Operator-valued Frames on C*-Modules

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. Introduction Frames on a Hilbert space are collections of vectors satisfying the condition a‖ξ‖ ≤ ∑ j∈J | < ξ, ξj > | 2 ≤ b‖ξ‖ for some positive constants a and b and all vectors ξ. This notion has been naturally extended by Frank and Larson [5] to countable collections of vectors in a Hilbert C*-module for a unital C*-algebra satisfying an analogous defining property (see below 1.1 for the definitions). Most properties of frames on a Hilbert space hold also for Hilbert C*-modules, often have quite different proofs, but new phenomena do arise. A different generalization where frames are no longer vectors in a Hilbert space but operators on a Hilbert space is given in [8] with the purpose of providing a natural framework for multiframes, especially for those obtained from a unitary system, e.g, a discrete group representation. Operator-valued frames both generalize vector frames and can be decomposed into vector frames. The goal of this article is to introduce the notion of operator-valued frame on a Hilbert C*-module. Since the frame transform of Frank and Larson permits to identify a vector frame on an arbitrary Hilbert C*-module with a vector frame on the standard Hilbert C*-module l(A) of the associated C*-algebra A, for simplicity’s 2000 Mathematics Subject Classification. Primary 46L05 . 1 2 VICTOR KAFTAL, DAVID LARSON, SHUANG ZHANG* sake we confine our definition directly to frames on l(A). When the associated C*algebra is σ-unital, it well known (see [9] ) that the algebra of bounded adjointable operators on l(A) can be identified with the multiplier algebraM(A⊗K) of A⊗K, about which a good deal is known. Since reference are mainly formulated in terms of right-modules, we treat l(A) as a right module (i.e., as ‘row vectors’). A frame on l(A) is thus defined as a collection of operators {Aj}j∈J with Aj ∈ E0M(A⊗K) for a fixed projection E0 ∈ M(A⊗K) for which aI ≤ ∑