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B ∗ ‐Algebras and their Representation
Author(s) -
Apostol Constantin
Publication year - 1971
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-3.1.30
Subject(s) - hausdorff space , mathematics , lemma (botany) , locally compact space , commutative property , unit (ring theory) , topology (electrical circuits) , algebra over a field , pure mathematics , topological group , space (punctuation) , discrete mathematics , combinatorics , computer science , ecology , mathematics education , poaceae , biology , operating system
Let A be a complete lmc‐∗‐algebra with unit whose topology is given by a family P of submultiplicative pseudonorms p such that p ( x ∗ x ) = p ( x ) 2 , x ∈ A (i.e. A is a b ∗ ‐algebra). We study the general properties of A (§2) and especially its representation in the commutative case. The description of the form of submultiplicative pseudonorms p such that p ( x ∗ x ) = p ( x ) 2 , x ∈ A (Lemma 3.2) enables us to obtain some structure theorems (§4). Thus we give an abstract characterisation of the algebra of all continuous complex functions on a locally compact Hausdorff space, endowed with the compact‐open topology [Th. 4.1, Th. 4.2].