[Paper Review] The role of C*-algebras in infinite dimensional numerical linear algebra

William Arveson|ArXiv.org|Jun 8, 1993

Advanced Operator Algebra ResearchMathematics8 references19 citations

TL;DR

This paper presents a C*-algebraic framework for computing the essential spectrum of bounded self-adjoint operators on separable Hilbert spaces using finite-dimensional matrix approximations. By leveraging tridiagonal operators and unique tracial states in simple C*-algebras, it establishes that eigenvalue distributions of finite truncations converge weakly to the spectral measure of the infinite operator, enabling reliable numerical approximation of the essential spectrum.

ABSTRACT

This is a survey of four recent papers which deal with the relationship of simple C*-algebras to the problem of computing the spectra of self-adjoint operators in the general case, especially when the spectrum is not discrete. It is an expanded version of a talk presented at the 50 year C*-algebra celebration, held at the annual meeting of the AMS in San Antonio during January, 1993.

Motivation & Objective

• To develop a mathematically rigorous method for computing the essential spectrum of self-adjoint operators in infinite-dimensional Hilbert spaces.
• To address the lack of proven techniques for approximating spectra of operators not given in diagonal form.
• To establish conditions under which finite-dimensional truncations of operators converge to the correct spectral limit.
• To clarify the role of C*-algebras and tracial states in the convergence of eigenvalue distributions of matrix approximations.

Proposed method

• Define a filtration of the Hilbert space via orthonormal bases, with projections onto finite-dimensional subspaces H_n.
• Introduce the notion of degree of an operator relative to a filtration, generalizing band-limited matrices.
• Construct a Banach *-algebra of operators decomposable into finite-degree components satisfying a summability condition.
• Use the unique tracial state τ on a C*-algebra A to define a spectral measure μ_T via the Riesz-Markov theorem.
• Prove that the eigenvalue distribution of n×n truncations of a tridiagonal operator T converges weakly to μ_T as n→∞.
• Establish an isomorphism between the quotient algebra A_+/K_+ and the original C*-algebra A, ensuring structural consistency.

Experimental results

Research questions

• RQ1 Under what conditions do finite-dimensional matrix truncations of an infinite-dimensional self-adjoint operator converge to the correct essential spectrum?
• RQ2 How can the spectral measure of an operator be recovered from the eigenvalue distributions of its finite truncations?
• RQ3 What role does the unique tracial state of a C*-algebra play in characterizing the limit of eigenvalue distributions?
• RQ4 In what sense is the essential spectrum preserved under the C*-algebraic structure of the operator algebra?
• RQ5 How does the choice of orthonormal basis affect the convergence of approximations to the spectrum?

Key findings

• The eigenvalue distribution of n×n truncations of a tridiagonal operator T converges weakly to the spectral measure μ_T associated with the unique tracial state of the C*-algebra.
• For every f ∈ C₀(ℝ), the average of f over the eigenvalues of the n×n truncation converges to ∫ f(x) dμ_T(x) as n→∞.
• The essential spectrum of the operator T is equal to the closed support of the spectral measure μ_T.
• If λ ∉ σ(T), then the number of eigenvalues of the n×n truncation in a small neighborhood of λ remains bounded as n→∞.
• The C*-algebra A_+ of truncated operators modulo compact operators is isomorphic to the original C*-algebra A, preserving spectral structure.