C*-algebra of bounded operators - Imported Knowlpedia Content

Definition

Let $H$ be a complex Hilbert space. The $C^*$ -algebra of bounded operators $B(H)$ consists of all bounded linear operators $T:H\to H$ . Addition and scalar multiplication are pointwise, multiplication is composition, and the involution is the Hilbert-space adjoint $T\mapsto T^*$ . With the operator norm

\lVert T\rVert=\sup_{\lVert\xi\rVert\leq1}\lVert T\xi\rVert,

$B(H)$ is a unital concrete $C^*$ -algebra whose identity is $I_H$ . It is the ambient algebra for concrete representations of $C^*$ -algebras and von Neumann algebras.

Algebraic and norm structure

The defining identity

\lVert T^*T\rVert=\lVert T\rVert^2

follows from Hilbert-space geometry, and completeness follows from completeness of the operator norm. Every bounded operator has a bounded adjoint, so $B(H)$ is closed under its involution. If $H$ is finite-dimensional, choosing an orthonormal basis identifies $B(H)$ with a full matrix algebra. If $H$ is infinite-dimensional, $B(H)$ is nonseparable in operator norm even when $H$ is separable Murphy, §2.1.

Operator topologies and commutants

Besides its norm topology, $B(H)$ carries the strong, weak, ultraweak, and ultrastrong operator topologies. These topologies are generally distinct on infinite-dimensional $H$ , and none changes the underlying $C^*$ -algebra operations. The commutant of a set of operators is computed inside $B(H)$ ; von Neumann algebras are exactly the unital $*$ -subalgebras of $B(H)$ that are closed in the weak operator topology, equivalently equal to their bicommutant.

Conventions and nearby spaces

$B(H,K)$ denotes the Banach space of bounded operators from $H$ to another Hilbert space $K$ . Unless $H=K$ , composition does not make $B(H,K)$ an algebra, so it should not be called a bounded-operator $C^*$ -algebra. Some authors write $\mathcal B(H)$ or $\mathscr L(H)$ . When $H=\{0\}$ , $B(H)$ is the zero algebra; whether it is called unital depends on the author's convention for the zero algebra.

References

Gerard J. Murphy, C-Algebras and Operator Theory*, Academic Press, 1990. DOI record. Relevant: §2.1 on bounded operators and concrete $C^*$ -algebras.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997. AMS record. Relevant: Chapter 2 on Hilbert-space operators and operator topologies.