Tensor products of maximal abelian subalgebras of C*-algebras

TL;DR

This paper investigates conditions under which tensor products of maximal abelian subalgebras (masas) in C*-algebras remain maximal abelian in various tensor product completions, providing new insights into the Kadison-Singer extension problem.

Contribution

It establishes that tensor products of masas are maximal abelian under specific conditions involving the extension property and approximate identities, answering a longstanding open question.

Findings

Tensor products of masas are maximal abelian if one has the extension property and the other contains an approximate identity.

Counterexamples show failure of maximal abelianness without the extension property.

Results have implications for the Kadison-Singer extension problem.

Abstract

It is shown that if $C_1$ and $C_2$ are maximal abelian self-adjoint subalgebras (masas) of C*-algebras $A_1$ and $A_2$, respectively, then the completion $C_1\otimes C_2$ of the algebraic tensor product $C_1\odot C_2$ of $C_1$ and $C_2$ in any C*-tensor product $A_1\otimes_{\beta} A_2$ is maximal abelian provided that $C_1$ has the extension property of Kadison and Singer and $C_2$ contains an approximate identity for $A_2$. An example is given to show that $C_1\otimes C_2$ can fail to be a masa in $A_1\otimes_{\beta} A_2$ with $A_1$ and $A_2$ unital if neither $C_1$ nor $C_2$ has the extension property. This gives an answer to a long-standing question, but leaves open some other interesting problems, one of which turns out to have a potentially intriguing implication for the Kadison-Singer extension problem.