nLab extended C*-algebra

Idea

An extension of the notion of a C*-algebra to unbounded operators.

In complete analogy to abstract and concrete definitions of a C*-algebra, there are abstract and concrete definitions of an extended C*-algebra.

Definition

An extended C*-algebra is a complex (or real) locally convex *-algebra $A$ such that

• the poset of subsets $B\subset A$ that are closed, absolutely convex, bounded, $1\in B$, and $B^2\subset B$ has a maximal element $B_0$;

• $A$ is pseudocomplete: for every $B$ as above, the Minkowski functional of $B$ induces a complete norm on $A$;

• $A$ is symmetric: for every $x\in A$, the element $(1+x^*x)^{-1}$ exists and belongs to $A_0$, the bounded part of $A$, comprising elements $a\in A$ such that there is a nonzero $\lambda$ for which the set $\{(\lambda x)^n\mid n\ge0\}$ is bounded.

Concrete definition

An extended C*-algebra is a complex (or real) *-algebra $A$ of closed densely defined unbounded operators on a Hilbert space closed under the operations of strong sum, strong multiplication, passing to adjoints that contains all scalar multiples of the identity operator and for every $x\in A$ we have $(1+x^*x)^{-1}\in A$.

Related concepts

• extended von Neumann algebra

References

• J. B. Cooper, Extended C*-algebras and W*-algebras, Proceedings of the Symposium on Functional Analysis (Istanbul, 1973), 75–84. Publications of the Mathematical Research Institute Istanbul 1. Mathematical Research Institute, Istanbul, 1974. PDF.

  • an expository article

• G. R. Allan, On a class of locally convex algebras, Proceedings of the London Mathematical Society s3-17:1 (1967), 91-114. DOI.

  • defines abstract extended C-algebras (as GB-algebras) in Definition 2.5
  • develops Gelfand duality

• P. G. Dixon, Generalized B*-algebras, Proceedings of the London Mathematical Society s3-21:4 (1970), 693-715. DOI.

  • establishes functional calculus for commutative extended C*-algebras and Gelfand-Neumark theorem for noncommutative extended C*-algebras
  • defines concrete C*-algebras in Definition 7.1
  • shows that abstract and concrete extended C*-algebras are equivalent in Theorem 7.13