Self-adjoint element of a C*-algebra - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. An element $a\in A$ is self-adjoint if

a^*=a.

The self-adjoint elements form a closed real vector subspace $A_{\mathrm{sa}}$ of $A$ , not generally a complex subalgebra. Every $x\in A$ has the unique decomposition

x=\frac{x+x^*}{2}+i\,\frac{x-x^*}{2i},

whose two summands before and after the factor $i$ are self-adjoint. In a concrete $C^*$ -algebra of operators on a Hilbert space, this definition is exactly the usual condition that an operator equal its Hilbert-space adjoint.

Spectral characterizations

An element of a $C^*$ -algebra is self-adjoint exactly when its spectrum is contained in $\mathbb R$ . Equivalently, $\|\exp(ita)\|=1$ for every real $t$ , with the exponential computed in the unitization when the algebra is nonunital. These equivalences use the $C^*$ -identity and functional calculus; they fail as stated in a general involutive Banach algebra Murphy, §2.2.

Order and functional calculus

A self-adjoint element is positive when its spectrum lies in $[0,\infty)$ ; self-adjointness alone does not imply positivity. The order on $A_{\mathrm{sa}}$ is defined by $a\leq b$ when $b-a$ is positive. Continuous real-valued functions on the spectrum of $a$ produce self-adjoint elements by functional calculus, while nonnegative functions produce positive elements. In particular, the positive and negative parts recover $a=a_+-a_-$ with $a_+a_-=0$ .

Examples and boundary cases

Hermitian matrices are the self-adjoint elements of $M_n(\mathbb C)$ . Multiplication by an essentially bounded real-valued function is self-adjoint in the commutative operator algebra on $L^2$ . A unitary element need not be self-adjoint; it is self-adjoint precisely when its square is the identity. For unbounded operators, the equation $T=T^*$ includes equality of domains and belongs to a different theory than self-adjoint elements of a $C^*$ -algebra, whose elements are bounded.

References

Gerard J. Murphy, $C^*$ -Algebras and Operator Theory, Academic Press, 1990. DOI record. Relevant: §2.2 on self-adjoint elements, positivity, and continuous functional calculus.
Gert K. Pedersen, $C^*$ -Algebras and Their Automorphism Groups, 2nd ed., Academic Press, 2018. DOI record. Relevant: §1.2 on self-adjoint and positive elements.