Unitary element of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a unital $C^*$ -algebra. A unitary element is an element $u\in A$ satisfying

u^*u=uu^*=1_A.

Equivalently, $u$ is invertible and $u^{-1}=u^*$ . If $A$ is faithfully represented on a Hilbert space, $u$ acts as a surjective inner-product preserving operator. For a nonunital $C^*$ -algebra there are no literal unitaries in this sense because $1_A$ is absent; one must specify unitaries in a unitization or in the multiplier algebra instead.

Basic properties

Every unitary has norm one unless the algebra is the zero algebra, and its spectrum is contained in the unit circle. Products and adjoints of unitaries are unitary. If $h=h^*$ , continuous functional calculus gives the unitary $e^{ih}$ . Conversely, a unitary need not admit a logarithm in the algebra; the obstruction is related to its homotopy class.

Stable homotopy and K-theory

Unitary elements in the matrix algebras $M_n(A)$ form groups under multiplication. After stabilization and homotopy, they define $K_1(A)$ . For nonunital $A$ , one uses unitaries in matrices over the unitization whose image in the scalar quotient is the identity. Thus a $K_1$ -representative is often written $1+a$ , but only when that element is actually unitary.

Multiplier convention

A unitary multiplier of a nonunital algebra $A$ means a unitary element of $M(A)$ , the unital multiplier algebra. It need not belong to $A$ . Nondegenerate representations of $A$ extend uniquely to $M(A)$ , so unitary multipliers act naturally in representation theory. The phrases “unitary of $A$ ” and “unitary multiplier of $A$ ” are therefore not interchangeable without stating the ambient algebra.

References

Bruce Blackadar, K-Theory for Operator Algebras, 2nd ed., Cambridge University Press, 1998. Publisher record. Relevant: Chapter IV, especially §8, on stable unitary groups and $K_1$ .
Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §§2.1–2.2 on unital $C^*$ -algebras, spectra, and functional calculus.