C*-Algebras

A C*-algebra is a complete normed star algebra whose norm and adjoint satisfy the C*-identity. This ties algebraic structure to spectral size strongly enough to define positivity and continuous functions of observables without first choosing coordinates on a Hilbert space. Both commutative classical observable algebras and noncommutative quantum algebras fit this definition.

Required background. Operator Algebra Minimum introduces star operations and positivity. Functional Calculus supplies the operator version of functions of an observable.

The C*-identity constrains both algebra and norm

Let $\mathcal A$ be a complex associative algebra with an antilinear involution $a\mapsto a^*$ satisfying $(ab)^*=b^*a^*$. It is a C*-algebra if it is complete in a submultiplicative norm and

$$\|a^*a\|=\|a\|^2.$$

An identity element is optional in the definition. This page uses a nonzero unital algebra when discussing spectra without further qualification. For a nonunital algebra, spectra are computed in its unitization. The compact operators $\mathcal K(\mathcal H)$ on an infinite-dimensional Hilbert space are a standard nonunital example.

Matrices $M_n(\mathbb C)$ and bounded operators $\mathcal B(\mathcal H)$ are concrete examples. For a compact Hausdorff space $X$, continuous functions $C(X)$ form a commutative example with pointwise products, complex conjugation, and supremum norm.

Completeness matters. Polynomials on $[0,1]$ are a star algebra in the supremum norm but are not complete; their closure is $C([0,1])$. Calculations stable under uniform approximation belong naturally to the completion.

Spectral size determines the norm

For a unital Banach algebra, $\sigma(a)$ consists of complex $\lambda$ for which $a-\lambda1$ has no inverse in the algebra. Its spectral radius satisfies

$$r(a)=\lim_{n\to\infty}\|a^n\|^{1/n}.$$

For a self-adjoint element $a$, the C*-identity repeatedly gives $\|a^{2^k}\|=\|a\|^{2^k}$. Combining this subsequence with the spectral-radius formula yields $r(a)=\|a\|$. For arbitrary $b$ this implies

$$\|b\|^2=\|b^*b\|=r(b^*b).$$

Thus the norm is fixed by the star-algebra structure; it is not an independent adjustable scale once the C*-axioms hold. In particular, star-isomorphisms between C*-algebras preserve norms.

The nontrivial background here is the Banach-algebra spectral-radius theorem and the C*-spectral theorem, including real spectrum for self-adjoint elements. Bratteli and Robinson develop these results. A star algebra with an arbitrary chosen norm need not possess them.

Continuous functional calculus stays inside the algebra

If $a$ is normal, $a^*a=aa^*$, there is an isometric unital star-isomorphism

$$C(\sigma(a))\longrightarrow C^*(1,a),\qquad f\longmapsto f(a),$$

sending the coordinate function to $a$. The algebra on the right is the norm-closed star algebra generated by $1$ and $a$.

For $a=a^*$, this supplies positive and negative parts, square roots of positive elements, and continuous cutoff functions. Positivity can be defined intrinsically by $a=a^*$ and $\sigma(a)\subset[0,\infty)$; equivalently, $a=b^*b$ for some $b$.

A sharp spectral indicator need not be continuous. Consequently $1_\Delta(a)$ need not belong to $C^*(1,a)$ even though it exists in a Hilbert-space spectral representation. For multiplication by $x$ on $L^2([0,1])$, the generated unital C*-algebra is multiplication by continuous functions. Multiplication by $1_{[0,1/2]}$ is absent. The von Neumann closure includes such measurable functions.

A closed star subalgebra has the correct spectrum

Let $\mathcal A\subset\mathcal B$ be unital C*-algebras with the same identity. Spectral permanence says that an element of $\mathcal A$ has the same spectrum in either algebra.

A useful proof mechanism is inverse closure. If $a\in\mathcal A$ is invertible in $\mathcal B$, then $a^*a$ is bounded below by a positive multiple of $1$. A convergent Neumann series, or continuous functional calculus, constructs $(a^*a)^{-1}$ inside $\mathcal A$. Hence

$$a^{-1}=(a^*a)^{-1}a^*\in\mathcal A.$$

The reverse inclusion of spectra follows immediately from containment of algebras. The shared identity and C*-closedness are essential hypotheses; arbitrary nonclosed or non-self-adjoint subalgebras need not be inverse closed.

Finally, the Gelfand–Naimark representation theorem states that every C*-algebra admits a faithful representation by bounded operators. It establishes compatibility with Hilbert-space quantum mechanics. Its construction uses sufficiently many states and their GNS representations; it does not prescribe one privileged physical state.

Exercise: what does one projection generate?

Let $P$ be a projection with $P\ne0,I$. Determine $C^*(I,P)$ and its positive elements.

Solution

Since $P^2=P=P^*$, every polynomial in $P$ reduces to $aP+b(I-P)$. This two-dimensional algebra is already closed and is star-isomorphic to $\mathbb C\oplus\mathbb C$. The element is positive precisely when $a,b$ are real and nonnegative, and its norm is $\max(|a|,|b|)$. One yes-or-no observable therefore generates a commutative algebra even when the full system algebra is noncommutative.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987, doi:10.1007/978-3-662-02520-8, Chapter 2 — C*-spectral theory, continuous functional calculus, and representations.
• Naaijkens, Pieter, Quantum Spin Systems on Infinite Lattices, Lecture Notes in Physics 933, Springer, 2017, arXiv:1311.2717 — abstract C*-algebras and their quantum-mechanical interpretation.