[Paper Review] Nuclear dimension for an inclusion of unital C*-algebras

Hiroyuki Osaka, Tamotsu Teruya|arXiv (Cornell University)|Nov 8, 2011

Advanced Operator Algebra Research32 references3 citations

TL;DR

This paper establishes that if a unital C*-algebra is locally in a class of algebras with decomposition rank ≤ n or nuclear dimension ≤ n, then it belongs to that class. The key contribution is proving stability of finite decomposition rank and nuclear dimension under inclusions with finite Watatani index and the Rokhlin property, affirming Problem 9.4 in Winter and Zacharias (2010) and extending results to crossed products and basic constructions.

ABSTRACT

Let $P \subset A$ be an inclusion of separable unital C*-algebras with finite Watatani index. Suppose that $E \colon A ightarrow P$ has the Rokhlin property, that is, there is a projection $e \in A' \cap A^\infty$ such that $E^\infty(e) = ({ m Index}E)^{-1}1$. We show that if $A$ has nuclear dimension $n$, then $P$ has nuclear dimension less than or equal to $n$. In particular, if an action $α$ of a finite group $G$ on $A$ has the Rokhlin property, then the nuclear dimension of the crossed product algebra $A times_αG$ is less than or equal to that of $A$.

Motivation & Objective

• To establish that local C*-algebras in classes of finite decomposition rank or nuclear dimension are globally in those classes.
• To resolve Problem 9.4 from Winter and Zacharias (2010) by showing that finite nuclear dimension is preserved under Rokhlin-type inclusions.
• To extend stability results to crossed products and basic constructions under the Rokhlin property.
• To investigate the inheritance of pureness (strict comparison and almost divisible Cuntz semigroup) under Rokhlin conditional expectations.
• To clarify the relationship between trace classes and order structure in inclusions of finite index type.

Proposed method

• Introduces the concept of a C*-algebra being locally in a class C_n (resp. C_nuc_n) via approximation by subalgebras in C_n (resp. C_nuc_n).
• Uses the Rokhlin property for conditional expectations E: A → P, defined via existence of a projection e ∈ A' ∩ A^∞ with E^∞(e) = (Index E)^{-1}1.
• Applies results from Osaka and Phillips (2011) on finite saturation of C_n and C_nuc_n classes to deduce global membership.
• Employs the Cuntz semigroup W(A) and its almost divisibility to characterize purity and stability under inclusions.
• Uses weak semiprojectivity of dimension drop algebras Z_{k,k+1} to lift homomorphisms from A to P and the basic construction.
• Applies Winter’s characterization of purity via strict comparison and almost divisibility of W(A), and the equivalence of strict comparison to trace-determined order when A is exact and has stable rank one.

Experimental results

Research questions

• RQ1 Does a C*-algebra that is locally in the class of algebras with decomposition rank ≤ n necessarily belong to that class?
• RQ2 Is nuclear dimension preserved under inclusions of unital C*-algebras with finite Watatani index and the Rokhlin property?
• RQ3 Does the Rokhlin property for group actions preserve finite decomposition rank and nuclear dimension in fixed-point algebras and crossed products?
• RQ4 Under what conditions is the fixed-point algebra P of a finite group action on a pure C*-algebra also pure?
• RQ5 How are traces and the order of projections related in inclusions of finite index type with the Rokhlin property?

Key findings

• If A is a local C_n (resp. C_nuc_n) C*-algebra, then A ∈ C_n (resp. C_nuc_n), confirming that these classes are closed under local approximation.
• For an inclusion P ⊂ A with finite Watatani index and a conditional expectation E with the Rokhlin property, if dr(A) ≤ n (resp. dim_nuc(A) ≤ n), then dr(P) ≤ n (resp. dim_nuc(P) ≤ n).
• If A is exact, pure, and has stable rank one, and E: A → P has the Rokhlin property, then P is pure.
• For finite group actions α with the Rokhlin property, if dr(A) ≤ n, then dr(A^α) ≤ n and dr(A ⋊_α G) ≤ n.
• If dim_nuc(A) ≤ n and α has the Rokhlin property, then dim_nuc(A^α) ≤ n and dim_nuc(A ⋊_α G) ≤ n.
• If A is separable, unital, exact, pure, and of stable rank one with α having the Rokhlin property, then both A^α and A ⋊_α G are pure.