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In recent years it has
become clear that the study of C∗-algebras without a unit element is more than just
a mildly interesting extension of the “typical” case of a C∗-algebra with unit. A
number of important examples of C∗-algebras rarely have a unit, for example the
group C∗-algebras and algebras of the form I ∩ I∗ where I is a closed left ideal of a
C∗-algebra. J. Dixmier’s book, Les C∗-algebras et leurs representations, carries
through all the basic theory of C∗-algebras for the no-unit case, and his main tool is
the approximate identity which such algebras have. Many C∗-algebra questions can
be answered for a C∗-algebra without unit by embedding such an algebra in a
C∗-algebra with unit. Some problems, especially those which involve approximate
units, are not susceptible to this approach. This paper will study some problems of
this type.
Theorem 1.1 states that if 𝒰 is a norm separable C∗-algebra and {f1,⋯fn} is a
finite set of orthogonal pure states of 𝒰 (i.e., ∥fi −fj∥ = 2 if i≠j), then there exists a
maximal abelian C∗-subalgebra A of 𝒰 such that fk|A is pure (k = 1,⋯,n) and fk|A
has unique pure state extension to 𝒰(k = 1,⋯,n). This extends the prototype result
of Aarnes and Kadison by (a) allowing a finite number of pure states instead of just
one, (b) dropping the assumption that 1 ∈𝒰, and (c) proving uniqueness of the pure
state extension. In §2 two examples are constructed which show that the uniqueness
assertion of Theorem 1.1 cannot be extended to the nonseparable case, and
that even in the separable case the subalgebra A must be carefully chosen
to insure uniqueness of pure state extension. Theorem 1.2 and Example
2.3 show that a very desirable majorization property of approximate units
does not quite carry over from the abelian case to the general case. (If it
did, several important problems, including the Stone-Weierstrass problem
would have been solved.) Theorem 1.3 extends the author’ characterization of
approximate units of C∗-algebras to approximate right units for left ideals of
C∗-algebras.
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