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Norms of derivations of ℒ(X)
Abstract
If x is a real or complex Banach space and L(Ӿ)is the algebra of bounded linear endomorphisms of Ӿ then each element T of L(Ӿ) defines an operator (Formula presented) and Stampfli has shown that when c is a complex Hilbert space equality holds. In this paper it is shown, by methods which apply to a large class of uniformly convex spaces, that this formula for (Formula presented) spaces the formula is true in the real case but not in the complex case when the space has dimension 3 or more.
... It is verified by examples in Section 5 that the resulting identities fail to hold already in L(l p ) (1<p<, p{2) or for restrictions to Banach ideals of L(l 2 ). Our examples are infinite-dimensional modifications of those due to Johnson [15] and Fialkow [9], that concern the norms of derivations on L(l p ) and on Banach ideals of L(l 2 ). We are indebted to J. Eschmeier for pointing out a gap in our original proof of Theorem 10. ...
... Our first example demonstrates that (4.5) does not extend to the inner derivations on L(l p ) when 1<p<, p{2. We use for this purpose an infinite dimensional version of the one-dimensional operators considered by Johnson [15]. Our example is formulated for complex scalars, but the argument is similar in the real case. ...
... The 2-dimensional geometric construction in l p from [15, p. 466] yields normalized elements x, y # l p and f # l p$ satisfying conditions (i)(iii) as well as f (x)=1. The uniform estimate in (iv), somewhat stronger than the corresponding one from [15], is conveniently verified using the modulus of uniform convexity of l p : ...
LetA=(A1, …, An) and (B1, …, Bn) ben-tuples of bounded linear operators on a Banach spaceE. The corresponding elementary operator A, Bis the mapS↦∑ni=1 AiSBionL(E), and a, bdenotes the induced operators↦∑ni=1 aisbion the Calkin algebra (E)=L(E)/K(E). Heret=T+K(E) forT∈L(E). We establish that ifEhas a 1-unconditional basis, thenfor all elementary operators A, BonL(E), whereW(·) stands for the weakly compact operators. There is equality throughout ifE=ℓp, 1<p<∞. Our results extend and improve a corresponding structural result of Apostol and Fialkow (Canad. J. Math.38(1986), 1485–1524), which they proved forE=ℓ2using the non-commutative Weyl–von Neumann theorem due to Voiculescu. By contrast, our arguments are based on subsequence techniques from Banach space theory. As a byproduct we obtain a positive answer to the generalized Fong–Sourour conjecture for a large class of Banach spaces. We also explicitly compute the norm of the generalized derivations↦as−sbon (ℓ2) (this improves a result due to Fong) and show that the resulting formula fails to hold on (ℓp).
... For a normal operator T , δ T 0 can be expressed as the geometry of the spectrum of T 0 . Johnson [21] established methods which apply to a uniformly convex spaces with a large class, i.e the formula δ T is false in l p and L p (0, 1) 1 < p < ∞, p = 2. For L 1 space the formula is true for a real case and not for a complex case whose space dimension is 3 or more. ...
In this note, we provide detailed characterization of operators in terms of norm-attainability and norm estimates in Banach algebras. In particular, we establish the necessary and sufficient conditions for norm-attainability of the derivations and also give their norm bounds in the norm-attainable classes.
... ). B. E. Johnson noted that the above formula is no longer true in the general case. If X is a uniformly convex Banach space the validity of Stampfli´s formula is a necessary and sufficient condition in order that X be a Hilbert space (see [4] and [7]). ...
We review recent advances and some problems related to our research about
bounded derivations on non amenable nuclear Banach algebras.
... One way is to calculate the distance d(a, Z(A)) from a to Z(A), the centre of A; the other way is to compute the operator norm of ad(a), the inner derivation of A induced by a. For von Neumann algebras and some closely related algebras one has the interesting result that these two methods yield essentially the same answer: for all elements a in the algebra, ad(a) = 2d(a, Z(A)) [5,7,8,[10][11][12]16,20,21]. The unital C * -algebras for which this equality holds have been characterized in [19] as those C * -algebras A for which the ideal P ∩ Q ∩ R is primal whenever P , Q, and R are primitive ideals of A such that P ∩ Z(A) = Q ∩ Z(A) = R ∩ Z(A). ...
It is well known that if A is a von Neumann algebra then the norm of any inner derivation ad(a) is equal to twice the distance from the element a to the centre Z(A) of the algebra. More generally, this property holds in a unital C∗-algebra if and only if the ideal P∩Q∩R is primal whenever P, Q, and R are primitive ideals of A such that P∩Z(A)=Q∩Z(A)=R∩Z(A). In this paper we give a characterization, in terms of ideal structure, of those unital C∗-algebras A for which the norm of any inner derivation ad(a) at least dominates the distance from a to the centre Z(A). In doing so, we show that if A does not have this property then it necessarily contains an element a, with ‖ad(a)‖=1, whose distance from Z(A) is greater than or equal to 3+8214. We also show how this number is related to the numbers 415 and 12+13 which have previously arisen in the study of norms of inner derivations. The techniques used in this work include spectral theory, the theory of primitive and primal ideals, and constrained geometrical optimisation.
... Apparently, the only elementary operators on a Hilbert space for which the norm is computed are the basic ones and generalized derivations [10]. We refer to [2,[4][5][6][7][8][9][10][11] for an intensive study of norms of elementary operators. ...
Let L(H) be the algebra of bounded linear operators on a Hilbert space H. For A,B∈L(H), define the elementary operator MA,B by MA,B(X)=AXB (X∈L(H)). We give necessary and sufficient conditions for any pair of operators A and B to satisfy the equation ‖I+MA,B‖=1+‖A‖‖B‖, where I is the identity operator on H.
... Stampfli [11] showed that (1) holds when B(H) is a primitive C * -algebra with an identity and in particular when B(H) is the algebra of bounded linear operators in a Banach space. Johnson [5] and Kyle [7] showed that the equality holds and sometimes does not. For quotients of W * -algebras, Sommerset [10] showed that (1) holds while recently, Bonyo and Agure [3] showed that when J is a proper two-sided ideal then (1) is true. ...
We investigate the relationship between inner derivations imple-mented by a norm-attainable element of a C*-algebra to those of ideals and primitive ideals. Moreover, we give related results on the relation-ship between the constants C(Ω) and C s (Ω) of C * -algebras to those of ideals and primitive ideals.
... This result has been recently extended to unital prime C * -algebras (see [3, Proposition 2.23] or [10, Corollary 2.9]). For Banach algebras of all bounded linear operators on a Banach space B. E. Johnson [4] showed that γ = 2 is not always 2034 M. CABRERA AND J. MART´INEZMART´MART´INEZ possible, and J. Kyle [5] showed that the equality γ = 2 characterizes Hilbert spaces among all uniformly convex Banach spaces. The main result (Theorem 2) in this paper solves the existence problem of a positive constant γ for ultraprime Banach algebras, thus extending the aforementioned result for unital prime C * -algebras. ...
We show that the set of all inner derivations of an ultraprime real Banach algebra is closed within all bounded derivations.
More concretely, we show that for such an algebra A there exists a positive number γ (depending only on the “constant of ultraprimeness” of A) satisfying γ ∥ a+Z(A) ∥≦∥ D
a
∥ for all a in A, where Z(A) denotes the centre of A and D
a
denotes the inner derivation on A induced by a. This result is an extension of the corresponding complex version obtained by the authors in [Proc. Amer. Math. Soc., to appear]. The proof relies on the following theorem: ultraproducts of a family of central ultraprime real Banach algebras
with a unit and with constant of ultraprimeness greater than or equal to a fixed positive constant K are central ultraprime Banach algebras with a unit. This fact is obained via a general result for real Banach algebras that
reads as follows: If A is a central real Banach algebra with a unit 1, then for every a in A satisfying ∥ 1+a
2 ∥<1 we have [1+√1−||1+1a
2||]2≦2(|⋎l+M
a
||+||D
a
||) where M
a
denotes the two-sided multiplication operator by a on A.
We give an expository survey of different results about the norm of a derivation on a Banach space, with particular emphasis on the special case of derivations having the same norm when they are restricted to any symmetric norm ideal.KeywordsDerivationsNormNumerical rangeMathematics Subject Classification (MSC2020)47A1247A3047B47
This chapter is devoted to \textcolor{red}{the survey of} quaternion restricted two-sided matrix equation $\mathbf{AXB}=\mathbf{D}$ and approximation problems related with it. Unique solutions to the considered approximation matrix problems and the restricted quaternion two-sided matrix equations with specific constraints are expressed in terms of the core-EP inverse and the dual core-EP inverse, the MPCEP and $*$CEPMP inverses, and the DMP and MPD inverses. The MPCEP--$*$CEPMP inverses and the DMP--MPD inverses are generalized inverses obtained by combining the Moore-Penrose (MP-)inverse with the core-EP (CEP-)inverse and the MP-inverse with the Drazin (D-)inverse, respectively. Several particular cases of these equations and approximation matrix problems are presented too. Cramer's rules for solving these constrained quaternion matrix equations and approximation matrix problems with their particular cases are developed by using of noncommutative row-column determinants. As a consequence, Cramer's rules for solving these constrained matrix equations with complex matrices are derived as well. Numerical examples are given to illustrate gained results.
Let n = n1 + • • • + nr, where r ³ 2 and the ni’s are positive integers. Then every element of G = SL (n, C) can be written as a block matrix (gij)1 £i, j£r where each block gij is a ni X nj matrix. Let Gn1,…, nr denote the subgroup of all diagonal block matrices, i.e., gij is the 0-matrix for i¹j. Let Tx be any element of the non-degenerate principal series of G. The main purpose of this paper is to decompose the restriction of Tx to Gn1,…,nr into irreducible representations.
n X j=1 LAjRBj. This concrete class includes many important operators on spaces of operators, such as the two-sided multiplicationsLARB, the commutators (or inner derivations)LA RA, and the intertwining maps (or generalized derivations)LA RB for givenA,B 2 L(X). Elementary operators also induce bounded operators between operator ideals, as well as between quotient algebras such as the Calkin algebra L(X)/K(X), where K(X) are the compact operators on X. Note also that definition (1.1) makes sense in the more general framework of Banach algebras. Elementary operators first appeared in a series of notes by Sylvester (Sy1884) in the 1880's, in which he computed the eigenvalues of the matrix operators correspond- ing to EA,B on the n ◊ n-matrices. The term elementary operator was coined by Lumer and Rosenblum (LR59) in the late 50's. The literature related to elementary operators is by now very large, and there are many excellent surveys and expositions of certain aspects. Elementary operators on C -algebras were extensively treated by Ara and Mathieu in (AM03, Chapter 5). Curto (Cu92) gives an exhaustive overview of spectral properties of elementary operators, Fialkow (Fi92) comprehensively dis- cusses their structural properties (with an emphasis on Hilbert space aspects and methods), and Bhatia and Rosenthal (BR97) deals with their applications to operator equations and linear algebra. Mathieu (Ma01b), (Ma01a) surveys some recent topics in the computation of the norm of elementary operators, and elementary operators on the Calkin algebra. These references also contain a number of applications, and we also note the survey by Carl and Schiebold (CS00), where they describe an in- triguing approach to certain nonlinear equations from soliton physics which involves some elementary operators (among many other tools).
Let G be a locally compact group. The notion of “totally real” unitary representation of G is defined and investigated in §1. In particular, if K is a compact subgroup of G, it is shown that every closed G-invariant subspace of L2(G/K) is spanned by real-valued functions if, and only if, KgK=Kg-1xK for every geG. In §2 the coset space X=G/K is specialized to a Riemannian symmetric space, where the double coset condition is replaced by a simple Weyl group condition.
The results in this paper reveal a dichotomy in regard to the existence of fixed points for smooth real maps and biholomorphic maps in Hilbert space. Kakutani has shown that there exists a homeomorphism of the closed unit sphere of Hilbert space onto itself which has no fixed point. A slight modification of his example shows that there is a diffeomor- phism having the same property. Our results show that in the complex case every biholomorphic map of the unit ball onto itself in Hilbert space has a fixed point.
The determination of the function spaces X which are intermediate in the weak sense between Lp and Lq has been shown, by the author, to depend on a pair of numbers (α,β) called the indices of the space. The indices depend on the function norm of X and on the properties of the underlying measure space: Whether it has finite or infinite measure, is non-atomic or atomic. In this paper, formulas are given for the indices of an Orlicz space in case the measure space is non-atomic with finite or infinite measure, or else is purely atomic with atoms of equal measure. The indices for an Orlicz space over a non-atomic finite measure space turn out to be the reciprocals of the exponents of the space as introduced by Matuszewska and Orlicz, and generalized by Shimogaki. Some new results concerning submultiplicative functions are used in the proof of the main result.
A set A and an integer n > 1 are given. S is any family of subsets of An. Necessary and sufficient conditions are found for the existence of a set F of finitary partial operations on A such that S is the set of all subalgebras of ˂A; F˃n. As a corollary, a family E of equivalence relations on A is the set of all congruences on ˂A; F˃ for some F if and only if E is an algebraic closure system on A2.
Let n = n1 + • • • + nr, where r ³ 2 and the ni’s are positive integers. Then every element of G = SL (n, C) can be written as a block matrix (gij)1 £i, j£r where each block gij is a ni X nj matrix. Let Gn1,…, nr denote the subgroup of all diagonal block matrices, i.e., gij is the 0-matrix for i¹j. Let Tx be any element of the non-degenerate principal series of G. The main purpose of this paper is to decompose the restriction of Tx to Gn1,…,nr into irreducible representations.
Hisahiro Tamano asked the following question that motivated his Some properties of the Stone-Čech compactifica- tion, J. Math. Soc. Japan, vol. 12, 1 (1960) 104-117. “What is a necessary and sufficient condition for a uniform space to have a paracompact completion?”. Although he did not arrive at a solution, he obtained characterizations of completeness, paracompactness, and the structure of the completion of a uniform space by means of the radical. This paper provides an answer to the above question and related ones. Also included is a development of other types of completeness in both uniform and non-uniform topological spaces. These other “complete” spaces yield characterizations of paracompactness and the Lindelöf property as well as necessary and sufficient conditions for paracompact spaces to be Lindelöf and entirely normal spaces to be paracompact.
The purpose of this paper is to present a definition of meromorphic annular functions which includes the definition of holomorphic annular functions. Several equivalent conditions for meromorphic annular functions are given.
Recent studies of the complex bordism homology theory (Formula Presented) have shown that for a finite complex X the integer hom-dim (Formula Presented) provides a useful numerical invariant measuring certain types of complexity in X. Associated to an element (Formula Presented) (X) one has the annihilator ideal (Formula Presented) Numerous relations between A (a) and hom-dim (Formula Presented) are known. In attempting to deal with these invariants it is of course useful to study special cases, and families of special cases. In this note we study the annihilator ideal of the canonical element (Formula Presented) where X is a complex of the form (Formula Presented) an odd prime. We show that (Formula Presented) where (Formula Presented) is a Milnor manifold for the prime p. This provides another piece of evidence that for such a complex X, hom-dim (Formula Presented) is 1 or 2.
A form of the Hamburger moment problem for the real line is generalized and solved for an arbitrary Lie group. The solution relates the unitary representations of the Lie group to certain symmetric representations of the associated universal enveloping algebra.
The first Radon-nikodým theorem for the Bochner integral was proven by Dunford and Pettis in 1940. In 1943, Phillips proved an extension of the Dunford and Pettis result. Then in 1968-69, three results appeared. One of these, due to Me ti vier, bears a direct resemblence to the earlier Phillips theorem. The remaining two were proven by Rieffel and seem to stand independent of the others. This paper is an attempt to put these apparently diverse theorems in some perspective by showing their connections, by simplifying some proofs and by providing some modest extensions of these results. In particular, it will be shown that the Dunford and Pettis theorem together with Rieffel’s theorem directly imply Phillips’ result. Also, it will be shown that, with almost no sacrifice of economy of effort, the theorems here can be stated in the setting of the Pettis integral.
In this paper, preordered topological spaces obeying two natural conditions, known as bicontinuous preordered spaces, are studied. The relationship between the topology of such a space and two associated convex topologies is examined.
In this paper we solve the triple integral equations (FORMULA RESENTED) where α,β,(FORMULA PRESENT), are real parameters, f2(x) is a known function,∅ (s) is to be determined and (FORMULA PRESENTED) denote the Mellin transform of h(x) and its inversion formula respectively.
Let F denote the class of Blaschke products B(z, {zn}) such that (FORMULA PRESENTED) is bounded for 0 < r < 1. The distributions of the zeros of Blaschke products in F are examined, and extensions are made to earlier results of MacLane and Rubel.
In this paper we examine the Köthe-Toeplitz reflexivity of certain sequence spaces and we characterize some classes of matrix transformations defined on them. The results are used to prove a generalization of a theorem by V. G. Iyer, concerning the equivalence of the notions of strong and weak convergence on the space of all integral functions, and also to generalize some theorems by Ch. Rao.
An axiomatic setting for the theory of convexity is provided by taking an arbitrary set X and constructing a family C of subsets of X which is closed under intersections. The pair consisting of any ordered vector space and its family of convex subsets thus become the prototype for all such pairs (.X, C). In this connection, Levi proved that a Radon number r for C implies a Helly number h ≦ r — 1; it is shown in this paper that exactly one additional relationship among the Carathéodory, Helly, and Radon numbers is true, namely, that if C has Carathéodory number o and Helly number h then C has Radon number r ≦ ch+1. Further, characterizations of (finite) Carathéodory, Helly, and Radon numbers are obtained in terms of separation properties, from which emerges a new proof of Levi’s theorem, and finally, axiomatic foundations for convexity in euclidean space are discussed, resulting in a theorem of the type proved by Dvoretzky.
In potential theory on Riemann surfaces three kernels are considered: The Green’s kernel on hyperbolic Riemann surfaces; the Evans kernel on parabolic Riemann surfaces; and the Sario kernel on arbitrary Riemann surfaces. Since the Sario kernel has no restriction on the domain surface, in contrast with the two other kernels, its potential theory enjoys the advantage of full generality. From the point of view of Riemannian spaces potential theory on Riemann surfaces is included in that on Riemannian spaces. The object of this note is to construct the Sario kernel and to develop the corresponding theory of Sario kernel on Riemannian spaces of dimension n ≧ 3. The Sario kernel, which is positive, symmetric and jointly continuous, posseses the property of Riez type decomposition (Theorem 1). The continuity principle, unicity principle, Frostman’s maximum principle, energy principle and capacity principle are valid for potentials with respect to the Sario kernel. It is also shown that a set of capacity zero with respect to the Sario kernel is, considered locally, of Newtonian capacity zero (Theorem 7), and so the relation of capacity function and the equilibrium Newtonian potential in Euclidean n-space is obtained.
The purpose of this note is to give a simple condition which is sufficient for a function on a real interval to be the boundary value of a schlicht (univalent) analytic mapping of the upper half plane into itself. This condition leads to a simple transformation which takes (possibly) non-schlicht mappings into schlicht ones. The methods used have applications to probability theory as well; they yield an interesting class of infinitely divisible characteristic functions.
Here, it is shown that if Mn is an n-manifold triangulated as a locally finite simplicial complex and Nk is a closed subcomplex of int Mn that is also a topological k-manifold, then Nk is topologically locally flat in Mn provided n-k ≠ 2 and each of Nk and Mn is a simplicial homotopy manifold. This result not only generalizes all known results to date, but also either includes the most general case, where no further assumptions on the triangulations are made, or the general case is false in a very strong sense. That is, if some triangulated topological n-manifold is not a simplicial homotopy n-manifold, then there exist, for some m, a triangulated m-sphere I and PL (m-l)-and (m+1)- spheres S and T respectively, such that I is a subcomplex of Sf, S is a subcomplex of (FORMULA PRESENT), where U is homeomorphic to Em+1 but p1 (V) ¹ 0, and S bounds a (FORMULA PRESENT) The main result is obtained by noting some results related to double suspensions of homotopy 3-and 4-spheres and showing that each open simplex of such a triangulation, as above, is topologically flat in the given manifold.
Necessary and sufficient conditions are given for the generalized convolution of two arithmetic functions which are units with respect to a basic sequence B to be again a unit on B. These conditions are then investigated from function-theoretic and combinatorial points of view.
Let M be a smooth real two dimensional submanifold of C2. Near a point of M where its tangent space is complex linear it may, after a certain local biholomorphic change of coordinates, be represented as the graph near 0 in C2 of a smooth function f(z) = Q(z) + o(|z| 2) in which z is the first coordinate of C2 and Q is a real valued quadratic form. This paper is concerned with the polynomialy convex hull of a small compact set K in M near 0 and associated descriptions of the Banach algebra P(M) of continuous functions uniformly approximable on K by polynomials in two complex variables. It treats the very special case where f has rank ≦ 1 near 0 (as an R2-valued map). It is shown that if Q has nonzero eigenvalues of opposite sign, then all sufficiently small compact sets K are polynomialy convex, and P(K) is the full algebra of continuous functions. If Q has nonzero eigenvalues of the same sign, the polynomial hull of K is described in terms of a foliation by certain simple analytic sets in C2, and P(K) is isomorphic to the algebra of continuous functions on the hull whose restriction to the interior of each analytic set is holomorphic.
The purpose of this paper is twofold: First, to determine the full translation groups for all n-uniform translation affine Hjelmslev planes for all positive integers n; and second, to prove that all such groups occur as the full translation groups of Pappian Hjelmslev Planes.
In this note, among other things, we shall establish the following result: Theorem. Let (H, X,π) be an effective transformation group, where X is a compact 3-dim, connected manifold, H is a connected locally compact but non-compact topological group and H acts on X almost periodically. Then H must be a Lie group.
Let R be a simple anti-flexible ring of characteristic distinct from 2 and 3. Anderson and Outcalt have proved that R+ is a commutative associative ring. The same authors have also shown that a commutative associative ring P of characteristic not 2 gives rise to a simple anti-flexible ring provided P has a suitably defined symmetric belinear form on it. The purpose of this paper is to give an explicit construction of such a symmetric bilinear form and determine the suitable commutative associative rings.
Consider all ordinary arithmetic and geometric means of n real nonnegative numbers taken k at a time. Let αk be the geometric mean of all the arithmetic means and γk the arithmetic mean of all the geometric means. It is proved that αk increases with k, γk decreases, and γk ≦ αk if h +k > n. These results are generalized to mixed means of any real orders. Comparison of αk and γk with elementary symmetric functions suggests a conjecture.
This paper studies the matrix equation f(Z) = D + λJ where f is a polynomial, Z a square (0, l)-matrix, D is diagonal, λ ≠ 0 and J is the matrix of ones. If Z is thought of as the incidence matrix of a digraph G, the equation implies various path length properties for G. It is shown that such a graph is an amalgamation of regular subgraphs with similar path length properties. Necessary and sufficient parameter conditions on the matrix Z are given in order that it satisfy such an equation for a fixed polynomial f and all non-regular digraphs corresponding to quadratic polynomials f are found.
In this paper conditions are obtained under which two closed homotopic embeddings of a locally compact separable metric space into a separable metric manifold modeled on the Hilbert cube are ambient isotopic.
441 A. A. Iskander, Subalgebra systems of powers of partial universal algebras
- Hideo Imai
- ............................................................................. Sario Potentials On Riemannian Spaces
Hideo Imai, Sario potentials on Riemannian spaces....................... 441
A. A. Iskander, Subalgebra systems of powers of partial universal
algebras......................................................... 457