TY - JOUR

T1 - Sylvester matrix rank functions on crossed products

AU - Ara, Pere

AU - Claramunt, Joan

PY - 2019/1/1

Y1 - 2019/1/1

N2 - © Cambridge University Press, 2019. In this paper we consider the algebraic crossed product induced by a homeomorphism on the Cantor set, where is an arbitrary field with involution and denotes the -algebra of locally constant -valued functions on. We investigate the possible Sylvester matrix rank functions that one can construct on by means of full ergodic -invariant probability measures on. To do so, we present a general construction of an approximating sequence of -subalgebras which are embeddable into a (possibly infinite) product of matrix algebras over. This enables us to obtain a specific embedding of the whole -algebra into, the well-known von Neumann continuous factor over, thus obtaining a Sylvester matrix rank function on by restricting the unique one defined on. This process gives a way to obtain a Sylvester matrix rank function on, unique with respect to a certain compatibility property concerning the measure, namely that the rank of a characteristic function of a clopen subset must equal the measure of.

AB - © Cambridge University Press, 2019. In this paper we consider the algebraic crossed product induced by a homeomorphism on the Cantor set, where is an arbitrary field with involution and denotes the -algebra of locally constant -valued functions on. We investigate the possible Sylvester matrix rank functions that one can construct on by means of full ergodic -invariant probability measures on. To do so, we present a general construction of an approximating sequence of -subalgebras which are embeddable into a (possibly infinite) product of matrix algebras over. This enables us to obtain a specific embedding of the whole -algebra into, the well-known von Neumann continuous factor over, thus obtaining a Sylvester matrix rank function on by restricting the unique one defined on. This process gives a way to obtain a Sylvester matrix rank function on, unique with respect to a certain compatibility property concerning the measure, namely that the rank of a characteristic function of a clopen subset must equal the measure of.

KW - 16D70 (Secondary)

KW - 16E50 (Primary)

KW - 16S35

KW - 2010 Mathematics Subject Classification

KW - 37A05

UR - https://ddd.uab.cat/record/207875

U2 - https://doi.org/10.1017/etds.2019.37

DO - https://doi.org/10.1017/etds.2019.37

M3 - Article

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

ER -