Singular value functions for C\(^*\)-algebras

Abstract: We introduce \textit{singular value functions} for C(^*)-algebras, generalizing the singular values of compact operators on Hilbert spaces. We also establish several fundamental properties of these singular value functions and present examples of singular value functions for certain classes of C(^*)-algebras.

• The paper introduces a generalization of singular value functions to arbitrary C*-algebras using the ordered K-theory framework.
• It establishes key analytical properties, including subadditivity and multiplicative inequalities, with rigorous proofs and explicit computations.
• The work highlights both positive realizations and counterexamples, delineating the boundaries for these invariants in noncommutative settings.

Singular Value Functions in the Context of C$^*$-Algebras

Introduction

This paper introduces a comprehensive generalization of the theory of singular value functions, extending the classical notion for compact operators on Hilbert spaces and their established generalizations to semifinite von Neumann algebras, to arbitrary C$^*$-algebras (2605.17235). The core construction leverages ordered K-theory, particularly the positive cone of the $K_0$-group, as the index set for these singular value functions, replacing traditional rank or trace correspondences. The work systematically analyzes foundational properties, analogues of classical inequalities, continuity behaviors, and realization theorems, providing both explicit computations in key cases and counterexamples clarifying the limitations of this apparatus in the C$^*$-algebraic setting.

Definition and Fundamental Properties

The singular value function for an element $a$ in a C$^*$-algebra $A$ is defined for $g$ in $K_0(A)_+$ by

$$s_g(a) := \inf \{ \| a - a p \| \mid p \in P(A),\ [p]_0 \leq g \},$$

where $^*$0 denotes projections in $^*$1, and $^*$2 denotes the $^*$3-class. This function is shown to be nonnegative and decreasing in $^*$4. For the algebra $^*$5 of compact operators, this recovers the standard singular values sequence indexed by rank. Importantly, for positive elements with finite spectrum, the associated singular value functions are step functions with steps determined by the order structure of $^*$6.

A suite of core properties is established, assuming the C$^*$7-algebra has cancellation, real rank zero, and $^*$8 (the dimension range):

• $^*$9
• $K_0$0, $K_0$1 for scalar $K_0$2 and $K_0$3
• Lipchitz continuity: $K_0$4
• $K_0$5
• Subadditivity: $K_0$6
• Multiplicative inequality: $K_0$7
• Monotonicity: $K_0$8 implies $K_0$9
• Functional calculus: for continuous increasing $^*$0 with $^*$1, $^*$2 for $^*$3

The subadditivity and multiplicative inequalities are direct analogues of classical Ky Fan-type inequalities for finite- and infinite-dimensional settings, and their proofs require intricate approximation and comparison arguments due to the lack of lattice structure in $^*$4. The $^*$5-invariance (i.e., $^*$6) is shown under a density assumption on invertibles, e.g., for algebras of stable rank one or stable C$^*$7-algebras.

Realization and Continuity of Singular Value Functions

For C$^*$8-algebras whose ordered $^*$9-group is simple and totally ordered (in particular, Archimedean), every right-continuous, decreasing function on $a$0 vanishing at infinity (or at the unit, in the unital case) and with countable jump set can be realized as the singular value function $a$1 for some $a$2. Moreover, for such algebras with real rank zero, every $a$3 is right-continuous and vanishes at infinity. These results generalize classical characterizations for II$a$4 factors and semifinite von Neumann algebras and establish that the appropriate topological and order-theoretic hypotheses suffice to recover the familiar analytical structure of singular value sequences in this noncommutative setting.

However, salient counterexamples are constructed when $a$5 lacks total order or when the group is equipped with the product topology coming from infinitesimal and affine state components: in these cases, singular value functions may fail to be lower semicontinuous.

Explicit Computations and Examples

The paper provides explicit forms for singular value functions in several key cases. For projections $a$6, $a$7 is an indicator function for whether $a$8. For positive finite-spectrum elements, singular value functions are piecewise constant, dictated by the $a$9-classes of minimal projections supporting different spectral values (see Proposition 3.7). These calculations underline the operational similarity with classic eigenvalue counting functions, but indexed by the algebraic and order-theoretic structure of projections in $^*$0.

Implications and Future Perspectives

By generalizing singular value functions to arbitrary C$^*$1-algebras and rigorously establishing their core analytical, order-theoretic, and continuity properties, this work creates a robust bridge between noncommutative integration, operator inequalities, and ordered K-theory. Practically, this framework provides a new toolkit for analyzing spectral invariants and variational properties in the context of C$^*$2-algebraic functional calculus, with applications to structure theory, classification, and the study of traces and dimension functions in non-classical settings. The negative results regarding continuity and lower semicontinuity in more general (e.g., AF) settings underscores the technical subtleties that arise outside the field of simple, totally ordered, Archimedean $^*$3-groups, indicating precise boundaries for effective generalization.

Looking forward, possible directions include exploiting these singular value functions for further C$^*$4-algebraic invariants in classification programs, quantitative comparison of positive elements, and potential applications to noncommutative metric geometry and quantum information, where operator variational inequalities play a central role.

Conclusion

This study presents a definitive framework for singular value functions indexed by the $^*$5-group in arbitrary C$^*$6-algebraic contexts, proving analogues of classical inequalities, characterizing continuity and realization properties, and providing structural insight into the interplay between operator theory and ordered K-theory. The results both solidify and circumscribe the reach of singular value function techniques, positioning them as precise spectral and order-theoretic invariants in the analysis of C$^*$7-algebras (2605.17235).