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)
Bounded module maps and pure completely positive maps. (English) Zbl 0791.46032

From the author’s introduction: Let A and B be two C * -algebras, φ a completely positive map from B to B. The map φ gives a Hilbert B-module H φ and a *-homomorphism π φ from A into the C * -algebra of all bounded B-module maps with adjoints on H φ . In the case that B=, it is well known that φ is pure if and only if π φ is irreducible. It is natural to ask whether it is also true for general C * -algebras B.

In this note we give a negative answer to the problem in general. We also show that for many C * -algebras B, the maps φ are never pure and π φ are never irreducible. However, for some C * -algebras B, the purity of φ does imply the irreducibility of π φ and for some C * -algebras B, the irreducibility of π φ implies φ is pure.

MSC:
46L05General theory of C * -algebras
46H25Normed modules and Banach modules, topological modules