DOCSLIB.ORG

Solutions to the Exercises of Lecture 13

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Solutions to the Exercises of Lecture 13 Solutions to the Exercises of Lecture 13 Team Dresden d d Exercise 13.1. Let (Vn)n in L1(R ) and V 2 L1(R ). Characterise convergence of d Vn(m) ! V (m) in the strong operator topology of L(L2(R )) in terms of convergence of (Vn)n similar to as was done in Proposition 13.2.1. d Solution 13.1. Vn(m) ! V (m) in the strong operator topology of L(L2(R )) if and only d d if (Vn) is bounded in L1(R ) and Vn ! V in L1;loc(R ). d Proof. Let Vn(m) ! V (m) in the strong operator topology of L(L2(R )). Then (Vn(m))n d is bounded in L(L2;loc(R )) by the uniform boundedness principle and as in the proof of d d 1 d Proposition 13.2.1 (Vn) is bounded in L1(R ). So let K ⊂ R ) compact, then K 2 L2(R ) 1 1 d and Vn(m) K ! V (m) K in L2(K) and in L1(K). Hence Vn ! V in L1;loc(R ), since K is arbitrary. d d Assume Vn ! V in L1;loc(R ) and Vn;V bounded. Let f 2 L2(R ) with compact support. Then by the compactness of supp f we can interpret Vn;V as operators on some finite mea- d sure space; so we can apply Proposition 13.2.1 and derive kVn(m)f − V (m)fkL2(R ) ! 0. d By density we obtain Vn(m) ! V (m) in the strong operator topology of L(L2(R )). Exercise 13.2. Show that there exists an unbounded sequence (Vn)n in L1((0; 1)) and V 2 L1((0; 1)) with Vn ! V in L1((0; 1)). Solution 13.2. Let Vn = n · 1[1;n−2] and V = 0. n 1 1 n2 Then kVnkL1((0;1)) = n; so the sequence is unbounded in L1((0; 1)) and 1 kV − V k = ! 0; n L1((0;1)) n so Vn ! V in L1((0; 1)). 1 Exercise 13.3. Let (Ω; Σ; µ) be a finite measure space, (Vn)n a bounded sequence in L1(µ) and assume that Vn ! V in L1(µ) for some V 2 L1(µ). Show that there exists ν > 0 such that −1 −1 (@t,ν + Vn(m)) ! (@t,ν + V (m)) 1 in the strong operator topology of L L2,ν(R; L2(µ));Hν (R; L2(µ)) . Solution 13.3. First, we rewrite −1 −1 −1 (@t,ν + Vn(m)) = @t,ν @t,ν(@t,ν + Vn(m)) −1 = @t,ν Mn(@t,ν) −1 −1 where the functions Mn : dom(Mn) ! L(L2(µ)) are defined by Mn(z) = (1+z Vn(m)) for n 2 N. Now, if ν > supn2N jjVnjjL1(µ) we see that (Mn)n is a bounded sequence of material laws. By proposition 13.2.1(b) we obtain Vn(m) ! V (m) in the strong operator −1 topology of L(L2(µ)), which implies that (Mn(z))n converges to M(z) := (1 + z V (m)) in the strong operator topology of L(L2(µ)) for all z 2 CRe>ν. Then one has Mn(@t,ν) ! M(@t,ν) strongly by Thereom 13.1.1. and the claim follows. Exercise 13.4. Let D := S [n + 1=2; n + 1] and V := 1 (n·). For suitable ν > 0 n2Z n D compute the limit of −1 (@t,ν + Vn(m)) n in the weak operator topolgy of L2,ν R; L2((0; 1)) . Solution 13.4. Here is the graph of Vn. 1 n 1 n k By Theorem 13.2.3, Vn ! 2 in the weak* topology of L1(R ). Note that Vn = Vn for k 1 n every k 2 N. Hence, Vn ! 2 in the weak* topology of L1(R ) for every k 2 N. k 1 By Proposition 13.2.1, this implies that, Vn (m) ! 2 Id for every k 2 N in the weak operator topology of L2((0; 1)). Hence for ν > 1, Theorem 13.3.1 yields 1 X 1 (@ + V (m))−1 ! (−@−1)k @−1 t,ν n t,ν 2 t,ν k=0 1 1 X = @−1 (−@−1)k 2 t,ν t,ν k=0 1 = (@ + 1)−1: 2 t,ν 2 ∗ Exercise 13.5. Let H be a Hilbert space, c > 0 and c 6 Bn = Bn 2 L(H) for all n 2 N. Characterise, in terms of convergence of (Bn)n in a suitable sense, that −1 (@t,νBn) n converges in the weak operator topology. In the case of convergence, find its limit and a sufficient condition for which there exists a B 2 L(H) such that −1 −1 (@t,νBn) ! (@t,νB) in the weak operator topology. Solution 13.5. We first proof the following lemma. Lemma. Let H be a Hilbert space, (Bn)n be a sequence in L(H) such that Re Bn > c −1 (n 2 N); supn kBnk 6 d for some c; d > 0 and Bn ! C in L(H) in the weak operator topology. Then C is invertible in L(H): −1 −1 Proof. We know that Bn is invertible in L(H) and kBn k 6 1=c: Let y 2 H; x := Bn y (then kyk = kBnxk 6 dkxk): We have c RehB−1y; yi = Rehx; B xi ckxk2 kyk2 n n > > d2 c 2 for all n 2 N: This implies that RehCy; yi > d2 kyk (for all y 2 H), and therefore C is −1 2 invertible (with kC k 6 d =c). Let H be a Hilbert space, ν > 0 and (Bn)n be a sequence in L(H) such that Re Bn > c −1 for some c > 0. Then ((@t,νBn) )n converges in the weak operator topology if and only −1 if (Bn )n converges in the weak operator topology. Moreover, if sup kBnk 6 d for some −1 d > 0 and Bn ! C 2 L(H) in the weak operator topology, then C is invertible and −1 −1 −1 (@t,νBn) ! (@t,νB) (where B := C ). −1 −1 Proof. We know that Bn is invertible in L(H) and kBn k 6 1=c: Let Mn(z) := (zBn) for z 2 CRe>ν: Then (Mn)n is a bounded sequence in M(H; ν): The equivalence of con- −1 −1 vergence of ((@t,νBn) )n and (Bn )n in the weak operator topology now follows from Theorem 13.1.1. −1 Now assume sup kBnk 6 d for some d > 0 and Bn ! C 2 L(H) in the weak oper- ator topology. Then by the above Lemma we obtain that C is invertible. Let B = C−1: −1 −1 Applying Theorem 13.1.1 gives (@t,νBn) ! (@t,νB) in the weak operator topology. Exercise 13.6. Let H be a Hilbert space. Show that BL(H) := fT 2 L(H); kT k 6 1g is a compact subset under the weak operator topology. If, in addition, H is separable, show that BL(H) is also metrisable under the weak operator topology. Solution 13.6. The proof will be given in two parts: (a) for the compactness, (b) for the metrisability. The proof of (a) is remindful of the proof of the Banach–Alaoglu theorem. (a) The weak operator topology τw is defined as the initial topology with respect to the family of mappings (L(H) 3 A 7! hx; Ayi 2 K)x;y2H , i.e., τw is the coarsest topology on L(H) making all the mappings A 7! hx; Ayi 2 K continuous. Equivalently, τw is defined by the set fpx;y; x; y 2 Hg of seminorms, where px;y(A) := jhx; Ayij (A 2 L(H); x; y 2 H): 3 Let S denote the set of sesquilinear mappings b: H × H ! K. Then S is a closed subset of KH×H (with the product topology). Indeed, the mappings H×H ∗ 'λ,x,y;z : K ! K; b 7! b(λx + y; z) − λ b(x; z) − b(y; z); H×H λ,x,y;z : K ! K; b 7! b(x; λy + z) − λb(x; y) − b(x; z) are continuous. (The reason is, that the canonical projections KH×H 3 b 7! b(x; y) 2 K are continuous, for all (x; y) 2 H × H.) Therefore \ −1 \ −1 S = 'λ,x,y;z(0) \ λ,x,y;z(0) λ2K;x;y;z2H λ2K;x;y;z2H is a closed subset of KH×H . H×H We define the mapping η : L(H) ! K by η(A) := (hx; Ayi)(x;y)2H×H ; then obvi- ously η(L(H)) ⊆ S. (Note that τw is the initial topology with respect to the mapping η.) Q H×H The set C := (x;y)2H×H BK[0; kxkkyk] is a compact subset of K , by Tikhonov’s theorem; hence the closed subset C \ S of C is compact as well. We show that η(BL(H)) = C \ S.‘⊆’ follows from jhx; Ayij 6 kAkkxkkyk 6 kxkkyk (A 2 BL(H); x; y 2 H). ‘⊇’: b 2 C \ S means that b is sesquilinear and jb(x; y)j 6 kxkkyk (x; y 2 H). This implies that there exists A 2 L(H) with kAk 6 1 such that b(x; y) = hx; Ayi (x; y 2 H), i.e., b = η(A). As both topologies, τw on BL(H) and the topology of C \S, are induced by the product H×H topology on K , it follows that η :(BL(H); τw) ! C \ S is a homeomorphism; hence (BL(H); τw) is compact.
Load more

Recommended publications
  • A Topology for Operator Modules Over W*-Algebras Bojan Magajna
    Journal of Functional AnalysisFU3203 journal of functional analysis 154, 1741 (1998) article no. FU973203 A Topology for Operator Modules over W*-Algebras Bojan Magajna Department of Mathematics, University of Ljubljana, Jadranska 19, Ljubljana 1000, Slovenia E-mail: Bojan.MagajnaÄuni-lj.si Received July 23, 1996; revised February 11, 1997; accepted August 18, 1997 dedicated to professor ivan vidav in honor of his eightieth birthday Given a von Neumann algebra R on a Hilbert space H, the so-called R-topology is introduced into B(H), which is weaker than the norm and stronger than the COREultrastrong operator topology. A right R-submodule X of B(H) is closed in the Metadata, citation and similar papers at core.ac.uk Provided byR Elsevier-topology - Publisher if and Connector only if for each b #B(H) the right ideal, consisting of all a # R such that ba # X, is weak* closed in R. Equivalently, X is closed in the R-topology if and only if for each b #B(H) and each orthogonal family of projections ei in R with the sum 1 the condition bei # X for all i implies that b # X. 1998 Academic Press 1. INTRODUCTION Given a C*-algebra R on a Hilbert space H, a concrete operator right R-module is a subspace X of B(H) (the algebra of all bounded linear operators on H) such that XRX. Such modules can be characterized abstractly as L -matricially normed spaces in the sense of Ruan [21], [11] which are equipped with a completely contractive R-module multi- plication (see [6] and [9]).
    [Show full text]
  • 7.3 Topological Vector Spaces, the Weak and Weak⇤ Topology on Banach Spaces
    138 CHAPTER 7. ELEMENTS OF FUNCTIONAL ANALYSIS 7.3 Topological Vector spaces, the weak and weak⇤ topology on Banach spaces The following generalizes normed Vector space. Definition 7.3.1. Let X be a vector space over K, K = C, or K = R, and assume that is a topology on X. we say that X is a topological vector T space (with respect to ), if (X X and K X are endowed with the T ⇥ ⇥ respective product topology) +:X X X, (x, y) x + y is continuous ⇥ ! 7! : K X (λ, x) λ x is continuous. · ⇥ 7! · A topological vector space X is called locally convex,ifeveryx X has a 2 neighborhood basis consisting of convex sets, where a set A X is called ⇢ convex if for all x, y A, and 0 <t<1, it follows that tx +1 t)y A. 2 − 2 In order to define a topology on a vector space E which turns E into a topological vector space we (only) need to define an appropriate neighbor- hood basis of 0. Proposition 7.3.2. Assume that (E, ) is a topological vector space. And T let = U , 0 U . U0 { 2T 2 } Then a) For all x E, x + = x + U : U is a neighborhood basis of x, 2 U0 { 2U0} b) for all U there is a V so that V + V U, 2U0 2U0 ⇢ c) for all U and all R>0 there is a V ,sothat 2U0 2U0 λ K : λ <R V U, { 2 | | }· ⇢ d) for all U and x E there is an ">0,sothatλx U,forall 2U0 2 2 λ K with λ <", 2 | | e) if (E, ) is Hausdor↵, then for every x E, x =0, there is a U T 2 6 2U0 with x U, 62 f) if E is locally convex, then for all U there is a convex V , 2U0 2T with V U.
    [Show full text]
  • AND the DOUBLE COMMUTANT THEOREM Recall
    Egbert Rijke Utrecht University [email protected] THE STRONG OPERATOR TOPOLOGY ON B(H) AND THE DOUBLE COMMUTANT THEOREM ABSTRACT. These are the notes for a presentation on the strong and weak operator topolo- gies on B(H) and on commutants of unital self-adjoint subalgebras of B(H) in the seminar on von Neumann algebras in Utrecht. The main goal for this talk was to prove the double commutant theorem of von Neumann. We will also give a proof of Vigiers theorem and we will work out several useful properties of the commutant. Recall that a seminorm on a vector space V is a map p : V ! [0;¥) with the properties that (i) p(lx) = jljp(x) for every vector x 2 V and every scalar l and (ii) p(x + y) ≤ p(x) + p(y) for every pair of vectors x;y 2 V. If P is a family of seminorms on V there is a topology generated by P of which the subbasis is defined by the sets fv 2 V : p(v − x) < eg; where e > 0, p 2 P and x 2 V. Hence a subset U of V is open if and only if for every x 2 U there exist p1;:::; pn 2 P, and e > 0 with the property that n \ fv 2 V : pi(v − x) < eg ⊂ U: i=1 A family P of seminorms on V is called separating if, for every non-zero vector x, there exists a seminorm p in P such that p(x) 6= 0.
    [Show full text]
  • Weak Operator Topology, Operator Ranges and Operator Equations Via Kolmogorov Widths
    Weak operator topology, operator ranges and operator equations via Kolmogorov widths M. I. Ostrovskii and V. S. Shulman Abstract. Let K be an absolutely convex infinite-dimensional compact in a Banach space X . The set of all bounded linear operators T on X satisfying TK ⊃ K is denoted by G(K). Our starting point is the study of the closure WG(K) of G(K) in the weak operator topology. We prove that WG(K) contains the algebra of all operators leaving lin(K) invariant. More precise results are obtained in terms of the Kolmogorov n-widths of the compact K. The obtained results are used in the study of operator ranges and operator equations. Mathematics Subject Classification (2000). Primary 47A05; Secondary 41A46, 47A30, 47A62. Keywords. Banach space, bounded linear operator, Hilbert space, Kolmogorov width, operator equation, operator range, strong operator topology, weak op- erator topology. 1. Introduction Let K be a subset in a Banach space X . We say (with some abuse of the language) that an operator D 2 L(X ) covers K, if DK ⊃ K. The set of all operators covering K will be denoted by G(K). It is a semigroup with a unit since the identity operator is in G(K). It is easy to check that if K is compact then G(K) is closed in the norm topology and, moreover, sequentially closed in the weak operator topology (WOT). It is somewhat surprising that for each absolutely convex infinite- dimensional compact K the WOT-closure of G(K) is much larger than G(K) itself, and in many cases it coincides with the algebra L(X ) of all operators on X .
    [Show full text]
  • Spaces of Compact Operators N
    Math. Ann. 208, 267--278 (1974) Q by Springer-Verlag 1974 Spaces of Compact Operators N. J. Kalton 1. Introduction In this paper we study the structure of the Banach space K(E, F) of all compact linear operators between two Banach spaces E and F. We study three distinct problems: weak compactness in K(E, F), subspaces isomorphic to l~ and complementation of K(E, F) in L(E, F), the space of bounded linear operators. In § 2 we derive a simple characterization of the weakly compact subsets of K(E, F) using a criterion of Grothendieck. This enables us to study reflexivity and weak sequential convergence. In § 3 a rather dif- ferent problem is investigated from the same angle. Recent results of Tong [20] indicate that we should consider when K(E, F) may have a subspace isomorphic to l~. Although L(E, F) often has this property (e.g. take E = F =/2) it turns out that K(E, F) can only contain a copy of l~o if it inherits one from either E* or F. In § 4 these results are applied to improve the results obtained by Tong and also to approach the problem investigated by Tong and Wilken [21] of whether K(E, F) can be non-trivially complemented in L(E,F) (see also Thorp [19] and Arterburn and Whitley [2]). It should be pointed out that the general trend of this paper is to indicate that K(E, F) accurately reflects the structure of E and F, in the sense that it has few properties which are not directly inherited from E and F.
    [Show full text]
  • Topological Results About the Set of Generalized Rhaly Matrices 1
    Gen. Math. Notes, Vol. 17, No. 1, July 2013, pp. 1-7 ISSN 2219-7184; Copyright c ICSRS Publication, 2013 www.i-csrs.org Available free online at http://www.geman.in Topological Results About the Set of Generalized Rhaly Matrices Nuh Durna1 and Mustafa Yildirim2 1;2 Cumhuriyet University, Faculty of Sciences Department of Mathematics, 58140 Sivas, Turkey 1 E-mail: [email protected] 2 E-mail: [email protected]; [email protected] (Received: 1-2-12 / Accepted: 15-5-13) Abstract In this paper, we will investigate some topological properties of Rb set of all bounded generalized Rhally matrices given by b sequences in H2. We will show 2 that Rb is closed subspace of B (H ) and we will define an operator from T to Rb and show that this operator is injective and continuous in strong-operator topology. Keywords: Generalized Rhaly operators, weak-operator topology, strong- operator topology. 1 Introduction A Hilbert space has two useful topologies (weak and strong); the space of operators on a Hilbert space has several. The metric topology induced by the norm is one of them; to distinguish it from the others, it is usually called the norm topology or the uniform topology. The next two are natural outgrowths for operators of the strong and weak topologies for vectors. A subbase for the strong operator topology is the collection of all sets of the form fA : k(A − A0) fk < g ; correspondingly a base is the collection of all sets of the form fA : k(A − A0) fik < , i = 1; 2; : : : ; kg 2 Nuh Durna et al.
    [Show full text]
  • An Introduction to Some Aspects of Functional Analysis, 7: Convergence of Operators
    An introduction to some aspects of functional analysis, 7: Convergence of operators Stephen Semmes Rice University Abstract Here we look at strong and weak operator topologies on spaces of bounded linear mappings, and convergence of sequences of operators with respect to these topologies in particular. Contents I The strong operator topology 2 1 Seminorms 2 2 Bounded linear mappings 4 3 The strong operator topology 5 4 Shift operators 6 5 Multiplication operators 7 6 Dense sets 9 7 Shift operators, 2 10 8 Other operators 11 9 Unitary operators 12 10 Measure-preserving transformations 13 II The weak operator topology 15 11 Definitions 15 1 12 Multiplication operators, 2 16 13 Dual linear mappings 17 14 Shift operators, 3 19 15 Uniform boundedness 21 16 Continuous linear functionals 22 17 Bilinear functionals 23 18 Compactness 24 19 Other operators, 2 26 20 Composition operators 27 21 Continuity properties 30 References 32 Part I The strong operator topology 1 Seminorms Let V be a vector space over the real or complex numbers. A nonnegative real-valued function N(v) on V is said to be a seminorm on V if (1.1) N(tv) = |t| N(v) for every v ∈ V and t ∈ R or C, as appropriate, and (1.2) N(v + w) ≤ N(v) + N(w) for every v, w ∈ V . Here |t| denotes the absolute value of a real number t, or the modulus of a complex number t. If N(v) > 0 when v 6= 0, then N(v) is a norm on V , and (1.3) d(v, w) = N(v − w) defines a metric on V .
    [Show full text]
  • Note on Operator Algebras
    Note on Operator Algebras Takahiro Sagawa Department of Physics, The University of Tokyo 15 December 2010 Contents 1 General Topology 2 2 Hilbert Spaces and Operator Algebras 5 2.1 Hilbert Space . 5 2.2 Bounded Operators . 6 2.3 Trace Class Operators . 8 2.4 von Neumann Algebras . 10 2.5 Maps on von Neumann Algebras . 12 3 Abstract Operator Algebras 13 3.1 C∗-Algebras . 13 3.2 W ∗-algebras . 14 1 Chapter 1 General Topology Topology is an abstract structure that can be built on the set theory. We start with introducing the topological structure by open stets, which is the most standard way. A topological space is a set Ω together with O, a collection of subsets of Ω, satisfying the following properties: ∙ 휙 2 O and Ω 2 O. ∙ If O1 2 O and O2 2 O, then O1 \ O2 2 O. ∙ If O훼 2 O (훼 2 I) for arbitrary set of suffixes, then [훼2I O훼 2 O. An element of O is called an open set. In general, a set may have several topologies. If two topologies satisfy O1 ⊂ O2, then O1 is called weaker than O2, or smaller than O2. Topological structure can be generated by a subset of open spaces. Let B be a collection of subsets of a set Ω. The weakest topology O such that B ⊂ O is called generated by B. We note that such O does not always exist for an arbitrary B. Figure 1.1: An open set and a compact set. We review some important concepts in topological spaces: 2 ∙ If O is an open set, then Ω n O is called a closed set.
    [Show full text]
  • Tutorial on Von Neumann Algebras
    Tutorial on von Neumann algebras Adrian Ioana University of California, San Diego Model Theory and Operator Algebras UC Irvine September 20-22, 2017 1/174 the adjoint T ⇤ B(H) given by T ⇠,⌘ = ⇠,T ⇤⌘ ,forall⇠,⌘ H. 2 h i h i 2 the spectrum of T is σ(T )= λ C T λ I not invertible . { 2 | − } Fact: σ(T ) is a compact non-empty subset of C. Definition An operator T B(H)iscalled 2 1 self-adjoint if T = T ⇤. 2 positive if T ⇠,⇠ 0, for all ⇠ H. (positive self-adjoint) h i 2 ) 3 a projection if T = T = T 2 T is the orthogonal projection onto ⇤ , a closed subspace K H. ⇢ 4 a unitary if TT ⇤ = T ⇤T = I . The algebra of bounded linear operators H separable Hilbert space, e.g. `2(N), L2([0, 1], Leb). B(H) algebra of bounded linear operators T : H H, ! i.e. such that T =sup T ⇠ ⇠ =1 < . k k {k k|k k } 1 2/174 the spectrum of T is σ(T )= λ C T λ I not invertible . { 2 | − } Fact: σ(T ) is a compact non-empty subset of C. Definition An operator T B(H)iscalled 2 1 self-adjoint if T = T ⇤. 2 positive if T ⇠,⇠ 0, for all ⇠ H. (positive self-adjoint) h i 2 ) 3 a projection if T = T = T 2 T is the orthogonal projection onto ⇤ , a closed subspace K H. ⇢ 4 a unitary if TT ⇤ = T ⇤T = I . The algebra of bounded linear operators H separable Hilbert space, e.g.
    [Show full text]
  • Stability of Operators and C0-Semigroups
    Stability of operators and C0-semigroups DISSERTATION der Mathematischen Fakult¨at der Eberhard–Karls–Universit¨atT¨ubingen zur Erlangung des Grades eines Doktors der Naturwissenschaften Vorgelegt von TATJANA EISNER aus Charkow 2007 Tag der m¨undlichen Qualifikation: 20. Juni 2007 Dekan: Prof. Dr. Nils Schopohl 1. Berichterstatter: Prof. Dr. Rainer Nagel 2. Berichterstatter: Prof. Dr. Wolfgang Arendt To my teachers Rainer Nagel and Anna M. Vishnyakova Zusammenfassung in deutscher Sprache In dieser Arbeit betrachten wir die Potenzen T n eines linearen, beschr¨ankten Operators T und stark stetige Operatorhalbgruppen (T (t))t≥0 auf einem Banachraum X. Daf¨ur suchen wir nach Bedingungen, die “Stabilit¨at” garantieren, d.h. lim T n = 0 bzw. lim T (t) = 0 n→∞ t→∞ bez¨uglich einer der nat¨urlichen Topologie. Dazu gehen wir wie folgt vor. In Kapitel 1 stellen wir die (nichttrivialen) funktionalanalytischen Methoden zusam- men, wie z.B. das Jacobs–Glicksberg–de Leeuw Zerlegungstheorem, spektrale Abbildungs- s¨atzeund eine inverse Laplacetransformation. In Kapitel 2 diskutieren wir den “zeitdiskreten” Fall und beschreiben zuerst polyno- miale Beschr¨anktheit und Potenzbeschr¨ankheiteines Operators T . In Abschnitt 2 wird die Stabilit¨atbez¨uglich der starken Operatortopologie behandelt. Schwache und fast schwache Stabilit¨atwird in den Abschnitten 3, 4 und 5 untersucht und durch abstrakte Charakterisierungen und konkrete Beispiele erl¨autet.Wir zeigen insbesondere, dass eine “typische” Kontraktion sowie ein “typischer” unit¨areroder isometrischer Operator auf einem unendlich-dimensionalen separablen Hilbertraum fast schwach aber nicht schwach stabil ist. Analog gehen wir in Kapitel 3 f¨ureine C0-Halbgruppe (T (t))t≥0 vor. Zun¨achst wird Beschr¨anktheit bzw.
    [Show full text]
  • The Order Bicommutant
    B. de Rijk The Order Bicommutant A study of analogues of the von Neumann Bicommutant Theorem, reflexivity results and Schur's Lemma for operator algebras on Dedekind complete Riesz spaces Master's thesis, defended on August 20, 2012 Thesis advisor: Dr. M.F.E. de Jeu Mathematisch Instituut, Universiteit Leiden Abstract It this thesis we investigate whether an analogue of the von Neumann Bicommutant Theorem and related results are valid for Riesz spaces. Let H be a Hilbert space and D ⊂ Lb(H) a ∗-invariant subset. The bicommutant D00 equals P(D0)0, where P(D0) denotes the set of pro- jections in D0. Since the sets D00 and P(D0)0 agree, there are multiple possibilities to define an analogue of bicommutant for Riesz spaces. Let E be a Dedekind complete Riesz space and A ⊂ Ln(E) a subset. Since the band generated by the projections in Ln(E) is given by Orth(E) and order projections in the commutant correspond bijectively to reducing bands, our approach is to define the bicommutant of A on E by U := (A 0 \ Orth(E))0. Our first result is that the bicommutant U equals fT 2 Ln(E): T is reduced by every A - reducing bandg. Hence U is fully characterized by its reducing bands. This is the analogue of the fact that each von Neumann algebra in Lb(H) is reflexive. This result is based on the following two observations. Firstly, in Riesz spaces there is a one-to-one correspondence between bands and order projections, instead of a one-to-one correspondence between closed subspaces and projections.
    [Show full text]
  • Topological Properties of the Unitary Group 3
    TOPOLOGICAL PROPERTIES OF THE UNITARY GROUP JESUS´ ESPINOZA AND BERNARDO URIBE Abstract. We show that the strong operator topology, the weak operator topology and the compact-open topology agree on the space of unitary opera- tors of a infinite dimensional separable Hilbert space. Moreover, we show that the unitary group endowed with any of these topologies is a Polish group. Introduction The purpose of this short note is to settle some topological properties of the unitary group U( ) of an infinite dimensional separable Hilbert space , whenever the group is endowedH with the compact open topology. H When dealing with equivariant Hilbert bundles and its relation with its associ- ated unitary principal equivariant bundles (see [1]), one is obliged to consider the compact-open topology on the structural group U( ). In one of the foundational papers for twisted equivariantH K-theory, Atiyah and Segal claimed that the unitary group endowed with the compact-open topology was not a topological group [1, Page 40], based on the fact that the inverse map on GL( ) is not continuous when GL( ) is endowed with the strong operator topol- ogy.H This unfortunate claim obligedH Atiyah and Segal to device a set of ingenious constructions in order to make U( ) into a topological group with the desired topo- logical properties suited for the classificationH of Fredholm bundles. Nevertheless, these ingenious constructions of Atiyah and Segal added difficulties on the quest of finding local cross sections for equivariant projective bundles, and therefore a clarification on the veracity of the claim was due. The purpose of this note is to show that the unitary group U( ) endowed with the compact-open topology is indeed a topological group, moreoveHr a Polish group, and that this topology agrees with the strong operator topology, as the weak op- arXiv:1407.1869v1 [math.AT] 7 Jul 2014 erator topology; this is the content of Theorem 1.2 which is the main result of this note.
    [Show full text]