zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Finite-dimensional Hilbert C * -modules. (English) Zbl 1195.46059

In studying perturbations of the Wigner equation in inner product C * -modules, J. Chmieliński, D. Ilišević, M. S. Moslehian and Gh. Sadeghi [J. Math. Phys. 49, No. 3, 033519, 8 p. (2008; Zbl 1153.81342)] introduced the condition [H] stating that, for every bounded sequence (v n ) in a Hilbert C * -module V, there are a subsequence (v n k ) of (v n ) and vV such that, for every yV, lim k y,v n k -y,v=0. They proved that condition [H] is satisfied in every Hilbert C * -module over a finite-dimensional C * -algebra. Later, Lj. Arambašić, D. Bakić and M. S. Moslehian [Oper. Matrices 3, No. 2, Article ID 14, 235–240 (2009; Zbl 1188.46036)] proved that, if a full Hilbert A-module satisfies condition [H], then A must be finite-dimensional.

In the paper under review, the authors characterize the finite-dimensional Hilbert C * -modules in terms of the convergence of certain sequences. More precisely, they prove that, if V is a full right Hilbert module over a C * -algebra A, then the following statements are mutually equivalent: (i) V is finite-dimensional; (ii) A and the C * -algebra K(V) of compact operators on V are finite-dimensional; (iii) for every bounded sequence (v n ) in V, there are a subsequence (v n k ) of (v n ) and vV such that lim k v n k a-va=0 (aA) and lim k y,v n k -y,v=0 (yV); (iv) K(V) is a unital C * -algebra, and for every bounded sequence (v n ) in V, there are a subsequence (v n k ) of (v n ) and vV such that lim k y,v n k -y,v=0 (yV); (v) A is a unital C * -algebra, and for every bounded sequence (v n ) in V, there are a subsequence (v n k ) of (v n ) and vV such that lim k v n k a-va=0(aA).

MSC:
46L08C * -modules
46L05General theory of C * -algebras
46C50Generalizations of inner products