e } = 1,,... such that Te = λ e for some λ C, that is, T is unarily equivalent to an infine diagonal matrix. Furthermore, when λ s are arranged in such a way that λ s are in decreasing order, then we must have lim λ 0... Miscellaneous Operators = (i) An operator T is called self-adjoint ift = T, that is, Tξ, ξ R forξ H. A selfadjoint operator S is called posive ( S 0) if Sξ, ξ 0 for ξ H. For selfadjoint operators S 1, S we consider the order S 1 S defined by S 1 S 0. For an operator S the following condions are equivalent: (a) S is posive, (b) S = A Awh some A B( H), (c) S = B wh a posive operator B. In the last condion a posive operator B is uniquely determined, and denoted by S (the square root of S ). (ii) For a closed subspace K in H, K = { η H; η, ξ = 0for each ξ K } is called the orthogonal complement of K. Any element ξ H can be wrten uniquely as ξ = ξ1+ ξ wh ξ1 K and ξ K. The operator P ( = PK) : ξ H ξ1 satisfies P = P= P. Such an operator is called a projection. Every projection P arises in this way wh K= PH so that the projections and the closed subspaces are in bijective correspondence.

6 (iii) An operator U is called a partial isometry if U U is a projection (or equivalently UU is a projection). The projections UU, UU are called the inial and final projections of U. The name partial isometry is justified by the following properties: (a) U sends UUH onto isometrically, (b) U acts as the zero operator in the orthogonal complement ( UU H). (iv) When U U SS = UU = 1, U is called a unary operator. On the other hand, when = 1, is called an isometry. Note that SS = 1does not imply SS = 1(unless dim H <). A typical example is the unilateral shift operator S 0 : S 0 e = e + 1 wh an orthonormal basis e } = 0,1,. In fact, SSis 0 0 the projection onto the closed subspace (of co-dimension one) generated by e } = 1,,.3. Polar Decomposion and Spectral Decomposion For an operator T we can find a partial isometry U and a posive operator S satisfying T = US in such a way that the support projection of S (i.e, the smallest projection P satisfying PS = S) coincides wh the inial projection U U. The above partial isometry U and posive operator T are unique subject to the condions mentioned so far. This decomposion is called the polar decomposion of T. The posive operator S (usually denoted by T ) is actually the posive square root For a self-adjoint operator T we can find a family { E } (i) λ R E is increasing, E Ε if λ μ λ (ii) Eλ = 0 ( λ < T ) and E = 1( T ) (iii) λ R E λ (iv)t = λdε λ. λ μ λ λ R. TT of projections satisfying λ λ, is continuous from the right in the strong operator topology, From (i) and (iii) the function λ Eλξ, ξ = Eλξ is increasing and continuous from the right for eachξ H, and (iv) (spectral decomposion of T) means Tξ, ξ = λ λ d E ξ (for eachξ H ), where the right side is understood in the Stieltjes sense. The family { Eλ} λ R is called a spectral family. This technique enables us to perform functional calculus f (T) of T for a bounded Borel function f on R via = λ f( T) ξ, ξ f( λ) d E ξ. For unbounded selfadjoint operators spectral families are also available, however the condion (ii) is replaced by the more general condion: limλ Eλ = 0and limλ Eλ = 1 in the strong operator topology.

7 .4. Spectrum A complex number λ is called a resolvent of T B( H) when the operator λ1 T is invertible in B( H ). The complement of the set of all resolvents is denoted by σ ( T ), the spectrum of T. Note that the H is fine-dimensional σ ( T ) is simply the set of all eigenvalues of a matrix T. However, the behavior of σ ( T ) ( C) for an infine dimensional operator is much more subtle. Nevertheless, the spectrum serves as a fundamental tool for handling operators. The spectrum σ ( T ) is nown to be a non-empty closed set sting in the closed dis of radius T. On the other hand, the spectrum σ e( T ) defined similarly at the level of the quotient C -algebra B( H)/ K( H)(see Section 4.1) is referred to as the essential spectrum of T. The classical Weyl-von Neumann theorem states that self-adjoint operators T 1, T are unarily equivalent modulo the compact operators K (.., ie T1 UTU K for some unary operator U) iffσ ( T1) = σ ( T). e e The same characterization remains valid for normal operators (Berg and Sionia independently, 1971). UU H 3. Operator Theory Operator theory studies individual operators, and is very diverse. Some selected topics are briefly outlined in the following Dilation Theory Let T B( H) be a contraction (i.e., T 1). Sz. Nagy (1960) showed that one can find a Hilbert space L containing H and a unary operator U on L such that T n n = P H U H wh the projection P B( L) (for each n = 0,1,,.) H onto H. Furthermore, the above unary U (called a unary dilation of T) is unique under a certain minimaly condion. Various results on dilations are nown, and such a dilation technique is useful in operator theory. For example, the following inequaly (originally due to von Neumann, 1951) is an immediate consequence of the preceding result: For a contraction T and a polynomial px ( ) we have p( T ) max{ p( λ) ; λ 1}.

8 - - - TO ACCESS ALL THE 5 PAGES OF THIS CHAPTER, Vis: Bibliography Blacadar B. (1998). K- theory for Operator Algebras (second edion), M.S.R.I Publications Vol. 5, Cambridge Univ. Press. [This presents a detailed description on K- theory as well as related remarable recent achievement in C -algebra theory.] Connes A. (1994). Non-commutative Geometry, Academic Press. [This boo demonstrates richness of non-commutative mathematics, and seems to indicate future direction of research on operator algebras.] Conway J.B. (1991). TheTheory of Subnormal Operators, Math. Surveys and Monographs Vol 36, Amer. Math. Soc. [This presents fruful interplay between operator theory and function theory, and the state of the art of subnormal and related operators can be found.] Evans D.E. and Kawahigashi Y. (1998). Quantum Symmetries on Operator Algebras, Oxford Math. Monographs, Clarendon Press. [This presents a detailed account on Jones index theory, and the state of the art of subfactor analysis can be found.] Gohberg I.C. and Krein M.G. (1969). Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Math. Monographs Vol. 18, Amer. Math. Soc. [Basic facts on compact operators can be found, and hard analysis on compact non-selfadjoint operators is presented.] Halmos P.R. (1974). A Hilbert Space Problem Boo, Springer-Verlag. [This boo is not intended to be an ordinary textboo, but is an excellent source for basic nowledge on Hilbert spaces and their linear operators.] Hardy G., Ltlewood J.E., and Pόlya G. (195). Inequalies (second edion), Cambridge Univ. Press. [This boo is a classic on inequalies and still very valuable source for nowledge.] Kadison R.V. and Ringrose J.R. (1983 and 1986). Fundamentals of the Theory of Operator Algebras I, II, Academic Press. [This is a self-contained style textboo on operator algebras.] Kato T. (1984). Perturbation Theory for Linear Operators (second edion), springer-verlag. [Here is a standard place to find various nowledge on unbounded operators, and is an encyclopedia for many worers.] Nagy B. Sz. and Foiaş C. (1970). Harmonic Analysis of Operators on Hilbert Spaces, North-Holland. [This is one of the standard textboos on operator theory and includes a very readable account on dilation technique.] Pedersen G.P. (1979). C -Algebras and Their Automorphism Groups, Academic Press. [This is one of the standard textboos on operator algebras wh main emphasis on C -algebras.] Saai s. (1971). C -Algebras and W -Algebras, Springer-Verlag. [This is one of the classics in the theory of operator algebras, and functional analysis viewpoint is emphasized.] Taesai M. (1979). Theory of Operator Algebras I, Springer-Verlag. [This presents an authorative account on basic theory of operator algebras.]

9 Voiculesce D.V., Dyema K.J., and Nica A. (1991). Free Random Variables, CRM Monograph Series Vol. 1, Amer. Math. Soc. [This boo is based on lecture notes by the founder of free probabily theory, and is a good starting point for the subject.] Biographical Setch Hidei Kosai was born in Received the degree of Ph D in June, 1980 from Universy of California, Los Angeles. Dr. Kosai specializes in functional analysis, especially the theory of operator algebras. Dr. Kosai has been woring on non-commutative integration theory and index theory for von Neumann algebras and is also interested in operator theory, and in fact has investigated various operator inequalies. Dr. Kosai held various posions: Centre de Physique Théorique, CNRS, Marseille; Universy of Kansas; Purdue Universy; Universy of Pennsylvania; Tulane Universy; MSRI, Bereley. Currently Dr. Kosai is since1994 Professor, Graduate School of Mathematics, Kyushu Universy