Frames on Hilbert C*-modules have been defined for unital C*algebras by Frank and Larson [5] and ... more Frames on Hilbert C*-modules have been defined for unital C*algebras by Frank and Larson [5] and operator-valued frames on a Hilbert space have been studied in [8]. The goal of this paper is to introduce operatorvalued frames on a Hilbert C*-module for a σ-unital C*-algebra. Theorem 1.4 reformulates the definition given in [5] in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator-valued frame are limits in the strict topology of a series in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator-valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes an one-to-one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operatorvalued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.
Transactions of the American Mathematical Society, 2013
We first prove that in a σ-finite von Neumann factor M , a positive element a with properly infin... more We first prove that in a σ-finite von Neumann factor M , a positive element a with properly infinite range projection Ra is a linear combination of projections with positive coefficients if and only if the essential norm a e with respect to the closed two-sided ideal J(M ) generated by the finite projections of M does not vanish. Then we show that if a e > 1, then a is a finite sum of projections. Both these results are extended to general properly infinite von Neumann algebras in terms of central essential spectra.
Q 1986 Birkh[user Verlag, Basel projections be used to introduce a notion of finiteness in M? In ... more Q 1986 Birkh[user Verlag, Basel projections be used to introduce a notion of finiteness in M? In general, can some notion of finiteness in M be inherited from the semifiniteness of this crossed product? This last question was investigated for the case of type III~ and type III 0 factors in [2,3]. A discrete decomposition {N, 8,~} of M was used instead of a continuous one, and a class J, analogous to the class of compact operators in semifinite algebras, was defined as the hereditary C*-algebra generated by the finite projections of N (identified Then Haagerup defined the LP spaces associated with M for 1 _< p _< oo as (I) LP(M) = {A ( ~ 1 8s(A) = e-S/PA, s ~ ~} L~176 = {A ~ ~ i @s (A) = A, s ~ ~}
This paper presents necessary and sufficient conditions for a positive bounded operator on a sepa... more This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum converging in the strong operator topology if the collection is infinite. A similar necessary condition is given when the operator and the projections are taken in a type II von Neumann factor, and the condition is proven to be also sufficient if the operator is "diagonalizable". A simpler necessary and sufficient condition is given in the type III factor case.
Main result: If a C*-algebra A is simple, σ-unital, has finitely many extremal traces, and has st... more Main result: If a C*-algebra A is simple, σ-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier M(A) also has strict comparison of positive elements by traces. The same results holds if "finitely many extremal traces" is replaced by "quasicontinuous scale".
In this paper we investigate whether positive elements in the multiplier algebras of certain fini... more In this paper we investigate whether positive elements in the multiplier algebras of certain finite C*-algebras can be written as finite linear combinations of projections with positive coefficients (PCP). Our focus is on the category of underlying C*-algebras that are separable, simple, with real rank zero, stable rank one, finitely many extreme traces, and strict comparison of projections by the traces. We prove that the strict comparison of projections holds also in the multiplier algebra M(A ⊗ K). Based on this result and under the additional hypothesis that M(A ⊗ K) has real rank zero, we characterize which positive elements of M(A ⊗ K) are of PCP.
This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals o... more This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or arithmetic mean at infinity stable principal ideals or of principal ideals with a generator that satisfies the Delta_1/2 condition, are all both upper and
The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arith... more The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we
This article - a part of a multipaper project investigating arithmetic mean ideals - investigates... more This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., ``How many traces can an ideal support?" We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced by
Majorization for Infinite Sequences, an Extension of the Schur-Horn Theorem, and Operator Ideals
The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: F... more The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequencesandthat converge to 0, there exists a compact operator A with eigenvalue listand diagonal sequenceif and only if Pn j=1 �jPn j=1 �j for every n if and only if � = Qfor some orthostochastic matrix Q. The similar result
Maximum Entropy and Joint Norm Bounds for Operator Extensions
New Aspects in Interpolation and Completion Theories, 1993
... norm :5 8. The condition II ( ~ ) II :5 8 implies the existence of a unique contraction 1 Par... more ... norm :5 8. The condition II ( ~ ) II :5 8 implies the existence of a unique contraction 1 Partially supported by Georgia State University. ... For the Nehari problem, this question was asked by Ciprian Foias and his collaborators and is still open. 4. An Application to Integral Operators. ...
Frames on Hilbert C*-modules have been defined for unital C*algebras by Frank and Larson [5] and ... more Frames on Hilbert C*-modules have been defined for unital C*algebras by Frank and Larson [5] and operator-valued frames on a Hilbert space have been studied in [8]. The goal of this paper is to introduce operatorvalued frames on a Hilbert C*-module for a σ-unital C*-algebra. Theorem 1.4 reformulates the definition given in [5] in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator-valued frame are limits in the strict topology of a series in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator-valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes an one-to-one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operatorvalued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.
Transactions of the American Mathematical Society, 2013
We first prove that in a σ-finite von Neumann factor M , a positive element a with properly infin... more We first prove that in a σ-finite von Neumann factor M , a positive element a with properly infinite range projection Ra is a linear combination of projections with positive coefficients if and only if the essential norm a e with respect to the closed two-sided ideal J(M ) generated by the finite projections of M does not vanish. Then we show that if a e > 1, then a is a finite sum of projections. Both these results are extended to general properly infinite von Neumann algebras in terms of central essential spectra.
Q 1986 Birkh[user Verlag, Basel projections be used to introduce a notion of finiteness in M? In ... more Q 1986 Birkh[user Verlag, Basel projections be used to introduce a notion of finiteness in M? In general, can some notion of finiteness in M be inherited from the semifiniteness of this crossed product? This last question was investigated for the case of type III~ and type III 0 factors in [2,3]. A discrete decomposition {N, 8,~} of M was used instead of a continuous one, and a class J, analogous to the class of compact operators in semifinite algebras, was defined as the hereditary C*-algebra generated by the finite projections of N (identified Then Haagerup defined the LP spaces associated with M for 1 _< p _< oo as (I) LP(M) = {A ( ~ 1 8s(A) = e-S/PA, s ~ ~} L~176 = {A ~ ~ i @s (A) = A, s ~ ~}
This paper presents necessary and sufficient conditions for a positive bounded operator on a sepa... more This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum converging in the strong operator topology if the collection is infinite. A similar necessary condition is given when the operator and the projections are taken in a type II von Neumann factor, and the condition is proven to be also sufficient if the operator is "diagonalizable". A simpler necessary and sufficient condition is given in the type III factor case.
Main result: If a C*-algebra A is simple, σ-unital, has finitely many extremal traces, and has st... more Main result: If a C*-algebra A is simple, σ-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier M(A) also has strict comparison of positive elements by traces. The same results holds if "finitely many extremal traces" is replaced by "quasicontinuous scale".
In this paper we investigate whether positive elements in the multiplier algebras of certain fini... more In this paper we investigate whether positive elements in the multiplier algebras of certain finite C*-algebras can be written as finite linear combinations of projections with positive coefficients (PCP). Our focus is on the category of underlying C*-algebras that are separable, simple, with real rank zero, stable rank one, finitely many extreme traces, and strict comparison of projections by the traces. We prove that the strict comparison of projections holds also in the multiplier algebra M(A ⊗ K). Based on this result and under the additional hypothesis that M(A ⊗ K) has real rank zero, we characterize which positive elements of M(A ⊗ K) are of PCP.
This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals o... more This part of a multi-paper project studies the lattice properties of the arithmetic mean ideals of B(H) introduced by Dykema, Figiel, Weiss, and Wodzicki. We prove: the lattices of all principal ideals, of arithmetic mean or arithmetic mean at infinity stable principal ideals or of principal ideals with a generator that satisfies the Delta_1/2 condition, are all both upper and
The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arith... more The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we
This article - a part of a multipaper project investigating arithmetic mean ideals - investigates... more This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., ``How many traces can an ideal support?" We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced by
Majorization for Infinite Sequences, an Extension of the Schur-Horn Theorem, and Operator Ideals
The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: F... more The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequencesandthat converge to 0, there exists a compact operator A with eigenvalue listand diagonal sequenceif and only if Pn j=1 �jPn j=1 �j for every n if and only if � = Qfor some orthostochastic matrix Q. The similar result
Maximum Entropy and Joint Norm Bounds for Operator Extensions
New Aspects in Interpolation and Completion Theories, 1993
... norm :5 8. The condition II ( ~ ) II :5 8 implies the existence of a unique contraction 1 Par... more ... norm :5 8. The condition II ( ~ ) II :5 8 implies the existence of a unique contraction 1 Partially supported by Georgia State University. ... For the Nehari problem, this question was asked by Ciprian Foias and his collaborators and is still open. 4. An Application to Integral Operators. ...
Uploads
Papers by Victor Kaftal