Homotopy of unitaries in simple C*-algebras with tracial rank one

Abstract

Let $\epsilon >0$ be a positive number. Is there a number $\delta >0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital simple $C^{\ast }$-algebra $A$ with $[v]=0$ in $K_1(A)$ for which $$ \|uv-vu\|<\dt, $$ there is a continuous path of unitaries $\{v(t):t\in [0,1]\}\subset A$ such that $$ v(0)=v, v(1)=1 \and \|uv(t)-v(t)u\|<ε\forall t\in [0,1]. $$ An answer is given to this question when $A$ is assumed to be a unital simple $C^{\ast }$-algebra with tracial rank no more than one. Let $C$ be a unital separable amenable simple $C^{\ast }$-algebra with tracial rank no more than one which also satisfies the UCT. Suppose that $\varphi :C\to A$ is a unital monomorphism and suppose that $v\in A$ is a unitary with $[v]=0$ in $K_1(A)$ such that $v$ almost commutes with $\varphi .$ It is shown that there is a continuous path of unitaries $\{v(t):t\in [0,1]\}$ in $A$ with $v(0)=v$ and $v(1)=1$ such that the entire path $v(t)$ almost commutes with $\varphi ,$ provided that an induced Bott map vanishes. Other versions of the so-called Basic Homotopy Lemma are also presented.

50 pages

References in corpus (2)

• Localizing the Elliott Conjecture at Strongly Self-absorbing C*-algebras --An Appendix
• Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras

Cited by in corpus (3)

• Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras
• Approximately diagonalizing matrices over C(Y)
• Unitaries in a Simple C*-algebra of Tracial Rank One